---
abstract: 'The concept of homology, originally developed as a useful tool in algebraic topology, has by now become pervasive in quite different branches of mathematics. The notion particularly carries over quite naturally to the setup of measure-preserving transformations arising from various group actions or, equivalently, the setup of stationary sequences considered in this paper. Our main result provides a sharp criterion which determines (and rules out) when two stationary processes belong to the same *null-homology equivalence class*. We also discuss some concrete cases where the notion of null-homology turns up in a relevant manner.'
address:
- |
    G. Alsmeyer\
    Inst. Math. Stochastics,\
    Department of Mathematics\
    and Computer Science\
    University of Münster\
    Orléans-Ring 10, D-48149\
    Münster, Germany\
    \
- |
    C. Mukherjee\
    Inst. Math. Stochastics,\
    Department of Mathematics\
    and Computer Science\
    University of Münster\
    Orléans-Ring 10, D-48149\
    Münster, Germany\
author:
- 
- 
title: 'On null-homology and stationary sequences'
---

Introduction and motivation {#sec:intro}
===========================

Homology is a notion that arises in various branches of mathematics. It was originally developed in algebraic topology in order to associate a sequence of algebraic objects. A typical fundamental question is the following: When does a $n$-cycle of a (simplical) complex form the boundary of a $(n+1)$-chain, or equivalently, when is its fundamental class a boundary for the singular homology? If such a requirement is fulfilled, the cycle is said to be *homologous to $0$* or *null-homologous*. In the present article, we provide a suitable criterion for null-homology in a different context, namely measure-preserving transformations arising from natural group actions on any complete and separable metric space. To formulate the question precisely, we recall some basic definitions.

Let $\bX= (X_{n})_{n\in \Z}$ be a sequence of random variables defined on a probability space with underlying probability measure $\Prob$ and such that the $X_{n}$’s take values in a complete separable metric space $\mathscr S$. Note that $\bX$ forms a stationary stochastic process if, for all $n\in\N$ and $m\in \Z$, $$\Prob\big((X_{1},\dots, X_{n}) \in \cdot\big)\ =\ \Prob\big((X_{m+1},\dots, X_{m+n})\in\cdot\big).$$ In other words, the joint law of $(X_{1},\dots, X_{n})$ for any $n$ coincides with the law of any of its “shifts" under the action of the additive group $\Z$ on the space of doubly-infinite sequences $\mathscr S^{\Z}$. There is a natural notion of *homology*, first coined by Lalley [@L86 p. 197] in this setup, that arises from the group action. Indeed, given any stationary sequence $\bX$ and measurable functions $F, G\colon \mathscr S^\Z\to \R^{d}$, we say that $F$ is *homologous* to $G$ (with respect to $\bX$ and $\Prob$) and write $F\sim G$ if there exists a function $\xi: \mathscr S^\Z\to \R^{d}$ such that $$\label{eq:F,G homologous}
F(X_{0}) - G(X_{0})\ =\ \xi (X_{1})- \xi(X_{0})\quad\Prob\text{-a.s.}$$ Then $\sim$ is an equivalence relation, and if $F\sim 0$, thus $$\label{eq:F null-homologous}
F(X_{0})\ =\ \xi (X_{1})- \xi(X_{0})\quad\Prob\text{-a.s.},$$ we say that $F$ is *null-homologous*. Now observe that, given any stationary process $\bX$ and a null-homologous function $F$, the process $(F(X_{n}))_{n\in\Z}$ is not only also stationary but in fact the incremental sequence of another stationary process, viz. $(\xi(X_{n}))_{n\in\Z}$. In view of this, the converse question which stationary processes are of this “incremental” type and therefore allowing a representation with respect to a null-homologous function appears to be natural. The main goal of the present article is to provide a sharp criterion for this fundamental property which is of interest for various reasons as will also be explained. Indeed, mere tightness of the partial sums $S_{n}=X_{1}+\cdots+X_{n}$, $n\in\N$, associated with the stationary process $\bX$ turns out to be the necessary and sufficient condition, see Theorem \[thm:main result\]. The proof, which does not even require ergodicity, is quite simple and relies on the construction of some *commutative maps* in a proper setup and an application of Schauder’s fixed point theorem. To put our work into context, we first discuss some concrete cases where null-homology turns up in a relevant way.

Markov random walks
-------------------

In the [@L86], Lalley considered random walks with increments from a fairly general class of stationary sequences, albeit restricted to the [*integrable set up*]{}, see Remark \[rem-Lalley\]. As a main result, he proved a Blackwell-type renewal theorem for which it was necessary to rule out a certain “lattice-type" behavior which is intimately connected to the notion of null-homology. In the following, we give a brief introduction of this notion within the framework of *Markov random walks* which are also called *Markov-additive processes* and indeed comprise random walks with stationary increments as explained below. Let $(\cS,\fS)$ be an arbitrary measurable space and $\cB(\R^{m})$ the Borel $\sigma$-field on $\R^{m}$ for $m{\geqslant}1$. Suppose that $(M_{n},X_{n})_{n{\geqslant}0}$ is a *Markov-modulated sequence* of $\cS\times\R^{d}$-valued random variables, where $\cS\times\R^{d}$ is endowed with the product $\sigma$-field $\fS\otimes\cB(\R^{d})$. This means that $X_{0},X_{1},\ldots$ are conditionally independent given the *driving chain* $(M_{n})_{n{\geqslant}0}$ and $$\begin{aligned}
\Prob(X_{i}\in B_{i},\,0{\leqslant}k{\leqslant}n|M_{j}=s_{j},\,j{\geqslant}0)\ =\ P_{0}(s_{0},B_{0})\prod_{i=1}^{n}P((s_{i-1},s_{i}),B_{i})\end{aligned}$$ for all $n\in\N_{0}$, $s_{0},\ldots,s_{n}\in\cS$, measurable $B_{0},\ldots,B_{n}\subset\R^{d}$ and suitable kernels $P_{0}$ and $P$ which describe the conditional laws of $X_{0}$ given $M_{0}$ and of $X_{n}$ given $(M_{n-1},M_{n})$ for $n{\geqslant}1$, respectively. We make the additional assumption that $(M_{n})_{n{\geqslant}0}$ *is ergodic with unique stationary distribution $\mu$*. Defining $S_{0}:=0$ and $$S_{n}\,:=\,\sum_{i=1}^{n}X_{i},\quad n=1,2,\ldots,$$ the bivariate sequence $(M_{n},S_{n})_{n{\geqslant}0}$ and also $(S_{n})_{n{\geqslant}0}$ are called *Markov random walk (MRW)* and $(M_{n})_{n{\geqslant}0}$ its *driving* or *modulating chain*. For our purposes, it is enough to study these objects in stationary regime, that is, under $\Prob_{\mu}:=\int_{\cS}\Prob(\cdot|M_{0}=s)\,\mu(\mathrm d s)$. We may then further assume the existence of a doubly infinite stationary extension $(M_{n},X_{n})_{n\in\Z}$ with associated doubly infinite random walk $$\begin{aligned}
S_{n}\ =\ \begin{cases} \sum_{i=1}^{n}X_{i}&\text{if }n{\geqslant}1,\\ 0,&\text{if }n=0,\\ -\sum_{i=n+1}^{0}X_{i}&\text{if }n<0. \end{cases}\end{aligned}$$

In this context, both $(M_{n},S_{n})_{n\in\Z}$ and $(M_{n},X_{n})_{n\in\Z}$ are called *null-homologous* if there exists a measurable function $\xi:\cS\to\R^{d}$ such that $$\begin{gathered}
X_{n}\ =\ \xi(M_{n})-\xi(M_{n-1})\quad\Prob_{\mu}\text{-a.s.}\label{eq:null-hom X}
\shortintertext{and thus}
S_{n}\ =\ \xi(M_{n})-\xi (M_{0})\quad\Prob_{\mu}\text{-a.s.}\label{eq:null-hom S}\end{gathered}$$ for all $n\in\Z$. The reader should note that the latter implies the stationarity of $(S_{n}+\xi(M_{0}))_{n\in\Z}$ and thus the “almost stationarity” of the random walk $(S_{n})_{n\in\Z}$ itself, in particular its tightness.

Now let $(X_{n})_{n\in\Z}$ be any doubly infinite stationary sequence of $\R^{m}$-valued random variables and put $$M_{n}\ :=\ (X_{i})_{i{\leqslant}n}$$ for $n\in\Z$. Observe that $(M_{n})_{n\in\Z}$ constitutes a stationary Markov chain (ergodic iff $(X_{n})_{n\in\Z}$ is ergodic) and $(M_{n},X_{n})_{n\in\Z}$ a Markov-modulated sequence. This shows that null-homology for stationary processes may indeed be viewed as a special instance of the very same notion within the framework of Markov-modulation under stationarity.

Null-homology arises also quite naturally in connection with the lattice-type of MRW’s. As before, let $(M_{n})_{n{\geqslant}0}$ be ergodic with unique stationary law $\mu$. Following Shurenkov [@S84], the MRW $(M_{n},S_{n})_{n{\geqslant}0}$ is called $d$-arithmetic if $d$ is the maximal positive number such that $$\Prob_{\mu}\big(X_{1}\in \xi(M_{1})-\xi(M_{0})+d\Z\big)\ =\ 1$$ for a suitable function $\xi:\cS\to [0,d)$, called *shift function*. If no such $d$ exists, it is called nonarithmetic. Equivalently, $(M_{n},S_{n})_{n{\geqslant}0}$ is $d$-arithmetic if $d>0$ is the maximal number such that $(M_{n},X_{n}-X_{n}')_{n\in\Z}$ is Markov-modulated and null-homologous for a sequence of $d\Z$-valued random variables $(X_{n}')_{n\in\Z}$. Namely, with $\xi$ denoting the shift function, $$X_{n}'\,:=\,X_{n}-\xi(M_{n})+\xi(M_{n-1})$$ for $n\in\Z$.

Stochastic homogenization
-------------------------

The notion of a *corrector* plays an important rôle in the context of *stochastic homogenization* of a random media. We will describe the setup and how null-homology comes into play for a particular instance of a random walk in random environment (in the reversible setup) known as the *random conductance model*. Let $$E_d\ =\ \big\{(x,y)\colon |x-y|=1, \, x,y \in \Z^{d}\big\}$$ be the set of nearest neighbor bonds in $\Z^{d}$ and $\Omega= [a,b]^{E_d}$ for any two fixed numbers $0<a<b$. We assume that $\Omega$ is equipped with the product $\sigma$-field $\mathcal B$ and carries a probability measure $\Prob$. For simplicity, we also assume that the canonical coordinates are i.i.d. variables under $\Prob$. Note that any $x\in\Z^{d}$ acts on $(\Omega,\mathcal B,\Prob)$ as a $\Prob$-preserving and ergodic transformation $\tau_{x}$, defined as the canonical translation $$\Omega\ \ni\ \omega(\cdot)\,\mapsto\,\omega(x+\cdot).$$ For any $\omega\in \Omega$, we then have a Markov chain $(S_{n})_{n{\geqslant}0}$ on $\Z^{d}$ under a family of probability measures $(P^{\pi,\omega}_{x})_{x\in\Z^{d}}$ such that $P^{\pi,\omega}_{x}(S_{0}=x)=1$ and transition probabilities are given by $$\label{pi}
\begin{aligned}
&P_{x,\omega}(S_{n+1}=y+e|S_{n}=y)\ =\ \pi_\omega(y,y+e)\\
&\hspace{.3cm}:=\ \frac {\omega((y,y+e))}{\sum_{|e^\prime|=1} \omega((y,y+e^\prime))}\ 
=\ \pi_{\tau_{y}\omega}(0,e)
\end{aligned}$$ for any $e$ with $|e|=1$ and $x\in\Z^{d}$. Furthermore, the sequence $\widehat \omega_{n}\stackrel{\mathrm{def}}=\tau_{S_{n}}\omega$ for $n{\geqslant}0$, with initial state $\omega$ and taking values in the “environment space" $\Omega$, is also a Markov chain with transition kernel $\widehat\Pi$ defined by $$(\widehat \Pi f)(\omega)\ :=\ \sum_{|e|=1} \pi_\omega(0,e) f(\tau_{e} \omega)$$ for all bounded and measurable $f$. It is called the *environment seen from the moving particle*, or simply the *environmental process*, and particularly useful in the following scenario: Suppose there is a probability density $\phi\in L^1(\Prob)$ (i.e. $\phi{\geqslant}0$ and $\int \phi\,\mathrm d\Prob=1$) such that $\mathrm d\Q=\phi\,\mathrm d\Prob$ is $\widehat\Pi$-invariant, i.e. $$\begin{gathered}
\label{invariance}
\langle \widehat \Pi f, \phi\rangle_{L^{2}(\Prob)}=\langle  f, \phi\rangle_{L^{2}(\Prob)},
\intertext{for all bounded and measurable $f$ or, equivalently,}
L^\star\phi\,=\,0\quad\mbox{if }L=\mathrm{Id}-\widehat\Pi.\nonumber\end{gathered}$$ It can be shown, see [@PV81; @K85; @KV86] and also [@BS02 Theorem 1.2], that such an invariant density $\phi$ if it exists is necessarily unique. Moreover, $\Prob$ and $\Q$ are then equivalent measures and $(\widehat\omega_{n})_{n{\geqslant}0}$ an ergodic process in equilibrium (under initial law $\Q$).

In the random conductance model with transition probabilities , the invariant density $\phi$ can easily be found by reversibility (solving the detailed balance equations), viz. $$\phi(\omega)\ =\ \frac 1C \sum_{|e|=1} \omega((0,e)),\quad\text{where}\quad C\ =\ \int\sum_{|e|=1}\omega((0,e))\ \Prob(\mathrm d\omega).$$ Reversibility further implies that $\widehat\Pi$ is self-adjoint on $L^{2}(\Q)$, that is $$\label{reversible}
\langle f, \widehat\Pi g\rangle_{L^{2}(\Q)}\ =\ \langle\widehat\Pi f,g\rangle_{L^{2}(\Q)}$$ for all bounded and measurable functions $f,g$. Returning to the Markov chain $(S_{n})_{n{\geqslant}0}$ under $P_{0,\omega}$, the ergodicity of $(\widehat\omega_{n})_{n{\geqslant}0}$ fairly easily provides a strong law of large numbers, viz. $S_{n}/n \to 0$ $P_{0,\omega}$-a.s. for $\Prob$-almost all $\omega$. To see this, let $$\mathrm d(x,\omega)\ =\ E^{\pi,\omega}_{x}[S_{1}-S_{0}]\ =\ \sum_{|e|=1}e\pi_{\omega}(x,x+e)\ =\ \mathrm d(0,\tau_{x}\omega)$$ denote the *local drift at $x$* under $P_{0,\omega}$. As $(\widehat\omega_{n})_{n{\geqslant}0}$ is ergodic under initial law $\Q$, Birkhoff’s ergodic theorem implies $$n^{-1} \sum_{j=0}^{n-1}\mathrm d(S_{j},\omega)\ =\ n^{-1} \sum_{j=0}^{n-1} \mathrm d(0,\widehat\omega_{j})\ \xrightarrow{n\to\infty}\ \mathbb \int\mathrm d(0,\omega')\ \Q(\mathrm d\omega')\ =\ 0$$ for $\Q$-almost all and thus $\Prob$-almost all $\omega$ (as $\Prob,\Q$ are equivalent), the right-hand side being 0 by reversibility (recall ) and the definition of $\Q$. Now observe that $Z_{n}= S_{n}-S_{0}-\sum_{j=0}^{n-1} \mathrm d(S_{j},\omega)$, $n{\geqslant}0$, is a $P_{0,\omega}$-martingale with bounded (uniformly in $\omega$) increments and therefore satisfies, by the Azuma-Hoeffding inequality, $$P_{0,\omega}\big(n^{-1}Z_{n}{\geqslant}n^{-\frac{1}{2}+\eps}\big)\ {\leqslant}\ \exp(-Cn^{2\eps})$$ for any $\eps>0$ and some $C>0$ (not depending on $\omega$). Finally, by an appeal to the Borel-Cantelli lemma, we infer that $Z_{n}/n\to 0$ holds $\Prob$-a.s., and since $\sum_{j=0}^{n-1} \mathrm d(S_{j},\omega)=o(n)$ a.s., it follows that $S_{n}/n\to 0$ $\Prob$-a.s., too.

As will be explained next, stochastic homogenization comes into play when turning to the more ambitious aim of deriving an almost sure central limit theorem (or an invariance principle) for the distribution (under the quenched measure $P^{\pi,\omega}_0$) of the random walk $(S_{n})_{n{\geqslant}0}$, and it leads to the notion of a *corrector*. Note that the local drift $\mathrm d$ is bounded and therefore particularly $\in L^{2}(\Prob)$. For any fixed $\eps>0$, let $g_{\eps}\in L^{2}(\Prob)$ be a solution to the *Poisson equation* $$((1+\eps)\textrm{Id}-\widehat\Pi)g_{\eps}\ =\ \mathrm d$$ The solution is well-defined and in fact given by the Neumann series $$g_{\eps}\ =\ d\ +\ \sum_{n{\geqslant}1} \frac{\wh{\Pi}^{n}d}{(1+\eps)^{n}}.$$ Putting $G_{\eps}(\omega,e):=(\nabla_e g_{\eps})\omega)=g_{\eps}(\tau_e\omega)-g_{\eps}(\omega)$ for any $e$ with $|e|=1$, we then have the result $$G_{\eps}(\cdot,e)\circ \tau_{x}\ \xrightarrow{L^{2}(\Prob)}\ G(\cdot,e)\circ\tau_{x}$$ for any $x\in\R^{d}$, see [@KV86 Theorem 1.3], where $G$ is a (divergence free) *gradient field*, i.e., it satisfies the *closed loop condition* $$\label{closed loop condition}
\sum_{j=1}^{n}G(\tau_{x_{j}}\omega, {x_{j+1}-x_{j}})\ =\ 0\quad\text{a.s.}$$ for any closed path $x_{0}\to x_{1}\to\dots\to x_{n}=x_{0}$ in $\R^{d}$. The last property allows us to define the *corrector* corresponding to $G$ as $$\label{corrector}
V_{G}(x,\omega)\,:=\,\sum_{j=1}^{n} G(\tau_{x_{j}}\omega, {x_{j+1}-x_{j}})$$ along any path $0\to x_{1}\to\ldots\to x_{n-1}\to x_{n}=x$, the particular choice of the path being irrelevant because of . It also follows that $V_{G}$ has stationary and $L^{2}$-bounded gradient in the sense that $$\begin{gathered}
V_{G}(x,\omega)-V_{G}(y,\omega)\ =\ V_{G}(x-y,\tau_y\omega)\quad\text{for all }x,y\in\Z^{d}
\shortintertext{and}
\sup_{x\in\Z^{d}}\|V_{G}(x+e,\cdot)-V_{G}(x,\cdot)\|_{L^{2}(\Prob)}\,<\,C,\end{gathered}$$ respectively. Even more importantly, the mapping $x\mapsto V_{G}(x,\omega)+x$ is harmonic with respect to the transition probabilities for $\Prob$-almost all $\omega$. This means that $(S_{n}+V_G(S_{n},\cdot))_{n{\geqslant}0}$ is a martingale with respect to $P^{\pi,\omega}_0$ so that the corrector $V_{G}$ expresses the “distance" (or the *deformation*) of the martingale from the random walk $(S_{n})_{n{\geqslant}0}$ itself. One can show that the contribution of this deformation grows at most sub-linearly at large distances (i.e. $\sup_{|x|\leq n} n^{-1} V_{G}(x,\cdot)\xrightarrow{n\to\infty} 0$ a.s.) whence, by the martingale central limit theorem, the laws $P^{\pi,\omega}_{0}(S_{n}/\sqrt n \in \cdot)$ converge weakly to a Gaussian law for almost every $\omega$, see [@SS04; @BB07; @MP07] for a detailed recount of the substantial progress made in this direction.

In order to finally make a connection with the notion of null-homology, let us note that the result just mentioned does not rule out the possibility that the corrector grows stochastically to infinity. Namely, although the gradient of $V_{G}$ is stationary and thus tight as pointed out above, the latter property may naturally fail for $V_{G}$ itself. On the other hand, a tight corrector means that the above martingale is just a “negligible” perturbation of the random walk $(S_{n})_{n{\geqslant}0}$ itself which is a much stronger statement than the above central limit theorem. Our main result, Theorem \[thm:main result\] below, establishes, as a further information, the equivalence of this property with the null-homology of its stationary gradient, under no extra assumptions. Now, for the random conductance model in dimension $d{\geqslant}3$, the tightness of $V_{G}$ has indeed been shown to hold, see [@GO15] and [@AKM17].

The main result
===============

We proceed with a description of the setup that allows us to define null homology in terms of probability measures rather than random variables. This appears to be more convenient to state and prove our main result.

Without any loss of generality, we work with $\mathscr S= \R^{d}$ and write $\Omega=(\R^{d})^{\otimes\Z}$ for the space of doubly infinite sequences ${\bf x}=(x_{n})_{n\in\Z}$ endowed with the Borel $\sigma$-field and $T:\Omega\to\Omega$ the (left) shift operator on $\Omega$, viz. $${\bf x}\ =\ (\ldots,x_{-1},x_{0},x_{1},\ldots)\ \mapsto\ (\ldots,x_{0},x_{1},x_{2},\ldots).$$ The coordinate mappings on $\Omega$ are denoted $X_{n}$ for $n\in\Z$, and we let $S_{n}$ be the mapping ${\bf x}\mapsto s_{n}$ on $\Omega$ for $n\in\Z$, where $$\begin{aligned}
s_{n}\ =\ \begin{cases} \hfill x_{1}+\ldots+x_{n}&\text{if }n{\geqslant}1,\\
\hfill 0,&\text{if }n=0,\\
-(x_{-n+1}+\ldots+x_{0})&\text{if }n{\leqslant}-1.
\end{cases}\end{aligned}$$ So $(X_{n})_{n\in\Z}$ forms a stationary sequence with associated random walk $(S_{n})_{n\in\Z}$ under any $T$-invariant probability measure on $\Omega$.

Next, we denote by $\cM(\Omega)$ the locally convex vector space of finite signed measures on $\Omega$ endowed with the topology of weak convergence and further by $\cM_{T}(\Omega)$ its subsets of $T$-invariant probability measures. Defining the map $D:\Omega\to\Omega$ by $${\bf x}\ \mapsto\ T{\bf x}-{\bf x}\ =\ (\ldots,x_{0}-x_{-1},x_{1}-x_{0},x_{2}-x_{1},\ldots),$$ we obviously have that $\Prob\in\cM_{T}(\Omega)$ implies $\Prob D^{-1}=\Prob(D\in\cdot)\in\cM_{T}(\Omega)$. Null homology for elements of $\cM_T(\Omega)$ can now be defined as follows.

Any $T$-invariant probability measure $\Prob\in\cM_T(\Omega)$ is called null-homologous if $\Prob=\Q D^{-1}$ for some $\Q\in\cM_T(\Omega)$.

Plainly, null homology of $\Prob$ is equivalent to the null homology of $(X_{n})_{n\in\Z}$ under $\Prob$.

The subsequent Theorem \[thm:main result\] provides a characterization of this property in terms of the laws of the $S_{n}$ under $\Prob$, thus $\{\Prob S_{n}^{-1}:n\in\Z\}$.

\[thm:main result\] Given any $\Prob\in\cM_T(\Omega)$, the following assertions are equivalent:

-   $\Prob$ is null-homologous.

-   $\{\Prob S_{n}^{-1}:n{\geqslant}0\}$ is tight.

-   $\{\Prob S_{-n}^{-1}:n{\geqslant}0\}$ is tight.

\[rem-Lalley\]In [@L86 Proposition 6], Lalley provided a criterion for null-homology within the subclass of integrable stationary sequences, called *$L^{1}$-null-homology*. It enabled him to rule out a certain lattice-type behavior for the derivation of a renewal theorem for certain stationary processes. In fact, he showed that $L^{1}$-null-homology is equivalent to the $L^1$-boundedness of the partial sums of the stationary sequence, i.e., of the associated random walk. Naturally, this is a much stronger requirement than the tightness appearing in our theorem above. Also, the proof of our result, which is based on an application of the Schauder fixed-point theorem in an appropriate context, differs entirely from the arguments used in [@L86].

Note that our criterion for null-homology holds for any $T$-invariant measure $\Prob\in \cM_T(\Omega)$ and is not restricted to the the ergodic ones, i.e., extremal points of $\cM_T(\Omega)$.

In the above context of $T$-invariant probability measures, we say that $\Prob\in\cM_T(\Omega)$ as well as the coordinate sequence $(X_{n})_{n\in\Z}$ (under $\Prob$) are *$L^{p}$-null-homologous* if they are null-homologous and $\Erw|X_{0}|^{p}<\infty$. Before giving the proof of Theorem \[thm:main result\], we provide as an immediate consequence the following corollary which characterizes *$L^{p}$-null-homology* for any $p>0$ and particularly comprises Lalley’s results for $p=1$ and $p=2$.

Given any $\Prob\in\cM_T(\Omega)$, the following assertions are equivalent for any $p>0$:

-   $\Prob$ is $L^{p}$-null-homologous.

-   $(S_{n})_{n{\geqslant}0}$ is $L^{p}$-bounded.

-   $(S_{-n})_{n{\geqslant}0}$ is $L^{p}$-bounded.

Obviously, it suffices to show that (b) implies (a). To this end, we consider the bivariate mappings $$\begin{gathered}
\Lambda_{k}:\Omega\to\Omega\times\Omega,\\
{\bf x}\ \mapsto\ \big(T^{k}{\bf x},{\bf x}+\ldots+T^{k-1}{\bf x}\big)\ =\ \big((x_{n+k})_{n\in\Z},(x_{n}+\ldots+x_{n+k-1})_{n\in\Z}\big)\end{gathered}$$ for $k\in\N$ and point out that (b) entails the tightness of the family $$\cP\ =\ \{\Prob\Lambda_{k}^{-1}:k\in\N\}.$$ We can lift the shift $T$ as well as the projections $X_{n}$ in a canonical way to mappings on $\Omega\times\Omega$ and, by slight abuse of notation, may call these mappings again $T$ and $X_{n}$. The projections on the $y$-components, namely $({\bf x},{\bf y})\mapsto y_{n}$ if ${\bf y}=(y_{k})_{k\in\Z}$, are denoted $Y_{n}$ for $n\in\Z$. Then the $T$-invariance of $\Prob$ implies the very same for the elements of $\cP$.

Now let $\cD$ be the closed convex hull of all weak limit points of $\cP$ which forms a compact convex subset of $\cM(\Omega\times\Omega)$. Consider the map $$S:\Omega\times\Omega\to\Omega\times\Omega,\quad \big({\bf x},{\bf y}\big)\ \mapsto\ \big(T{\bf x},{\bf x}+{\bf y}\big)$$ which is linear, continuous, commutes with $T$, i.e. $S\circ T=T\circ S$, and satisfies further $S\circ\Lambda_{n}=\Lambda_{n+1}$, thus $\Gamma_{n}S^{-1}=\Gamma_{n+1}$ for all $n\in\N$, where $\Gamma_{n}:=\Prob\Lambda_{n}^{-1}$. Then the last property entails that the set $\cD$ is $S$-invariant which in turn, by invoking Schauder’s fixed point theorem, allows us to conclude that $S$ has a fixed point, say $\Gamma$, in $\cD$. This means that $\Gamma S^{-1}=\Gamma$ or, equivalently, that $\Gamma$ is $S$-invariant.

Finally, by considering the map $$G=(X_{0},Y_{0}):\Omega\times\Omega\to\R^{d}\times\R^{d},\quad \big({\bf x},{\bf y}\big)\ \mapsto\  (x_{0},y_{0})$$ we have that $(X_{n}',Y_{n}'):=G\circ S^{n}=(X_{n},Y_{n}\circ S^{n})$, $n{\geqslant}0$, is stationary under $\Lambda$ and satisfies:

-   $(X_{n}')_{n{\geqslant}0}=(X_{n})_{n{\geqslant}0}$ and has law $\Prob$ under $\Lambda$, because this is the case under any element of $\cD$.

-   $Y_{n+1}'=Y_{0}+X_{0}+\ldots+X_{n}=Y_{n}'+X_{n}'$, thus $$X_{n}\ =\ X_{n}'\ =\ Y_{n+1}'-Y_{n}'$$ for all $n{\geqslant}0$.

Since $(Y_{n}')_{n{\geqslant}0}$ is stationary under $\Lambda$, (a) follows.

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