Exponential forgetting of smoothing distributions for pairwise Markov models

Abstract

We consider a bivariate Markov chain $Z={\{Z_k\}}_{k\ge 1}={\{(X_k,Y_k)\}}_{k\ge 1}$ taking values on product space $\mathcal{Z}=\mathcal{X}\times \mathcal{Y}$, where $\mathcal{X}$ is possibly uncountable space and $\mathcal{Y}=\{1,\dots ,|\mathcal{Y}|\}$ 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:

$$\underset{n\ge t}{sup}\Vert P(Y_{t:\mathrm{\infty }}\in \cdot |X_{l:n})-P(Y_{t:\mathrm{\infty }}\in \cdot |X_{s:n}){\Vert }_{\mathrm{TV}}\le K_s{\mathit{\alpha }}^t,\text{a.s.}$$

Here $t\ge s\ge l\ge 1$, $K_s$ is $\mathit{\sigma }(X_{s:\mathrm{\infty }})$-measurable finite random variable and $\mathit{\alpha }\in (0,1)$ 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.