ERGODICITY AND STABILITY OF THE CONDITIONAL DISTRIBUTIONS OF NONDEGENERATE MARKOV CHAINS

作者：Tong Xin Thomson^*; van Handel Ramon

来源：Annals of Applied Probability, 2012, 22(4): 1495-1540.

DOI：10.1214/11-AAP800

• Nonlinear filtering
• unique ergodicity
• asymptotic stability
• nondegenerate Markov chains
• exchange of intersection and supremum
• Markov chain in random environment

We consider a bivariate stationary Markov chain (X-n, Y-n)(n >= 0) in a Polish state space, where only the process (Y-n)(n >= 0) 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 >= 0), 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 >= 0) are inherited from the ergodicity of the unobserved process (X-n)(n >= 0) provided that the Markov chain (X-n, Y-n)(n >= 0) 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.

• 出版日期2012-8