Ergodicity and stability of the conditional distributions of nondegenerate Markov chains

Details Ergodicity and stability of the conditional distributions of nondegenerate Markov chains Ergodicity and stability of the conditional distributions of nondegenerate Markov chains

2011-01-10

arxiv.org/abs/1101.1822v1

Mathematics

Probability

46 pages

Scientific paper

We consider a bivariate stationary Markov chain (X_n,Y_n) in a Polish state space, where only the process (Y_n) is presumed to be observable. The goal of this paper is to investigate the ergodic theory and stability properties of the measure-valued process (\Pi_n), where \Pi_n is the conditional distribution of X_n given Y_0,...,Y_n. We show that the ergodic and stability properties of (\Pi_n) are inherited from the ergodicity of the unobserved process (X_n) provided that the Markov chain (X_n,Y_n) is nondegenerate, that is, its transition kernel is equivalent to the product of independent transition kernels. Our main results generalize, subsume and in some cases correct previous results on the ergodic theory of nonlinear filters.

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