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2021 Exponential forgetting of smoothing distributions for pairwise Markov models
Jüri Lember, Joonas Sova
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Electron. J. Probab. 26: 1-30 (2021). DOI: 10.1214/21-EJP628


We consider a bivariate Markov chain Z={Zk}k1={(Xk,Yk)}k1 taking values on product space Z=X×Y, where X is possibly uncountable space and Y={1,,|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:


Here tsl1, Ks is σ(Xs:)-measurable finite random variable and α(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.

Funding Statement

The research is supported by Estonian institutional research funding IUT34-5 and PRG 865.


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Jüri Lember. Joonas Sova. "Exponential forgetting of smoothing distributions for pairwise Markov models." Electron. J. Probab. 26 1 - 30, 2021.


Received: 27 February 2019; Accepted: 11 April 2021; Published: 2021
First available in Project Euclid: 25 May 2021

Digital Object Identifier: 10.1214/21-EJP628

Primary: 60J05 , 60J55

Keywords: Hidden Markov models , Markov models , smoothing probabilities


Vol.26 • 2021
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