Probability and Mathematical Statistics: Vol. 20, Fasc. 2

DISCRETE PROBABILITY MEASURES ON $2× 2$ STOCHASTIC MATRICES AND A FUNCTIONAL EQUATION ON $[0,1]$

A. Mukherjea
J. S. Ratti

Abstract: In this paper, we consider the following natural problem: suppose $m 1$ and $m 2$ are two probability measures with finite supports $S(m ),S(m ) 1 2$ respectively, such that $|S(m )|= |S(m )| 1 2$ and $S(m ) U S(m ) < 2× 2 1 2$ stochastic matrices, and $mn 1$ (the $n$ -th convolution power of $m 1$ under matrix multiplication), as well as $mn, 2$ converges weakly to the same probability measure $c,$ where $S(c) < 2× 2$ stochastic matrices with rank one. Then when does it follow that $m = m ? 1 2$ What if $S(m ) = S(m )? 1 2$ In other words, can two different random walks, in this context, have the same invariant probability measure? Here, we consider related problems.

1991 AMS Mathematics Subject Classification: Primary: -; Secondary: -;

Key words and phrases: -