title: Coast

text: The Earth has around of coastline. Coastal habitats, which extend to the margins of the continental shelves, make up about 7 percent of the Earth's oceans, but at least 85% of commercially harvested fish depend on coastal environments during at least part of their life cycle. As of October 2010, about 2.86% of exclusive economic zones were part of marine protected areas.


title: Coast

text: The definition of coasts varies. Marine scientists think of the "wet" (aquatic or intertidal) vegetated habitats as being coastal ecosystems (e.g. seagrass, salt marsh etc.) whilst some terrestrial scientist might only think of coastal ecosystems as purely terrestrial plants that live close to the seashore (see also estuaries and coastal ecosystems).


title: Coast

text: While there is general agreement in the scientific community regarding the definition of coast, in the political sphere, the delineation of the extents of a coast differ according to jurisdiction. Government authorities in various countries may define coast differently for economic and social policy reasons.


title: Coast

text: Tides often determine the range over which sediment is deposited or eroded. Areas with high tidal ranges allow waves to reach farther up the shore, and areas with lower tidal ranges produce deposition at a smaller elevation interval. The tidal range is influenced by the size and shape of the coastline. Tides do not typically cause erosion by themselves; however, tidal bores can erode as the waves surge up the river estuaries from the ocean.


title: Coast

text: Geologists classify coasts on the basis of tidal range into "macrotidal coasts" with a tidal range greater than ; "mesotidal coasts" with a tidal range of ; and "microtidal coasts" with a tidal range of less than . The distinction between macrotidal and mesotidal coasts is more important. Macrotidal coasts lack barrier islands and lagoons, and are characterized by funnel-shaped estuaries containing sand ridges aligned with tidal currents. Wave action is much more important for determining bedforms of sediments deposited along mesotidal and microtidal coasts than in macrotidal coasts.


title: Coast

text: Waves erode coastline as they break on shore releasing their energy; the larger the wave the more energy it releases and the more sediment it moves. Coastlines with longer shores have more room for the waves to disperse their energy, while coasts with cliffs and short shore faces give little room for the wave energy to be dispersed. In these areas, the wave energy breaking against the cliffs is higher, and air and water are compressed into cracks in the rock, forcing the rock apart, breaking it down. Sediment deposited by waves comes from eroded cliff faces and is moved along the coastline by the waves. This forms an abrasion or cliffed coast.


title: Coast

text: Sediment deposited by rivers is the dominant influence on the amount of sediment located in the case of coastlines that have estuaries. Today, riverine deposition at the coast is often blocked by dams and other human regulatory devices, which remove the sediment from the stream by causing it to be deposited inland. Coral reefs are a provider of sediment for coastlines of tropical islands.


title: Coast

text: Like the ocean which shapes them, coasts are a dynamic environment with constant change. The Earth's natural processes, particularly sea level rises, waves and various weather phenomena, have resulted in the erosion, accretion and reshaping of coasts as well as flooding and creation of continental shelves and drowned river valleys (rias).


title: Coast

text: More and more of the world's people live in coastal regions. According to a United Nations atlas, 44% of all people live within 150 km (93 mi) of the sea. Many major cities are on or near good harbors and have port facilities. Some landlocked places have achieved port status by building canals.


title: Coast

text: Nations defend their coasts against military invaders, smugglers and illegal migrants. Fixed coastal defenses have long been erected in many nations, and coastal countries typically have a navy and some form of coast guard.


title: Coast

text: Coasts, especially those with beaches and warm water, attract tourists often leading to the development of seaside resort communities. In many island nations such as those of the Mediterranean, South Pacific Ocean and Caribbean, tourism is central to the economy. Coasts offer recreational activities such as swimming, fishing, surfing, boating, and sunbathing.


title: Coast

text: Growth management and coastal management can be a challenge for coastal local authorities who often struggle to provide the infrastructure required by new residents, and poor management practices of construction often leave these communities and infrastructure vulnerable to processes like coastal erosion and sea level rise. In many of these communities, management practices such as beach nourishment or when the coastal infrastructure is no longer financially sustainable, managed retreat to remove communities from the coast.


title: Coast

text: According to one principle of classification, an emergent coastline is a coastline that has experienced a fall in sea level, because of either a global sea-level change, or local uplift. Emergent coastlines are identifiable by the coastal landforms, which are above the high tide mark, such as raised beaches. In contrast, a submergent coastline is one where the sea level has risen, due to a global sea-level change, local subsidence, or isostatic rebound. Submergent coastlines are identifiable by their submerged, or "drowned" landforms, such as rias (drowned valleys) and fjords


title: Coast

text: According to the second principle of classification, a concordant coastline is a coastline where bands of different rock types run parallel to the shore. These rock types are usually of varying resistance, so the coastline forms distinctive landforms, such as coves. Discordant coastlines feature distinctive landforms because the rocks are eroded by the ocean waves. The less resistant rocks erode faster, creating inlets or bay; the more resistant rocks erode more slowly, remaining as headlands or outcroppings.


title: Coast

text: "Coastal waters" (or "coastal seas") is a rather general term used differently in different contexts, ranging geographically from the waters within a few kilometers of the coast, through to the entire continental shelf which may stretch for more than a hundred kilometers from land. Thus the term coastal waters is used in a slightly different way in discussions of legal and economic boundaries (see territorial waters and international waters) or when considering the geography of coastal landforms or the ecological systems operating through the continental shelf (marine coastal ecosystems). The research on coastal waters often divides into these separate areas too.


title: Coast

text: The dynamic fluid nature of the ocean means that all components of the whole ocean system are ultimately connected, although certain regional classifications are useful and relevant. The waters of the continental shelves represent such a region. The term “coastal waters” has been used in a wide variety of different ways in different contexts. In European Union environmental management it extends from the coast to just a few nautical miles while in the United States the US EPA considers this region to extend much further offshore.


title: Coast

text: "Coastal waters" has specific meanings in the context of commercial coastal shipping, and somewhat different meanings in the context of naval littoral warfare. Oceanographers and marine biologists have yet other takes. Coastal waters have a wide range of marine habitats from enclosed estuaries to the open waters of the continental shelf.


title: Coast

text: Similarly, the term littoral zone has no single definition. It is the part of a sea, lake, or river that is close to the shore. In coastal environments, the littoral zone extends from the high water mark, which is rarely inundated, to shoreline areas that are permanently submerged.


title: Coast

text: The identification of bodies of rock formed from sediments deposited in shoreline and nearshore environments (shoreline and nearshore "facies") is extremely important to geologists. These provide vital clues for reconstructing the geography of ancient continents ("paleogeography"). The locations of these beds show the extent of ancient seas at particular points in geological time, and provide clues to the magnitudes of tides in the distant past.


title: Coast

text: Sediments deposited in the shoreface are preserved as lenses of sandstone in which the upper part of the sandstone is coarser than the lower part (a "coarsening upwards sequence"). Geologists refer to these are "parasequences". Each records an episode of retreat of the ocean from the shoreline over a period of 10,000 to 1,000,000 years. These often show laminations reflecting various kinds of tidal cycles.


title: Coast

text: Some of the best-studied shoreline deposits in the world are found along the former western shore of the Western Interior Seaway, a shallow sea that flooded central North America during the late Cretaceous Period (about 100 to 66 million years ago). These are beautifully exposed along the Book Cliffs of Utah and Colorado.


title: Coast

text: Larger animals that live in coastal areas include puffins, sea turtles and rockhopper penguins, among many others. Sea snails and various kinds of barnacles live on rocky coasts and scavenge on food deposited by the sea. Some coastal animals are used to humans in developed areas, such as dolphins and seagulls who eat food thrown for them by tourists. Since the coastal areas are all part of the littoral zone, there is a profusion of marine life found just off-coast, including sessile animals such as corals, sponges, starfish, mussels, seaweeds, fishes, and sea anemones.


title: Coast

text: There are many kinds of seabirds on various coasts. These include pelicans and cormorants, who join up with terns and oystercatchers to forage for fish and shellfish. There are sea lions on the coast of Wales and other countries.


title: Coast

text: Many coastal areas are famous for their kelp beds. Kelp is a fast-growing seaweed that can grow up to half a meter a day in ideal conditions. Mangroves, seagrasses, macroalgal beds, and salt marsh are important coastal vegetation types in tropical and temperate environments respectively. Restinga is another type of coastal vegetation.


title: Coast

text: Coasts also face many human-induced environmental impacts and coastal development hazards. The most important ones are:


title: Coast

text: The pollution of coastlines is connected to marine pollution which can occur from a number of sources: Marine debris (garbage and industrial debris); the transportation of petroleum in tankers, increasing the probability of large oil spills; small oil spills created by large and small vessels, which flush bilge water into the ocean.


title: Coast

text: International attention to address the threats of coasts has been captured in Sustainable Development Goal 14 "Life Below Water" which sets goals for international policy focused on preserving marine coastal ecosystems and supporting more sustainable economic practices for coastal communities. Likewise, the United Nations has declared 2021-2030 the UN Decade on Ecosystem Restoration, but restoration of coastal ecosystems has received insufficient attention.


title: Ion Iliescu

text: Ion Iliescu (; born 3 March 1930) is a Romanian politician and engineer who served as President of Romania from 1989 until 1996 and from 2000 until 2004. Between 1996 and 2000 and also from 2004 to 2008, the year in which he retired, Iliescu was a senator for the Social Democratic Party (PSD), of which he is the founder and honorary president to this day.


title: Ion Iliescu

text: Iliescu joined the Romanian Communist Party (PCR) in 1953 and became a member of its Central Committee in 1965. Beginning with 1971, he was gradually marginalized by Nicolae Ceaușescu. He had a leading role in the Romanian Revolution, becoming the country's president in December 1989. In May 1990, he became Romania's first freely elected head of state. After a new constitution was approved by popular referendum, he served a further two terms, firstly from 1992 to 1996 and then secondly from 2000 to 2004, separated by the presidency of Emil Constantinescu, who defeated him in 1996.


title: Ion Iliescu

text: In 2004, during his presidency, Romania joined NATO. In April 2018, Iliescu was charged in Romania with committing crimes against humanity by "approving military measures, some of which had an evidently diversionary character" during the deadly aftermath of the country's 1989 revolution. In 2020, a judge rejected the case due to irregularities in the indictment. Iliescu is currently the oldest living former Romanian president.


title: Ion Iliescu

text: He was born in Oltenița, the son of Alexandru Iliescu and Maria Dumitru Toma. His mother, who was originally from Bulgaria, abandoned him when he was an infant. His father, a railroad worker, had communist views, during a period in which the Romanian Communist Party was banned by the authorities. In 1931, he went to the Soviet Union to take part in the Congress of the Communist Party of the Soviet Union in Moscow. He remained in the USSR for the next four years and was arrested upon his return. He was imprisoned from June 1940 to August 1944 and died in August 1945. During his time in the Soviet Union, Alexandru Iliescu divorced and married Marița, a chambermaid. According to Iliescu's own statements, his grandfather, Vasili Ivanovici, was a Russian Jew, who, being persecuted by the tsarist authorities because of his socialist views, took refuge in Romania.


title: Ion Iliescu

text: Iliescu was raised by his stepmother and by his grandparents. At age 9 he was adopted by an aunt, Aristița, who worked as a cook for Ana Pauker. In 1948 he met Elena (Nina) Șerbănescu, when they were both 18-year old students, he at the Saint Sava High School and she at the Iulia Hasdeu High School, in Bucharest. The two were married on July 21, 1951; they have no children, not by choice but because they could not, as Nina had three miscarriages. He studied fluid mechanics at the Bucharest Polytechnic Institute and then as a foreign student at Moscow Power Engineering Institute. During his stay in Moscow, he was the secretary of the "Association of Romanian Students"; it is alleged that he met Mikhail Gorbachev, although Iliescu always denied this. However, years later, president Nicolae Ceaușescu probably believed that there was a connection between the two, since during Gorbachev's visit to Romania in July 1989, Iliescu was sent outside of Bucharest to prevent any contact.


title: Ion Iliescu

text: Iliescu learned to speak English after the 1989 Revolution; he also speaks Russian, French, and some Spanish.


title: Ion Iliescu

text: He joined the Union of Communist Youth in 1944 and the Communist Party in 1953 and made a career in the Communist nomenklatura, becoming a secretary of the Central Committee of the Union of Communist Youth in 1956 and a member of the Central Committee of the Romanian Communist Party in 1965. At one point, he served as the head of the Central Committee's Department of Propaganda. Iliescu later served as Minister for Youth-related Issues between 1967 and 1971.


title: Ion Iliescu

text: However, in 1971, Ceaușescu felt threatened by Iliescu, who was seen as Ceaușescu's heir apparent. He was marginalized and removed from all major political offices, becoming vice-president of the Timiș County Council (1971–1974), and later president of the Iași Council (1974–1979). Until 1989, he was in charge of the Editura Tehnică publishing house. For most of the 1980s (if not before), he was tailed by the Securitate (secret police), as he was known to oppose Ceaușescu's harsh rule.


title: Ion Iliescu

text: The Romanian Revolution began as a popular revolt in Timișoara. After Ceaușescu was overthrown on 22 December, the political vacuum was filled by an organization named National Salvation Front (FSN: "Frontul Salvării Naționale"), formed spontaneously by second-rank Communist party members opposed to the policies of Ceaușescu and non-affiliated participants in the revolt. Iliescu was quickly acknowledged as the leader of the organization and therefore of the provisional authority. He first learned of the revolution when he noticed the Securitate was no longer tailing him.


title: Ion Iliescu

text: The Ceaușescus were captured, hauled before a drumhead court-martial, and executed on Christmas Day. Years later, Iliescu conceded that the trial and execution were "quite shameful, but necessary" to end the chaos that had riven the country since Ceaușescu's overthrow.


title: Ion Iliescu

text: Iliescu proposed multi-party elections and an "original democracy". This is widely held to have meant the adoption of "Perestroika"-style reforms rather than the complete removal of existing institutions; it can be linked to the warm reception the new regime was given by Mikhail Gorbachev and the rest of the Soviet leadership, and the fact that the first post-revolutionary international agreement signed by Romania was with that country.


title: Ion Iliescu

text: Rumours abounded for years that Illiescu and other second-rank Communists had been planning to overthrow Ceaușescu, but the events of December 1989 overtook them. For instance, Nicolae Militaru, the new regime's first Defense Minister, said that Illiescu and others had planned to take Ceaușescu prisoner in February 1990 while he was out of the capital. However, Illiescu denies this, saying that the nature of the Ceaușescu regime—particularly the Securitate's ubiquity—made advance planning for a coup all but impossible.


title: Ion Iliescu

text: The National Salvation Front (FSN) subsequently decided to organize itself as a party and participate in the 1990 general election—the first free election held in the country in 53 years–with Illiescu as its presidential candidate. The FSN won a sweeping victory, taking strong majorities in both chambers. In the separate presidential election, Iliescu won handily, taking 85 percent of the vote, still the largest vote share for a free presidential election. He became Romania's first democratically elected head of state. To date, it is the only time since the Fall of Communism that a president has been elected in a single round.


title: Ion Iliescu

text: Iliescu and his supporters split from the Front and created the Democratic National Salvation Front (FDSN), which later evolved into the Party of Social Democracy in Romania (PDSR), then the Social Democratic Party (PSD) (see Social Democratic Party of Romania). Progressively, the Front lost its character as a national government or generic coalition, and became vulnerable to criticism for using its appeal as the first institution involved in power sharing, while engaging itself in political battles with forces that could not enjoy this status, nor the credibility.


title: Ion Iliescu

text: Under the pressure of the events that led to the Mineriads, his political stance has veered with time: from a proponent of "Perestroika", Iliescu recast himself as a Western European social democrat. The main debate around the subject of his commitment to such ideals is linked to the special conditions in Romania, and especially to the strong nationalist and autarkic attitude visible within the Ceaușescu regime. Critics have pointed out that, unlike most Communist-to-social democrat changes in the Eastern Bloc, Romania's tended to retain various cornerstones.


title: Ion Iliescu

text: Romania adopted its first post-Communist Constitution in 1991. In 1992, Iliescu won a second term when he received 61% of the vote in the second round. He immediately suspended his NSDF membership; the Constitution does not allow the president to be a formal member of a political party during his term. He ran for a third time in 1996 but, stripped of media monopoly, he lost in the second round to Emil Constantinescu, his second-round opponent in 1992. Over 1,000,000 votes were cancelled, leading to accusations of widespread fraud. Nevertheless, Iliescu conceded defeat within hours of polls closing, making him the only incumbent president to lose a bid for re-election since the end of Communism.


title: Ion Iliescu

text: In the 2000 presidential election Iliescu ran again and won in the run-off against the ultra-nationalist Corneliu Vadim Tudor. He began his third term on 20 December of that year, ending on 20 December 2004. The center-right was severely defeated during the 2000 elections due largely to public dissatisfaction with the harsh economic reforms of the previous four years as well as the political instability and infighting of the multiparty coalition. Tudor's extreme views also ensured that most urban voters either abstained or chose Iliescu.


title: Ion Iliescu

text: In the PSD elections of 21 April 2005, Iliescu lost the Party presidency to Mircea Geoană, but was elected as honorary president of the party in 2006, a position without official executive authority in the party.


title: Ion Iliescu

text: Though enjoying a certain popularity due to his opposition to Ceaușescu and image as a revolutionary, his political career after 1989 was characterized by multiple controversies and scandals. Public opinion regarding his tenure as president is still divided.


title: Ion Iliescu

text: Some alleged Iliescu had connections to the KGB; the allegations continued during 2003–2008, when Russian dissident Vladimir Bukovsky, who had been granted access to Soviet archives, declared that Iliescu and some of the NSF members were KGB agents, that Iliescu had been in close connection with Mikhail Gorbachev ever since they had allegedly met during Iliescu's stay in Moscow, and that the Romanian Revolution of 1989 was a plot organized by the KGB to regain control of the country's policies (gradually lost under Ceaușescu's rule). The only hard evidence published was a discussion between Gorbachev and Bulgaria's Aleksandar Lilov from 23 May 1990 (after Iliescu's victory in the 20 May elections) in which Gorbachev says that Iliescu holds a "calculated position", and that despite sharing common views with Iliescu, Gorbachev wanted to avoid sharing this impression with the public.


title: Ion Iliescu

text: Iliescu, along with other figures in the leading FSN, was allegedly responsible for calling the Jiu Valley miners to Bucharest on January (January 1990 Mineriad) and June (June 1990 Mineriad) 1990 to end the protests of the citizens gathered in University Square, Bucharest, protests aimed against the ex-Communist leaders of Romania (like himself). The pejorative term for this demonstration was the Golaniad (from the Romanian "golan", rascal). On 13 June, an attempt of the authorities to remove from the square around 100 protesters, which had remained in the street even after the May elections had confirmed Iliescu and the FSN, resulted in attacks against several state institutions, such as the Ministry of Interior, the Bucharest Police Headquarters and the National Television. Iliescu issued a call to the Romanian people to come and defend the government, prompting several groups of miners to descend on the capital, armed with wooden clubs and bats. They trashed the University of Bucharest, some newspaper offices and the headquarters of opposition parties, claiming that they were havens of decadence and immorality – drugs, firearms and munitions, "an automatic typewriter", and fake currency. The June 1990 Mineriad in particular was widely criticized both at home and internationally, with one historian (Andrei Pippidi) comparing the events to Nazi Germany's Kristallnacht. Government inquiries later established that the miners were infiltrated and instigated by former Securitate operatives. In February 1994 a Bucharest court "found two security officers, Colonel Ion Nicolae and warrant officer Corneliu Dumitrescu, guilty of ransacking the house of Ion Rațiu, a leading figure in the Christian Democratic National Peasants' Party, during the miners' incursion and stealing $100,000."


title: Ion Iliescu

text: In 1992, three years after the revolution which overthrew the Communist dictatorship, the Romanian government allowed King Michael I to return to his country for Easter celebrations, where he drew large crowds. In Bucharest, over a million people turned out to see him. Michael's popularity alarmed the government of President Ion Iliescu, so Michael was forbidden to visit Romania again for five years. In 1997, after Iliescu's defeat by Emil Constantinescu, the Romanian Government restored Michael's citizenship and again allowed him to visit the country.


title: Ion Iliescu

text: In December 2001, Iliescu pardoned three inmates convicted for bribery, including George Tănase, former Financial Guard head commissioner for Ialomița County. Iliescu had to revoke Tănase's pardon a few days later due to the media outcry, claiming that "a legal adviser was superficial in analyzing the case". Later, the humanitarian reasons invoked in the pardon were contradicted by another medical expert opinion. Another controversial pardon was that of Dan Tartagă, a businessman from Brașov who, while drunk, had run over and killed two people on a pedestrian crossing. He was sentenced to three years and a half but was pardoned after only a couple of months. Tartagă was later sentenced to a two-year sentence for fraud.


title: Ion Iliescu

text: Most controversial of all, on 15 December 2004, a few days before the end of his last term, Iliescu pardoned 47 convicts, including Miron Cozma, the leader of the miners during the early 1990s, who had been sentenced in 1999 to 18 years in prison in conjunction with the September 1991 Mineriad. This has attracted harsh criticism from all Romanian media. Many of the pardoned had been convicted for corruption or other economic crimes, while one had been imprisoned for his involvement in the attempts at suppressing the 1989 Revolution.


title: Ion Iliescu

text: In the last days of his president mandate, he awarded the Order of the Star of Romania (rank of ceremonial knighthood) to the controversial, ultra-nationalist politician Corneliu Vadim Tudor, a gesture which drew criticism in the press and prompted Nobel Peace Prize winner Elie Wiesel, fifteen Radio Free Europe journalists, Timișoara mayor Gheorghe Ciuhandu, songwriter Alexandru Andrieș, and historian Randolph Braham to return their Romanian honours in protest. The leader of the Democratic Alliance of Hungarians in Romania, Béla Markó, did not show up to claim the award he received on the same occasion. President Traian Băsescu revoked the award granted to Tudor on 24 May 2007, but a lawsuit is ongoing even after Băsescu's decree was declared constitutional.


title: Ion Iliescu

text: Ion Iliescu was mentioned in the report of the Council of Europe investigator into illegal activities of the CIA in Europe, Dick Marty. He was identified as one of the people who authorized or at least knew about and should stand accountable for the operation of a CIA black site at Mihail Kogălniceanu airbase from 2003 to 2005, in the context of the War on terror. In April 2015, Iliescu confirmed that he had granted a CIA request for a site in Romania, but was not aware of the nature of the site, describing it as a small gesture of goodwill to an ally in advance of Romania's eventual accession to NATO. Iliescu further stated that had he known of the intended use of the site, he would certainly not have approved the request.


title: Ion Iliescu

text: In 2016, a previously closed legal case regarding crimes against humanity committed by the interim government headed by Iliescu during the Romanian Revolution was reopened. In 2015, after 26 years of prolonged investigation, the authorities concluded that there was no evidence with which they could prosecute. In 2016, the case was ordered to be re-examined by the interim General Prosecutor. By 2017, military prosecutors had alleged that the events of 1989 were orchestrated by a misinformation campaign on the part of Iliescu's government, which were disseminated through broadcasting media. Reportedly, this investigation lead to speculation of whether the conflict of 1989 could be classified as a revolution, or else as a coup d'état.


title: Ion Iliescu

text: In April 2018, the General Prosecutor asked that Iliescu be put on trial. President Klaus Iohannis approved this request, as well as the proceeding of the prosecution of Petre Roman. Iliescu was charged for his alleged role in the killing of 862 people during the revolution, at which time he headed the National Salvation Front (FSN) interim government, as well as the spreading of misinformation. Allegations included Iliescu's apparent involvement in the Mineriad case, in which miners quashed protests against the government. The initial charges, brought forward in 2005, were shortly dropped, until 2014 when the European Court of Human Rights found Iliescu's lack of investigation into the events of Mineriad to be in violation of human rights to life, freedom from inhumane, and degrading treatment and demonstration, and again in 2015, when the Military Prosecutor's Section within the Prosecutor's Office and the Justice Office reopened investigations into the Mineriad protests, accusing Iliescu, along with other accused perpetrators, of coordinating a general and systematic attack against the civilian population during the events from 13 until 15 June 1990 in Bucharest. On 13 June 2017, the Prosecutor's Office indicted Iliescu for crimes against humanity for actions taken by Iliescu during the Mineriad protests. The statement released by the office claimed that the attack illegally involved forces of the Interior Ministry, Defence Ministry, Romanian Intelligence service, as well as the miners and other workers from various areas of the country. The office further alleged that attacks were also carried out against peaceful residents. The case was ultimately rejected in December 2020, as the judges found that the indictment was void and thus could not be used in a trial.


title: Ion Iliescu

text: On 8 April 2019, Iliescu was officially charged with crimes against humanity Iliescu's lawyer Adrian Georgescu complained that the file was illegitimate due to its lack of a prosecutor. In December 2019, Iliescu's trial began to focus on allegations that he had intentionally spread disinformation through the use of broadcast media with the aid of Aurel Dragoș Munteanu, a member of the FSN and the director of TVR during the revolution of 1989, meaning that he was greatly influential in the FSN's ability to foster support in the Romanian public. Among the claims investigated were Iliescu's broadcast claim that "unknown terrorists" were responsible for the deaths of Elena and Nicolae Ceaușescu.


title: Ion Iliescu

text: Amid candlelight vigils and other memorial services during the 30th anniversary of the Romanian Revolution, several survivors of the conflict spoke out against Iliescu's trial, with many claiming it is a publicity stunt on the part of Iohannis to gain popularity from the Romanian population that still seek the truth about the revolution. Iliescu's trial is not expected to reach a definitive conclusion. The trial was first postponed to February 2020 due to Iliescu's declining health and the slow pace of legal proceedings. This case was also rejected in June 2020, as a judge decided the indictment was not valid.


title: Romford

text: Romford is a large town in east London and the administrative centre of the London Borough of Havering. It is located northeast of Charing Cross and is one of the major metropolitan centres identified in the London Plan. Historically, Romford was a market town in the county of Essex, and formed the administrative centre of the liberty of Havering before that liberty was dissolved in 1892. Good road links to London and the opening of the railway station in 1839 were key to the development of the town. The economic history of Romford is characterised by a shift from agriculture to light industry and then to retail and commerce. As part of the suburban growth of London throughout the 20th century, Romford significantly expanded and increased in population, becoming a municipal borough in 1937 and was incorporated into Greater London in 1965. Today, it is one of the largest commercial, retail, entertainment and leisure districts in London and has a well-developed night-time economy as well. Its population, as of 2011, was 122,854.


title: Romford

text: Romford is first recorded in 1177 as "Romfort", which is formed from Old English 'rūm' and 'ford' and means "the wide or spacious ford". The naming of the River Rom is a local 'back-formation' from the name of the town; and the river is elsewhere known as the Beam. The ford most likely existed on the main London to Colchester road where it crossed that river.


title: Romford

text: The town developed in the Middle Ages on the main road to London and the regionally significant Romford Market was established in 1247. The original site of the town was to the south, in an area still known as Oldchurch. It was moved northwards to the present site in the later medieval period to avoid the frequent flooding of the River Rom. The first building on the new site was the 1410 Chapel of St Edward (since replaced by the 1850 Parish Church of St Edward the Confessor. The early history of Romford and the immediate area is agricultural and it is recorded as being the location of a number of mills used to grind corn. The area was a focus of the leather industry from the 15th to the early 19th centuries and there is record of a wide range of industries such as cloth making, weaving, charcoal burning, metal working and brewing. Communications played an important part in its development; the main road to London was maintained by the Middlesex and Essex Turnpike Trust from 1721 and Romford became a coaching town in the 18th century.


title: Romford

text: Several failed attempts were made in the early 19th century to connect the town to the Thames via a Romford Canal. It was initially intended to terminate at a basin near to the Star Brewery, to transport agricultural products to London and, eventually, to serve growing industrial sites in Romford. A later proposal included an extension to Collier Row, whereby timber from Hainault Forest could be transported to the Thames for use in the Royal Dockyards. Only two miles of canal were constructed and the canal company were unable to reach the town.


title: Romford

text: The development of the town was accelerated by the opening of the railway station in 1839 which stimulated the local economy and was key to the development of the Star Brewery. Initially Eastern Counties Railway services operated between Mile End and Romford, with extensions to Brentwood and to Shoreditch in 1840. A second station was opened on South Street in 1892 by the London, Tilbury and Southend Railway on the line to Upminster and Grays, giving Romford a rail connection to Tilbury Docks. The two stations were combined into one in 1934. Light industry slowly developed, reaching a peak in the 1970s with a number of factories on the edge of town, such as the Roneo Vickers office machinery company, Colvern manufacturers of wireless components, May's Sheet Metal Works and brush manufacturers Betterware. Suburban expansion increased the population and reinforced Romford's position as a significant regional town centre. The Liberty Shopping Centre was constructed in the 1960s, and has been modernised and supplemented with further shopping centres throughout the town, including The Mall, opened in 1990 (as 'Liberty 2'); and The Brewery, opened in 2000 on the site of the old Star Brewery.


title: Romford

text: Romford formed a chapelry in the large ancient parish of Hornchurch in the Becontree hundred of Essex; as well as the town it included the wards of Collier Row, Harold Wood, and Noak Hill. Through ancient custom the area enjoyed special status and a charter in 1465 removed the parish from the Becontree hundred and the county of Essex and it instead formed the independent liberty of Havering governed from a court house in the market place. Over time the vestry of Romford chapelry absorbed the local powers that would usually be held by the parish authorities in Hornchurch and in 1849 Romford became a separate parish within the liberty. Improvement commissioners were set up in 1819 for paving, lighting, watching, and cleansing of the marketplace and main streets. As the town grew this arrangement became ineffective at controlling sanitation and in 1851 a local board of health was set up for the parish; although its area was reduced in 1855 to cover only the town ward. The remainder of the parish became part of the Romford rural sanitary district in 1875. These changes and the introduction of the Romford Poor Law Union in 1836 eroded the powers of the liberty and it was finally abolished in 1892 and reincorporated into Essex.


title: Romford

text: The Local Government Act 1894 reformed local government and created the Romford Urban District and Romford Rural District to replace the local board and sanitary district; following which the Romford parish was split into "Romford Urban" and "Romford Rural" along the lines of the urban district. In 1900 the parish was recombined and the urban district expanded to cover all of the former area of the historic chapelry, except for Noak Hill which remained in the rural district and had become a parish in its own right in 1895. The enlarged urban district formed part of the London Traffic Area from 1924 and the London Passenger Transport Area from 1933. The suburban expansion of London caused an increase in population during the 1930s and the urban district was expanded further in 1934, taking in the parishes of Havering-atte-Bower and Noak Hill. It was incorporated as the Municipal Borough of Romford in 1937. In 1965 the municipal borough was abolished and its former area was combined with that of Hornchurch Urban District; it was again removed from Essex and since then has formed the northern part of the London Borough of Havering in Greater London.


title: Romford

text: There was early expansion in the 1840s when 200 cottages were built in the area formerly occupied by an army barracks; it was known as "New Romford". To acknowledge the military connection, when in 1961 these were in turn replaced with new housing the name "Waterloo Road Estate" was applied. To the east of the market place from 1850 middle class suburban housing was constructed with a much larger area of built-over to the south of the railway from 1851. Through a gradual process of selling off former manors, houses were built radiating from the town in all directions for about a mile. More significant growth occurred between 1910 and 1911 with the construction of Romford Garden Suburb, which included Raphael Park and Gidea Park railway station. Large sections of land to the north of the town at Collier Row were developed in the interwar period and after World War II, the London County Council built the Harold Hill estate to the north east from 1948 to 1958.


title: Romford

text: The right to supply electricity to the town was secured by the County of London Electricity Supply Company in 1913. Initially power was generated within the Star Brewery site, with the supply switching to Barking Power Station in 1925. Gas supply began in 1825 with gas works of constructed by 1938. Following the Telegraph Act 1899 Romford became part of the Post Office London telephone area and the Romford exchange was recorded as having 240 subscribers in 1916. The town water supply initially came from the Havering Well, and 1859 a new public well and pump was built at the east end of the market. The South Essex Waterworks Company started installing mains water supply in 1863 and had offices in South Street. By 1905 its supply was serving Ilford, Collier Row, Ardleigh Green, Brentwood, and Hornchurch. Sewage works were installed by the local board at Oldchurch in 1862, with further works built in Hornchurch in 1869.


title: Romford

text: The Romford UK Parliament constituency consists of the Havering wards of Brooklands, Havering Park, Hylands, Mawneys, Pettits, Romford Town, and Squirrel's Heath.


title: Romford

text: The current MP is Andrew Rosindell, a native of the town. Romford forms part of the Havering and Redbridge London Assembly constituency.


title: Romford

text: Each ward elects three councillors to Havering London Borough Council. As of the 2018 council elections, all the elected councillors for the wards in Romford constituency were Conservative.


title: Romford

text: Romford F.C., who currently play in the Essex Senior League, is the local football team. The London Raiders ice hockey team is based in Romford. Romford is home to the Romford and Gidea Park Rugby Football Club, which was established in 1927. In 2003, the club became one of the first in the country to have a ladies Rugby team.


title: Romford

text: The town centre is about above sea level on a gravel terrace rising from the River Thames. The north of the town has developed on London Clay and is situated as much as above sea level. A continuous gentle rise in the eastern suburbs towards Gidea Park and Harold Wood peaks around around the Harold Court. On the northern side, Harold Hill peaks at . The semi-rural area north of Collier Row and Harold Hill consists of many rolls of hills, with elevation peaking at the village of Havering-atte-Bower, .


title: Romford

text: The town centre is for the most part contained within a ring road formed of St Edwards Way, Mercury Gardens, Thurloe Gardens, Oldchurch Road and Waterloo Road. The market place and much of South Street and the High Street are pedestrianised. The railway cuts through the town from east to west on a viaduct, with the bulk of the central Romford area to its north. The River Rom flows through the town in underground channels and joins the Thames after flowing through Hornchurch; elsewhere along its course it is known as the River Beam and forms part of the strategic waterways Blue Ribbon Network. Romford has formed part of the continuously built-up area of London since the 1930s and is contiguous with Rush Green to the west, Collier Row to the north, Gidea Park to the east and Hornchurch to the south east.


title: Romford

text: The Romford post town covers all of the former municipal borough and extends over a much wider area, including parts of Barking and Dagenham and Epping Forest.


title: Romford

text: Neighbourhoods of Romford include: Collier Row, Gidea Park, Harold Hill, Harold Park, Harold Wood, Havering-atte-Bower, Rise Park and Rush Green.


title: Romford

text: Romford is located northeast of Charing Cross in central London; northeast of Ilford; north of Dagenham; northwest of Grays; south-west of Brentwood; west of Basildon; and southeast of Epping.


title: Romford

text: Climate data for Romford is taken from the nearest weather station at Greenwich, around southwest of the marketplace.


title: Romford

text: The Havering committee area for Romford is defined as the wards of Romford Town and Brooklands. Demographic data is produced by the Office for National Statistics for these wards. In 2001 the population of Romford Town was 13,200 and Brooklands was 13,024, giving a total population of 26,224. In contrast, the approximate population of the area within the 2005 Romford Urban Strategy was estimated to be 36,500. 71.52% in Romford Town and 70.48% in Brooklands report their religion as Christian, compared to 76.13% for Havering, 58.23% in London and 71.74% in England. 15.71% in Romford Town and 16.62% in Brooklands report having no religion, compared to 13.18% in Havering, 15.76% in London and 14.59% in England.


title: Romford

text: In 2011, the Romford Parliament constituency was 82% White British, 5.8% Asian, 5% Other White and 4.7% Black out of a total population of 95,894. The constituency is predominantly Christian with 64% of the residents reporting that religion.


title: Romford

text: Out of the wards that make up Romford overall, the highest male life expectancy was in Squirrel's Heath (80.7 years) while the highest female expectancy was in Romford Town (85.7 years). The lowest were Heaton (76.2 years) and Heaton and Gooshays (both 81.3 years) respectively.


title: Romford

text: The average house price as of 2014 was £225,000 in Romford Town ward. In the Pettits ward, 87.5% of houses were owned by households; the lowest figure, and the only minority one, was Gooshays ward with 48.6%.


title: Romford

text: Romford is recognised in the London Plan as one of 13 regionally significant metropolitan centres in Greater London, with a considerable catchment area. The total commercial floorspace in the town was in 2002, of which was retail space and was offices. The retail space is growing and in 2005 consisted of . The retail economy is complemented by a central business district close to the railway station, where the offices of employers such as Aon are located. Employment in the town centre was categorised in 2002 as approximately 40% commercial office, 40% comparison retail, 10% hospitality, 5% public sector, 2.5% service retail and 2.5% arts and entertainment. Compared to the similar east London areas of Ilford, Stratford and Barking, there is more comparison retail and commercial office employment in Romford and less public sector work. The total turnover of £413,395,000 in 2002 for Romford was larger than any other comparable town centre in east London and approximately 70% came from the commercial office businesses.


title: Romford

text: There is a developed night time economy, greater than in any other metropolitan centre in Greater London, with of cinemas, theatres and concert hall space; of bars and pubs; of cafés and restaurants; and of fast food and take away venues. The night time economy is almost as significant as the day economy with around 12,000 visits to Romford during the day and 11,000 visits to pubs, clubs and bars at night.


title: Romford

text: As of 2012, Romford has of total town centre floorspace (retail, leisure and vacant), placing it fifth in Greater London only behind the West End, Croydon, Kingston upon Thames and Stratford for "town centre vitality and viability".


title: Romford

text: The town is served by Romford railway station; it is situated on the Great Eastern Main Line and the Elizabeth line, in London fare zone 6. Elizabeth line trains call at the station, formed of high-frequency services between London Liverpool Street and Shenfield. Services will extend through central London to Reading when the line is opened fully.


title: Romford

text: Some Greater Anglia services to/from and also call at the station. A branch line shuttle on the Romford to Upminster Line is operated by London Overground.


title: Romford

text: Romford is a hub of the London Buses network, with services to Canning Town, Stratford, Leytonstone and Dagenham; there are also feeder services from the large housing developments at Collier Row and Harold Hill. There are night bus services to Stratford, Harold Hill and Paddington. Romford town centre has a very high Public Transport Accessibility Level score of 6.


title: Romford

text: The A12 trunk road passes to the north of Romford, while the A118 road from Stratford connects with it at Gallows Corner at the start of the A127 road to Southend.


title: Romford

text: Havering Council's urban strategy aims to make Romford a cultural destination, whilst recognising that Hornchurch forms the main cultural hub of the borough with a large theatre and arts spaces. As a former market and coaching town, Romford is well served by public houses and two that are located in the market place are listed buildings. The market and adjacent streets also form a conservation area.


title: Romford

text: Mass entertainment facilities in the town include the Brookside Theatre, Romford Greyhound Stadium, one of the few remaining dog racing tracks in London; 2 multi-screen cinemas; and until April 2013 Romford Ice Arena, which was home to the local Romford Raiders ice hockey team. The Dolphin Centre was a popular swimming and leisure facility located in the town from 1982 to 1995, but the site was redeveloped into the current Axis residential tower block and Asda superstore in the mid-2000s. There is also a Romford F.C. associated with the town. Romford Bowls Club is based in Lodge Farm Park. 1980s Post Punk bands Department S and Purple Hearts both have origins in Romford. The town is strongly associated with the electronic music group Underworld, who cite Romford in their hit "Born Slippy", affiliated to the movie "Trainspotting".


title: Romford

text: Romford's position as a focus for electronic music production was reinforced by the presence of the Strictly Underground and Suburban Base record labels, with Suburban Base developing from the Boogie Times record store. According to a Billboard article in 1992, Romford-produced dance music formed part of a trend favouring suburban and provincial "bedroom" record labels over those in central London. In 2013, the film "Death Walks" was filmed in Romford over a four-month period. The cult TV series "Garth Marenghi's Darkplace" was set in the fictional Darkplace Hospital, in Romford. The local newspapers for the town and the borough of Havering are the "Romford Recorder", "Romford and Havering Post" and "Romford Yellow Advertiser". Two radio stations are located in the area: Time 107.5 and Bedrock Radio (a community health and hospital radio station).


title: Poisson point process

text: In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one another. The Poisson point process is often called simply the Poisson process, but it is also called a Poisson random measure, Poisson random point field or Poisson point field. This point process has convenient mathematical properties, which has led to its being frequently defined in Euclidean space and used as a mathematical model for seemingly random processes in numerous disciplines such as astronomy, biology, ecology, geology, seismology, physics, economics, image processing, and telecommunications.


title: Poisson point process

text: The process is named after French mathematician Siméon Denis Poisson despite Poisson's never having studied the process. Its name derives from the fact that if a collection of random points in some space forms a Poisson process, then the number of points in a region of finite size is a random variable with a Poisson distribution. The process was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and insurance mathematics.


title: Poisson point process

text: The Poisson point process is often defined on the real line, where it can be considered as a stochastic process. In this setting, it is used, for example, in queueing theory to model random events, such as the arrival of customers at a store, phone calls at an exchange or occurrence of earthquakes, distributed in time. In the plane, the point process, also known as a spatial Poisson process, can represent the locations of scattered objects such as transmitters in a wireless network, particles colliding into a detector, or trees in a forest. In this setting, the process is often used in mathematical models and in the related fields of spatial point processes, stochastic geometry, spatial statistics and continuum percolation theory. The Poisson point process can be defined on more abstract spaces. Beyond applications, the Poisson point process is an object of mathematical study in its own right. In all settings, the Poisson point process has the property that each point is stochastically independent to all the other points in the process, which is why it is sometimes called a "purely" or "completely" random process. Despite its wide use as a stochastic model of phenomena representable as points, the inherent nature of the process implies that it does not adequately describe phenomena where there is sufficiently strong interaction between the points. This has inspired the proposal of other point processes, some of which are constructed with the Poisson point process, that seek to capture such interaction.


title: Poisson point process

text: The point process depends on a single mathematical object, which, depending on the context, may be a constant, a locally integrable function or, in more general settings, a Radon measure. In the first case, the constant, known as the rate or intensity, is the average density of the points in the Poisson process located in some region of space. The resulting point process is called a homogeneous or stationary Poisson point process. In the second case, the point process is called an inhomogeneous or nonhomogeneous Poisson point process, and the average density of points depend on the location of the underlying space of the Poisson point process. The word "point" is often omitted, but there are other "Poisson processes" of objects, which, instead of points, consist of more complicated mathematical objects such as lines and polygons, and such processes can be based on the Poisson point process. Both the homogeneous and nonhomogeneous Poisson point processes are particular cases of the generalized renewal process.


title: Poisson point process

text: Depending on the setting, the process has several equivalent definitions as well as definitions of varying generality owing to its many applications and characterizations. The Poisson point process can be defined, studied and used in one dimension, for example, on the real line, where it can be interpreted as a counting process or part of a queueing model; in higher dimensions such as the plane where it plays a role in stochastic geometry and spatial statistics; or on more general mathematical spaces. Consequently, the notation, terminology and level of mathematical rigour used to define and study the Poisson point process and points processes in general vary according to the context.


title: Poisson point process

text: Despite all this, the Poisson point process has two key properties—the Poisson property and the independence property— that play an essential role in all settings where the Poisson point process is used. The two properties are not logically independent; indeed, independence implies the Poisson distribution of point counts, but not the converse.


title: Poisson point process

text: A Poisson point process is characterized via the Poisson distribution. The Poisson distribution is the probability distribution of a random variable formula_1 (called a "Poisson random variable") such that the probability that formula_2 equals formula_3 is given by:


title: Poisson point process

text: where formula_5 denotes factorial and the parameter formula_6 determines the shape of the distribution. (In fact, formula_6 equals the expected value of formula_1.)


title: Poisson point process

text: By definition, a Poisson point process has the property that the number of points in a bounded region of the process's underlying space is a Poisson-distributed random variable.


title: Poisson point process

text: Consider a collection of disjoint and bounded subregions of the underlying space. By definition, the number of points of a Poisson point process in each bounded subregion will be completely independent of all the others.


title: Poisson point process

text: This property is known under several names such as "complete randomness", "complete independence", or "independent scattering" and is common to all Poisson point processes. In other words, there is a lack of interaction between different regions and the points in general, which motivates the Poisson process being sometimes called a "purely" or "completely" random process.


title: Poisson point process

text: If a Poisson point process has a parameter of the form formula_9, where formula_10 is Lebesgue measure (that is, it assigns length, area, or volume to sets) and formula_11 is a constant, then the point process is called a homogeneous or stationary Poisson point process. The parameter, called rate or intensity, is related to the expected (or average) number of Poisson points existing in some bounded region, where "rate" is usually used when the underlying space has one dimension. The parameter formula_11 can be interpreted as the average number of points per some unit of extent such as length, area, volume, or time, depending on the underlying mathematical space, and it is also called the "mean density" or "mean rate"; see Terminology.


title: Poisson point process

text: The homogeneous Poisson point process, when considered on the positive half-line, can be defined as a counting process, a type of stochastic process, which can be denoted as formula_13. A counting process represents the total number of occurrences or events that have happened up to and including time formula_14. A counting process is a homogeneous Poisson counting process with rate formula_15 if it has the following three properties:


title: Poisson point process

text: In other words, the probability of the random variable formula_20 being equal to formula_21 is given by:


title: Poisson point process

text: The Poisson counting process can also be defined by stating that the time differences between events of the counting process are exponential variables with mean formula_23. The time differences between the events or arrivals are known as interarrival or interoccurence times.


title: Poisson point process

text: Interpreted as a point process, a Poisson point process can be defined on the real line by considering the number of points of the process in the interval formula_24. For the homogeneous Poisson point process on the real line with parameter formula_15, the probability of this random number of points, written here as formula_26, being equal to some counting number formula_27 is given by:


title: Poisson point process

text: For some positive integer formula_29, the homogeneous Poisson point process has the finite-dimensional distribution given by:


title: Poisson point process

text: In other words, formula_26 is a Poisson random variable with mean formula_33, where formula_34. Furthermore, the number of points in any two disjoint intervals, say, formula_35 and formula_36 are independent of each other, and this extends to any finite number of disjoint intervals. In the queueing theory context, one can consider a point existing (in an interval) as an "event", but this is different to the word event in the probability theory sense. It follows that formula_11 is the expected number of "arrivals" that occur per unit of time.


title: Poisson point process

text: In other words, for any finite formula_40, the random variable formula_41 is independent of formula_14, so it is also called a stationary Poisson process.


title: Poisson point process

text: The quantity formula_43 can be interpreted as the expected or average number of points occurring in the interval formula_44, namely:


title: Poisson point process

text: where formula_46 denotes the expectation operator. In other words, the parameter formula_11 of the Poisson process coincides with the "density" of points. Furthermore, the homogeneous Poisson point process adheres to its own form of the (strong) law of large numbers. More specifically, with probability one:


title: Poisson point process

text: where formula_49 denotes the limit of a function, and formula_50 is expected number of arrivals occurred per unit of time.


title: Poisson point process

text: The distance between two consecutive points of a point process on the real line will be an exponential random variable with parameter formula_11 (or equivalently, mean formula_23). This implies that the points have the memoryless property: the existence of one point existing in a finite interval does not affect the probability (distribution) of other points existing, but this property has no natural equivalence when the Poisson process is defined on a space with higher dimensions.


title: Poisson point process

text: where little-o notation is being used. A point process is called a simple point process when the probability of any of its two points coinciding in the same position, on the underlying space, is zero. For point processes in general on the real line, the property of orderliness implies that the process is simple, which is the case for the homogeneous Poisson point process.


title: Poisson point process

text: On the real line, the homogeneous Poisson point process has a connection to the theory of martingales via the following characterization: a point process is the homogeneous Poisson point process if and only if


title: Poisson point process

text: On the real line, the Poisson process is a type of continuous-time Markov process known as a birth process, a special case of the birth–death process (with just births and zero deaths). More complicated processes with the Markov property, such as Markov arrival processes, have been defined where the Poisson process is a special case.


title: Poisson point process

text: If the homogeneous Poisson process is considered just on the half-line formula_55, which can be the case when formula_14 represents time then the resulting process is not truly invariant under translation. In that case the Poisson process is no longer stationary, according to some definitions of stationarity.


title: Poisson point process

text: There have been many applications of the homogeneous Poisson process on the real line in an attempt to model seemingly random and independent events occurring. It has a fundamental role in queueing theory, which is the probability field of developing suitable stochastic models to represent the random arrival and departure of certain phenomena. For example, customers arriving and being served or phone calls arriving at a phone exchange can be both studied with techniques from queueing theory.


title: Poisson point process

text: The homogeneous Poisson process on the real line is considered one of the simplest stochastic processes for counting random numbers of points. This process can be generalized in a number of ways. One possible generalization is to extend the distribution of interarrival times from the exponential distribution to other distributions, which introduces the stochastic process known as a renewal process. Another generalization is to define the Poisson point process on higher dimensional spaces such as the plane.


title: Poisson point process

text: A spatial Poisson process is a Poisson point process defined in the plane formula_57. For its mathematical definition, one first considers a bounded, open or closed (or more precisely, Borel measurable) region formula_58 of the plane. The number of points of a point process formula_2 existing in this region formula_60 is a random variable, denoted by formula_61. If the points belong to a homogeneous Poisson process with parameter formula_62, then the probability of formula_3 points existing in formula_64 is given by:


title: Poisson point process

text: For some finite integer formula_68, we can give the finite-dimensional distribution of the homogeneous Poisson point process by first considering a collection of disjoint, bounded Borel (measurable) sets formula_69. The number of points of the point process formula_70 existing in formula_71 can be written as formula_72. Then the homogeneous Poisson point process with parameter formula_62 has the finite-dimensional distribution:


title: Poisson point process

text: The spatial Poisson point process features prominently in spatial statistics, stochastic geometry, and continuum percolation theory. This point process is applied in various physical sciences such as a model developed for alpha particles being detected. In recent years, it has been frequently used to model seemingly disordered spatial configurations of certain wireless communication networks. For example, models for cellular or mobile phone networks have been developed where it is assumed the phone network transmitters, known as base stations, are positioned according to a homogeneous Poisson point process.


title: Poisson point process

text: The previous homogeneous Poisson point process immediately extends to higher dimensions by replacing the notion of area with (high dimensional) volume. For some bounded region formula_64 of Euclidean space formula_76, if the points form a homogeneous Poisson process with parameter formula_62, then the probability of formula_3 points existing in formula_79 is given by:


title: Poisson point process

text: where formula_66 now denotes the formula_82-dimensional volume of formula_64. Furthermore, for a collection of disjoint, bounded Borel sets formula_84, let formula_72 denote the number of points of formula_2 existing in formula_71. Then the corresponding homogeneous Poisson point process with parameter formula_62 has the finite-dimensional distribution:


title: Poisson point process

text: Homogeneous Poisson point processes do not depend on the position of the underlying space through its parameter formula_90, which implies it is both a stationary process (invariant to translation) and an isotropic (invariant to rotation) stochastic process. Similarly to the one-dimensional case, the homogeneous point process is restricted to some bounded subset of formula_91, then depending on some definitions of stationarity, the process is no longer stationary.


title: Poisson point process

text: If the homogeneous point process is defined on the real line as a mathematical model for occurrences of some phenomenon, then it has the characteristic that the positions of these occurrences or events on the real line (often interpreted as time) will be uniformly distributed. More specifically, if an event occurs (according to this process) in an interval formula_92 where formula_93, then its location will be a uniform random variable defined on that interval. Furthermore, the homogeneous point process is sometimes called the "uniform" Poisson point process (see Terminology). This uniformity property extends to higher dimensions in the Cartesian coordinate, but not in, for example, polar coordinates.


title: Poisson point process

text: The inhomogeneous or nonhomogeneous Poisson point process (see Terminology) is a Poisson point process with a Poisson parameter set as some location-dependent function in the underlying space on which the Poisson process is defined. For Euclidean space formula_76, this is achieved by introducing a locally integrable positive function formula_95, such that for every bounded region formula_64 the (formula_82-dimensional) volume integral of formula_98 over region formula_64 is finite. In other words, if this integral, denoted by formula_100, is:


title: Poisson point process

text: where formula_102 is a (formula_82-dimensional) volume element, then for every collection of disjoint bounded Borel measurable sets formula_69, an inhomogeneous Poisson process with (intensity) function formula_105 has the finite-dimensional distribution:


title: Poisson point process

text: Furthermore, formula_100 has the interpretation of being the expected number of points of the Poisson process located in the bounded region formula_64, namely


title: Poisson point process

text: On the real line, the inhomogeneous or non-homogeneous Poisson point process has mean measure given by a one-dimensional integral. For two real numbers formula_110 and formula_111, where formula_112, denote by formula_113 the number points of an inhomogeneous Poisson process with intensity function formula_114 occurring in the interval formula_92. The probability of formula_3 points existing in the above interval formula_92 is given by:


title: Poisson point process

text: A feature of the one-dimension setting, is that an inhomogeneous Poisson process can be transformed into a homogeneous by a monotone transformation or mapping, which is achieved with the inverse of formula_122.


title: Poisson point process

text: The inhomogeneous Poisson point process, when considered on the positive half-line, is also sometimes defined as a counting process. With this interpretation, the process, which is sometimes written as formula_123, represents the total number of occurrences or events that have happened up to and including time formula_124. A counting process is said to be an inhomogeneous Poisson counting process if it has the four properties:


title: Poisson point process

text: In the case of point processes with refractoriness (e.g., neural spike trains) a stronger version of property 4 applies: formula_131.


title: Poisson point process

text: The above properties imply that formula_132 is a Poisson random variable with the parameter (or mean)


title: Poisson point process

text: An inhomogeneous Poisson process defined in the plane formula_57 is called a spatial Poisson process It is defined with intensity function and its intensity measure is obtained performing a surface integral of its intensity function over some region. For example, its intensity function (as a function of Cartesian coordinates formula_136 and formula_137) can be


title: Poisson point process

text: In the plane, formula_142 corresponds to a surface integral while in formula_143 the integral becomes a (formula_144-dimensional) volume integral.


title: Poisson point process

text: When the real line is interpreted as time, the inhomogeneous process is used in the fields of counting processes and in queueing theory. Examples of phenomena which have been represented by or appear as an inhomogeneous Poisson point process include:


title: Poisson point process

text: In the plane, the Poisson point process is important in the related disciplines of stochastic geometry and spatial statistics. The intensity measure of this point process is dependent on the location of underlying space, which means it can be used to model phenomena with a density that varies over some region. In other words, the phenomena can be represented as points that have a location-dependent density. This processes has been used in various disciplines and uses include the study of salmon and sea lice in the oceans, forestry, and search problems.


title: Poisson point process

text: The Poisson intensity function formula_145 has an interpretation, considered intuitive, with the volume element formula_146 in the infinitesimal sense: formula_147 is the infinitesimal probability of a point of a Poisson point process existing in a region of space with volume formula_146 located at formula_136.


title: Poisson point process

text: For example, given a homogeneous Poisson point process on the real line, the probability of finding a single point of the process in a small interval of width formula_150 is approximately formula_151. In fact, such intuition is how the Poisson point process is sometimes introduced and its distribution derived.


title: Poisson point process

text: If a Poisson point process has an intensity measure that is a locally finite and diffuse (or non-atomic), then it is a simple point process. For a simple point process, the probability of a point existing at a single point or location in the underlying (state) space is either zero or one. This implies that, with probability one, no two (or more) points of a Poisson point process coincide in location in the underlying space.


title: Poisson point process

text: Simulating a Poisson point process on a computer is usually done in a bounded region of space, known as a simulation "window", and requires two steps: appropriately creating a random number of points and then suitably placing the points in a random manner. Both these two steps depend on the specific Poisson point process that is being simulated.


title: Poisson point process

text: The number of points formula_152 in the window, denoted here by formula_153, needs to be simulated, which is done by using a (pseudo)-random number generating function capable of simulating Poisson random variables.


title: Poisson point process

text: For the homogeneous case with the constant formula_154, the mean of the Poisson random variable formula_152 is set to formula_156 where formula_157 is the length, area or (formula_144-dimensional) volume of formula_153.


title: Poisson point process

text: For the inhomogeneous case, formula_156 is replaced with the (formula_144-dimensional) volume integral


title: Poisson point process

text: For the homogeneous case in one dimension, all points are uniformly and independently placed in the window or interval formula_164. For higher dimensions in a Cartesian coordinate system, each coordinate is uniformly and independently placed in the window formula_164. If the window is not a subspace of Cartesian space (for example, inside a unit sphere or on the surface of a unit sphere), then the points will not be uniformly placed in formula_164, and suitable change of coordinates (from Cartesian) are needed.


title: Poisson point process

text: For the inhomogeneous case, a couple of different methods can be used depending on the nature of the intensity function formula_105. If the intensity function is sufficiently simple, then independent and random non-uniform (Cartesian or other) coordinates of the points can be generated. For example, simulating a Poisson point process on a circular window can be done for an isotropic intensity function (in polar coordinates formula_169 and formula_170), implying it is rotationally variant or independent of formula_170 but dependent on formula_169, by a change of variable in formula_169 if the intensity function is sufficiently simple.


title: Poisson point process

text: For more complicated intensity functions, one can use an acceptance-rejection method, which consists of using (or 'accepting') only certain random points and not using (or 'rejecting') the other points, based on the ratio:


title: Poisson point process

text: The Poisson point process can be further generalized to what is sometimes known as the general Poisson point process or general Poisson process by using a Radon measure formula_176, which is locally-finite measure. In general, this Radon measure formula_176 can be atomic, which means multiple points of the Poisson point process can exist in the same location of the underlying space. In this situation, the number of points at formula_178 is a Poisson random variable with mean formula_179. But sometimes the converse is assumed, so the Radon measure formula_176 is diffuse or non-atomic.


title: Poisson point process

text: A point process formula_181 is a general Poisson point process with intensity formula_176 if it has the two following properties:


title: Poisson point process

text: The Radon measure formula_176 maintains its previous interpretation of being the expected number of points of formula_181 located in the bounded region formula_64, namely


title: Poisson point process

text: Furthermore, if formula_176 is absolutely continuous such that it has a density (which is the Radon–Nikodym density or derivative) with respect to the Lebesgue measure, then for all Borel sets formula_64 it can be written as:


title: Poisson point process

text: Despite its name, the Poisson point process was neither discovered nor studied by the French mathematician Siméon Denis Poisson; the name is cited as an example of Stigler's law. The name stems from its inherent relation to the Poisson distribution, derived by Poisson as a limiting case of the binomial distribution. This describes the probability of the sum of formula_3 Bernoulli trials with probability formula_201, often likened to the number of heads (or tails) after formula_3 biased flips of a coin with the probability of a head (or tail) occurring being formula_201. For some positive constant formula_204, as formula_3 increases towards infinity and formula_201 decreases towards zero such that the product formula_207 is fixed, the Poisson distribution more closely approximates that of the binomial.


title: Poisson point process

text: Poisson derived the Poisson distribution, published in 1841, by examining the binomial distribution in the limit of formula_201 (to zero) and formula_3 (to infinity). It only appears once in all of Poisson's work, and the result was not well known during his time. Over the following years a number of people used the distribution without citing Poisson, including Philipp Ludwig von Seidel and Ernst Abbe.


title: Poisson point process

text: There are a number of claims for early uses or discoveries of the Poisson point process. For example, John Michell in 1767, a decade before Poisson was born, was interested in the probability a star being within a certain region of another star under the assumption that the stars were "scattered by mere chance", and studied an example consisting of the six brightest stars in the Pleiades, without deriving the Poisson distribution. This work inspired Simon Newcomb to study the problem and to calculate the Poisson distribution as an


title: Poisson point process

text: At the beginning of the 20th century the Poisson process (in one dimension) would arise independently in different situations.


title: Poisson point process

text: In Sweden 1903, Filip Lundberg published a thesis containing work, now considered fundamental and pioneering, where he proposed to model insurance claims with a homogeneous Poisson process.


title: Poisson point process

text: In Denmark in 1909 another discovery occurred when A.K. Erlang derived the Poisson distribution when developing a mathematical model for the number of incoming phone calls in a finite time interval. Erlang was not at the time aware of Poisson's earlier work and assumed that the number phone calls arriving in each interval of time were independent to each other. He then found the limiting case, which is effectively recasting the Poisson distribution as a limit of the binomial distribution.


title: Poisson point process

text: In 1910 Ernest Rutherford and Hans Geiger published experimental results on counting alpha particles. Their experimental work had mathematical contributions from Harry Bateman, who derived Poisson probabilities as a solution to a family of differential equations, though the solution had been derived earlier, resulting in the independent discovery of the Poisson process. After this time there were many studies and applications of the Poisson process, but its early history is complicated, which has been explained by the various applications of the process in numerous fields by biologists, ecologists, engineers and various physical scientists.


title: Poisson point process

text: The years after 1909 led to a number of studies and applications of the Poisson point process, however, its early history is complex, which has been explained by the various applications of the process in numerous fields by biologists, ecologists, engineers and others working in the physical sciences. The early results were published in different languages and in different settings, with no standard terminology and notation used. For example, in 1922 Swedish chemist and Nobel Laureate Theodor Svedberg proposed a model in which a spatial Poisson point process is the underlying process to study how plants are distributed in plant communities. A number of mathematicians started studying the process in the early 1930s, and important contributions were made by Andrey Kolmogorov, William Feller and Aleksandr Khinchin, among others. In the field of teletraffic engineering, mathematicians and statisticians studied and used Poisson and other point processes.


title: Poisson point process

text: The Swede Conny Palm in his 1943 dissertation studied the Poisson and other point processes in the one-dimensional setting by examining them in terms of the statistical or stochastic dependence between the points in time. In his work exists the first known recorded use of the term "point processes" as "Punktprozesse" in German.


title: Poisson point process

text: It is believed that William Feller was the first in print to refer to it as the "Poisson process" in a 1940 paper. Although the Swede Ove Lundberg used the term "Poisson process" in his 1940 PhD dissertation, in which Feller was acknowledged as an influence, it has been claimed that Feller coined the term before 1940. It has been remarked that both Feller and Lundberg used the term as though it were well-known, implying it was already in spoken use by then. Feller worked from 1936 to 1939 alongside Harald Cramér at Stockholm University, where Lundberg was a PhD student under Cramér who did not use the term "Poisson process" in a book by him, finished in 1936, but did in subsequent editions, which his has led to the speculation that the term "Poisson process" was coined sometime between 1936 and 1939 at the Stockholm University.


title: Poisson point process

text: The terminology of point process theory in general has been criticized for being too varied. In addition to the word "point" often being omitted, the homogeneous Poisson (point) process is also called a "stationary" Poisson (point) process, as well as "uniform" Poisson (point) process. The inhomogeneous Poisson point process, as well as being called "nonhomogeneous", is also referred to as the "non-stationary" Poisson process.


title: Poisson point process

text: The term "point process" has been criticized, as the term "process" can suggest over time and space, so "random point field", resulting in the terms "Poisson random point field" or "Poisson point field" being also used. A point process is considered, and sometimes called, a random counting measure, hence the Poisson point process is also referred to as a "Poisson random measure", a term used in the study of Lévy processes, but some choose to use the two terms for Poisson points processes defined on two different underlying spaces.


title: Poisson point process

text: The underlying mathematical space of the Poisson point process is called a carrier space, or state space, though the latter term has a different meaning in the context of stochastic processes. In the context of point processes, the term "state space" can mean the space on which the point process is defined such as the real line, which corresponds to the index set or parameter set in stochastic process terminology.


title: Poisson point process

text: The measure formula_176 is called the "intensity measure", "mean measure", or "parameter measure", as there are no standard terms. If formula_176 has a derivative or density, denoted by formula_105, is called the "intensity function" of the Poisson point process. For the homogeneous Poisson point process, the derivative of the intensity measure is simply a constant formula_62, which can be referred to as the "rate", usually when the underlying space is the real line, or the "intensity". It is also called the "mean rate" or the "mean density" or "rate ". For formula_214, the corresponding process is sometimes referred to as the "standard Poisson" (point) process.


title: Poisson point process

text: The notation of the Poisson point process depends on its setting and the field it is being applied in. For example, on the real line, the Poisson process, both homogeneous or inhomogeneous, is sometimes interpreted as a counting process, and the notation formula_123 is used to represent the Poisson process.


title: Poisson point process

text: Another reason for varying notation is due to the theory of point processes, which has a couple of mathematical interpretations. For example, a simple Poisson point process may be considered as a random set, which suggests the notation formula_216, implying that formula_217 is a random point belonging to or being an element of the Poisson point process formula_2. Another, more general, interpretation is to consider a Poisson or any other point process as a random counting measure, so one can write the number of points of a Poisson point process formula_181 being found or located in some (Borel measurable) region formula_64 as formula_61, which is a random variable. These different interpretations results in notation being used from mathematical fields such as measure theory and set theory.


title: Poisson point process

text: For general point processes, sometimes a subscript on the point symbol, for example formula_217, is included so one writes (with set notation) formula_223 instead of formula_216, and formula_217 can be used for the bound variable in integral expressions such as Campbell's theorem, instead of denoting random points. Sometimes an uppercase letter denotes the point process, while a lowercase denotes a point from the process, so, for example, the point formula_217 or formula_175 belongs to or is a point of the point process formula_228, and be written with set notation as formula_229 or formula_230.


title: Poisson point process

text: Furthermore, the set theory and integral or measure theory notation can be used interchangeably. For example, for a point process formula_2 defined on the Euclidean state space formula_232 and a (measurable) function formula_233 on formula_76 , the expression


title: Poisson point process

text: demonstrates two different ways to write a summation over a point process (see also Campbell's theorem (probability)). More specifically, the integral notation on the left-hand side is interpreting the point process as a random counting measure while the sum on the right-hand side suggests a random set interpretation.


title: Poisson point process

text: In probability theory, operations are applied to random variables for different purposes. Sometimes these operations are regular expectations that produce the average or variance of a random variable. Others, such as characteristic functions (or Laplace transforms) of a random variable can be used to uniquely identify or characterize random variables and prove results like the central limit theorem. In the theory of point processes there exist analogous mathematical tools which usually exist in the forms of measures and functionals instead of moments and functions respectively.


title: Poisson point process

text: For a Poisson point process formula_2 with intensity measure formula_176, the Laplace functional is given by:


title: Poisson point process

text: The probability generating function of non-negative integer-valued random variable leads to the probability generating functional being defined analogously with respect to any non-negative bounded function formula_240 on formula_76 such that formula_242. For a point process formula_181 the probability generating functional is defined as:


title: Poisson point process

text: where the product is performed for all the points in formula_245. If the intensity measure formula_176 of formula_181 is locally finite, then the formula_248 is well-defined for any measurable function formula_249 on formula_76. For a Poisson point process with intensity measure formula_176 the generating functional is given by:


title: Poisson point process

text: For a general Poisson point process with intensity measure formula_176 the first moment measure is its intensity measure:


title: Poisson point process

text: where formula_66 is the length, area or volume (or more generally, the Lebesgue measure) of formula_64.


title: Poisson point process

text: The Mecke equation characterizes the Poisson point process. Let formula_260 be the space of all formula_261-finite measures on some general space formula_262. A point process formula_263 with intensity formula_264 on formula_262 is a Poisson point process if and only if for all measurable functions formula_266 the following holds


title: Poisson point process

text: For a general Poisson point process with intensity measure formula_176 the formula_3-th factorial moment measure is given by the expression:


title: Poisson point process

text: where formula_176 is the intensity measure or first moment measure of formula_181, which for some Borel set formula_64 is given by


title: Poisson point process

text: where formula_277 is the length, area, or volume (or more generally, the Lebesgue measure) of formula_71. Furthermore, the formula_3-th factorial moment density is:


title: Poisson point process

text: The avoidance function or void probability formula_240 of a point process formula_181 is defined in relation to some set formula_64, which is a subset of the underlying space formula_76, as the probability of no points of formula_181 existing in formula_64. More precisely, for a test set formula_64, the avoidance function is given by:


title: Poisson point process

text: For a general Poisson point process formula_181 with intensity measure formula_176, its avoidance function is given by:


title: Poisson point process

text: Simple point processes are completely characterized by their void probabilities. In other words, complete information of a simple point process is captured entirely in its void probabilities, and two simple point processes have the same void probabilities if and if only if they are the same point processes. The case for Poisson process is sometimes known as Rényi's theorem, which is named after Alfréd Rényi who discovered the result for the case of a homogeneous point process in one-dimension.


title: Poisson point process

text: In one form, the Rényi's theorem says for a diffuse (or non-atomic) Radon measure formula_176 on formula_76 and a set formula_294 is a finite union of rectangles (so not Borel) that if formula_2 is a countable subset of formula_76 such that:


title: Poisson point process

text: Mathematical operations can be performed on point processes to get new point processes and develop new mathematical models for the locations of certain objects. One example of an operation is known as thinning which entails deleting or removing the points of some point process according to a rule, creating a new process with the remaining points (the deleted points also form a point process).


title: Poisson point process

text: For the Poisson process, the independent formula_300-thinning operations results in another Poisson point process. More specifically, a formula_300-thinning operation applied to a Poisson point process with intensity measure formula_176 gives a point process of removed points that is also Poisson point process formula_303 with intensity measure formula_304, which for a bounded Borel set formula_64 is given by:


title: Poisson point process

text: This thinning result of the Poisson point process is sometimes known as Prekopa's theorem. Furthermore, after randomly thinning a Poisson point process, the kept or remaining points also form a Poisson point process, which has the intensity measure


title: Poisson point process

text: The two separate Poisson point processes formed respectively from the removed and kept points are stochastically independent of each other. In other words, if a region is known to contain formula_3 kept points (from the original Poisson point process), then this will have no influence on the random number of removed points in the same region. This ability to randomly create two independent Poisson point processes from one is sometimes known as "splitting" the Poisson point process.


title: Poisson point process

text: If there is a countable collection of point processes formula_309, then their superposition, or, in set theory language, their union, which is


title: Poisson point process

text: also forms a point process. In other words, any points located in any of the point processes formula_311 will also be located in the superposition of these point processes formula_181.


title: Poisson point process

text: The superposition theorem of the Poisson point process says that the superposition of independent Poisson point processes formula_311 with mean measures formula_314 will also be a Poisson point process with mean measure


title: Poisson point process

text: In other words, the union of two (or countably more) Poisson processes is another Poisson process. If a point formula_136 is sampled from a countable formula_21 union of Poisson processes, then the probability that the point formula_217 belongs to the formula_319th Poisson process formula_320 is given by:


title: Poisson point process

text: For two homogeneous Poisson processes with intensities formula_322, the two previous expressions reduce to


title: Poisson point process

text: The operation clustering is performed when each point formula_217 of some point process formula_181 is replaced by another (possibly different) point process. If the original process formula_181 is a Poisson point process, then the resulting process formula_328 is called a Poisson cluster point process.


title: Poisson point process

text: A mathematical model may require randomly moving points of a point process to other locations on the underlying mathematical space, which gives rise to a point process operation known as displacement or translation. The Poisson point process has been used to model, for example, the movement of plants between generations, owing to the displacement theorem, which loosely says that the random independent displacement of points of a Poisson point process (on the same underlying space) forms another Poisson point process.


title: Poisson point process

text: One version of the displacement theorem involves a Poisson point process formula_181 on formula_76 with intensity function formula_105. It is then assumed the points of formula_181 are randomly displaced somewhere else in formula_76 so that each point's displacement is independent and that the displacement of a point formerly at formula_217 is a random vector with a probability density formula_335. Then the new point process formula_336 is also a Poisson point process with intensity function


title: Poisson point process

text: If the Poisson process is homogeneous with formula_338 and if formula_339 is a function of formula_340, then


title: Poisson point process

text: In other words, after each random and independent displacement of points, the original Poisson point process still exists.


title: Poisson point process

text: The displacement theorem can be extended such that the Poisson points are randomly displaced from one Euclidean space formula_76 to another Euclidean space formula_343, where formula_344 is not necessarily equal to formula_82.


title: Poisson point process

text: Another property that is considered useful is the ability to map a Poisson point process from one underlying space to another space.


title: Poisson point process

text: If the mapping (or transformation) adheres to some conditions, then the resulting mapped (or transformed) collection of points also form a Poisson point process, and this result is sometimes referred to as the mapping theorem. The theorem involves some Poisson point process with mean measure formula_176 on some underlying space. If the locations of the points are mapped (that is, the point process is transformed) according to some function to another underlying space, then the resulting point process is also a Poisson point process but with a different mean measure formula_347.


title: Poisson point process

text: More specifically, one can consider a (Borel measurable) function formula_233 that maps a point process formula_181 with intensity measure formula_176 from one space formula_351, to another space formula_352 in such a manner so that the new point process formula_353 has the intensity measure:


title: Poisson point process

text: with no atoms, where formula_64 is a Borel set and formula_356 denotes the inverse of the function formula_233. If formula_181 is a Poisson point process, then the new process formula_353 is also a Poisson point process with the intensity measure formula_347.


title: Poisson point process

text: The tractability of the Poisson process means that sometimes it is convenient to approximate a non-Poisson point process with a Poisson one. The overall aim is to approximate both the number of points of some point process and the location of each point by a Poisson point process. There a number of methods that can be used to justify, informally or rigorously, approximating the occurrence of random events or phenomena with suitable Poisson point processes. The more rigorous methods involve deriving upper bounds on the probability metrics between the Poisson and non-Poisson point processes, while other methods can be justified by less formal heuristics.


title: Poisson point process

text: One method for approximating random events or phenomena with Poisson processes is called the clumping heuristic. The general heuristic or principle involves using the Poisson point process (or Poisson distribution) to approximate events, which are considered rare or unlikely, of some stochastic process. In some cases these rare events are close to being independent, hence a Poisson point process can be used. When the events are not independent, but tend to occur in clusters or "clumps", then if these clumps are suitably defined such that they are approximately independent of each other, then the number of clumps occurring will be close to a Poisson random variable and the locations of the clumps will be close to a Poisson process.


title: Poisson point process

text: Stein's method is a mathematical technique originally developed for approximating random variables such as Gaussian and Poisson variables, which has also been applied to point processes. Stein's method can be used to derive upper bounds on probability metrics, which give way to quantify how different two random mathematical objects vary stochastically. Upperbounds on probability metrics such as total variation and Wasserstein distance have been derived.


title: Poisson point process

text: Researchers have applied Stein's method to Poisson point processes in a number of ways, such as using Palm calculus. Techniques based on Stein's method have been developed to factor into the upper bounds the effects of certain point process operations such as thinning and superposition. Stein's method has also been used to derive upper bounds on metrics of Poisson and other processes such as the Cox point process, which is a Poisson process with a random intensity measure.