Kropf, Sara - Variance and Covariance of Several Simultaneous Outputs of a Markov Chain

dmtcs:1341 - Discrete Mathematics & Theoretical Computer Science, June 23, 2016, Vol. 18 no. 3
Variance and Covariance of Several Simultaneous Outputs of a Markov Chain

Authors: Kropf, Sara

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.

Volume: Vol. 18 no. 3
Section: Analysis of Algorithms
Published on: June 23, 2016
Submitted on: June 23, 2016
Keywords: Mathematics - Combinatorics,Mathematics - Probability,60C05, 60J10, 05C20, 05C30, 68Q45, 60F05, 05A16