2011/01/31 by Xin Thomson Tong, Ramon van Handel · 10 citations
Business, Management and Accounting · Economics, Econometrics and Finance · Mathematics · #Advanced Queuing Theory Analysis #Conditional probability distribution #Ergodic theory #Ergodicity #Kernel (algebra) #Markov Chains and Monte Carlo Methods #Markov chain #Markov kernel #Markov process #Stability (learning theory) #Stationary ergodic process #Stochastic processes and financial applications #math.PR
paper · pdf · doi:10.1214/11-aap800
published in The Annals of Applied Probability 22(4) (Institute of Mathematical Statistics) · 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)
openalex publication_date 2012/08/01 · arxiv created 2012/08/21 · arxiv updated 2012/08/22 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06
We consider a bivariate stationary Markov chain (Xn,Yn)n≥0 in a Polish state space, where only the process (Yn)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 (Πn)n≥0, where Πn is the conditional distribution of Xn given Y0,…,Yn. We show that the ergodic and stability properties of (Πn)n≥0 are inherited from the ergodicity of the unobserved process (Xn)n≥0 provided that the Markov chain (Xn,Yn)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.