Exponential forgetting of smoothing distributions for pairwise Markov models

We consider a bivariate Markov chain Z={Zk}k≥1={(Xk,Yk)}k≥1 taking values on product space Z=X×Y, where X is possibly uncountable and Y={1,…,|Y|} finite state-space. The purpose of the paper to find sufficient conditions that guarantee exponential convergence smoothing, filtering predictive probabilities: supn≥t‖P(Yt:∞∈⋅|Xl:n)−P(Yt:∞∈⋅|Xs:n)‖TV≤Ksαt,a.s. Here t≥s≥l≥1, Ks σ(Xs:∞)-measurable random variable α∈(0,1) fixed. In second part paper, we establish two-sided versions above-mentioned convergence. show desired convergences hold under fairly general conditions. A special case very model popular hidden (HMM). prove in HMM-case, our assumptions are more than all similar mixing-type encountered practice, yet relatively easy verify.