Abstract
This paper presents a general strong limit theorem for delayed sum of functions of random variables for a hidden time inhomogeneous Markov chain (HTIMC), and as corollaries, some strong laws of large numbers for HTIMC are established thereby.
Highlights
Hidden Markov chain is an important branch of Markov chain theory
Many new theories were introduced into hidden time inhomogeneous Markov chain (HTIMC) theory
Delayed sums of random variables were first discussed by Zygmund [3]
Summary
Hidden Markov chain is an important branch of Markov chain theory. A classical hidden Markov model was first introduced by Baum and Petrie [1]. Many new theories were introduced into hidden time inhomogeneous Markov chain (HTIMC) theory. G.Q. Yang et al [2] gave a law of large numbers for countable hidden time inhomogeneous Markov models. Wang and Yang [5] studied the generalized entropy ergodic theorem with a.e. and L1 convergence for time inhomogeneous Markov chains. We often encounter time series that appear to be “locally stationary”, so we can take an average of what has happened in some window of the recent past Based on this idea and the above researches, the main focus of this paper is to obtain a general strong limit theorem of delayed sums of functions of random variables for an HTIMC, and as corollaries, some strong laws of large numbers for HTIMC are established thereby.
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