Ergodicity and stability of the conditional distributions of nondegenerate Markov chains
Abstract
We consider a bivariate stationary Markov chain $(X_n,Y_n)_{n\ge0}$ in a Polish state space, where only the process $(Y_n)_{n\ge0}$ is presumed to be observable. The goal of this paper is to investigate the ergodic theory and stability properties of the measure-valued process $(\Pi_n)_{n\ge0}$, where $\Pi_n$ is the conditional distribution of $X_n$ given $Y_0,...,Y_n$. We show that the ergodic and stability properties of $(\Pi_n)_{n\ge0}$ are inherited from the ergodicity of the unobserved process $(X_n)_{n\ge0}$ provided that the Markov chain $(X_n,Y_n)_{n\ge0}$ 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.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2011
- DOI:
- 10.48550/arXiv.1101.1822
- arXiv:
- arXiv:1101.1822
- Bibcode:
- 2011arXiv1101.1822T
- Keywords:
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- Mathematics - Probability
- E-Print:
- Published in at http://dx.doi.org/10.1214/11-AAP800 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)