Variance and Covariance of Several Simultaneous Outputs of a Markov Chain

TL;DR

This paper characterizes when the partial sums and joint outputs of Markov chains have unbounded variance and covariance, providing combinatorial criteria and illustrating with examples like binary sequences and non-adjacent forms.

Contribution

It offers a combinatorial characterization of unbounded variance and covariance in Markov sources, extending to higher dimensions and providing practical examples.

Findings

Unbounded variance occurs under specific cycle conditions in the Markov chain graph.

Unbounded covariance implies dependence between coordinates; bounded covariance implies asymptotic independence.

Examples include block counts in sequences and Hamming weights in non-adjacent forms.

Abstract

The partial sum of the states of a Markov chain or more generally a Markov source is asymptotically normally distributed under suitable conditions. One of these conditions is that the variance is unbounded. A simple combinatorial characterization of Markov sources which satisfy this condition is given in terms of cycles of the underlying graph of the Markov chain. Also Markov sources with higher dimensional alphabets are considered. Furthermore, the case of an unbounded covariance between two coordinates of the Markov source is combinatorically characterized. If the covariance is bounded, then the two coordinates are asymptotically independent. The results are illustrated by several examples, like the number of specific blocks in $0$-$1$-sequences and the Hamming weight of the width-$w$ non-adjacent form.