A kind of asymptotic properties of moving averages for Markov chains in Markovian environments

Wang Zhong-Zhi

Communications in Statistics - Theory and Methods, 2017, vol. 46, issue 22, 10926-10940

Abstract: Consider a Markov chain with finite alphabets. In this paper, we study the asymptotic properties of moving average, harmonic mean, and strong deviation theorems (limit theorems expressed by inequalities) of moving geometric average of random transition probabilities and the generalized entropy ergodic theorem for Markov chains in single infinite Markovian environments. It is shown that, under some mild conditions, the sequence of the generalized relative entropy density fan,bn(ω)$f_{a_n,b_n}(\omega)$ converges almost surely and in L1$\mathcal {L}_1$. The trick of the proofs is the construction of random variables with a parameter and the application of Borel–Cantelli lemma.

Date: 2017
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DOI: 10.1080/03610926.2016.1252404

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