We consider a bivariate Markov chain taking values on product space , where is possibly uncountable space and is a finite state-space. The purpose of the paper is to find sufficient conditions that guarantee the exponential convergence of smoothing, filtering and predictive probabilities:
Here , is -measurable finite random variable and is fixed. In the second part of the paper, we establish two-sided versions of the above-mentioned convergence. We show that the desired convergences hold under fairly general conditions. A special case of above-mentioned very general model is popular hidden Markov model (HMM). We prove that in HMM-case, our assumptions are more general than all similar mixing-type of conditions encountered in practice, yet relatively easy to verify.
The research is supported by Estonian institutional research funding IUT34-5 and PRG 865.
"Exponential forgetting of smoothing distributions for pairwise Markov models." Electron. J. Probab. 26 1 - 30, 2021. https://doi.org/10.1214/21-EJP628