Limit distributions of random walks on stochastic matrices

Authors

Santanu Chakraborty, The University of Texas Rio Grande ValleyFollow
Arunava Mukherjea, University of South Florida

Document Type

Article

Publication Date

11-2014

Abstract

Problems similar to Ann. Prob. 22 (1994) 424–430 and J. Appl. Prob. 23 (1986) 1019–1024 are considered here. The limit distribution of the sequence XnXn−1 ··· X1, where (Xn)n≥1 is a sequence of i.i.d. 2 × 2 stochastic matrices with each Xn distributed as μ, is identified here in a number of discrete situations. A general method is presented and it covers the cases when the random components Cn and Dn (not necessarily independent), (Cn, Dn) being the first column of Xn, have the same (or different) Bernoulli distributions. Thus (Cn, Dn) is valued in {0, r}2, where r is a positive real number. If for a given positive real r, with 0 < r ≤ 1 2 , r−1Cn and r−1Dn are each Bernoulli with parameters p1 and p2 respectively, 0 < p1, p2 < 1 (which means Cn ∼ p1δ{r} + (1 − p1)δ{0} and Dn ∼ p2δ{r} + (1 − p2)δ{0}), then it is well known that the weak limit λ of the sequence μn exists whose support is contained in the set of all 2 × 2 rank one stochastic matrices. We show that S(λ), the support of λ, consists of the end points of a countable number of disjoint open intervals and we have calculated the λ-measure of each such point. To the best of our knowledge, these results are new.

Comments

Original published version available at https://doi.org/10.1007/s12044-014-0199-y.

Recommended Citation

CHAKRABORTY, S., MUKHERJEA, A. Limit distributions of random walks on stochastic matrices. Proc Math Sci 124, 603–612 (2014). https://doi.org/10.1007/s12044-014-0199-y

First Page

603

Last Page

612

Publication Title

Proceedings Mathematical Sciences

DOI

10.1007/s12044-014-0199-y