associated with N (or any other large number of samples) independent random samples X k i ( x k − − 1 ) , i = 1 , ⋯ ⋯ , N with the conditional distribution of the random state X k given X k − − 1 = x k − − 1 . The consistency of the resulting particle filter of this approximation and other extensions are developed in. In the above display δ δ a stands for the Dirac measure at a given state a.