**Sankhya:
The Indian Journal of Statistics**

2007, Volume 69, Pt. 2, 289--303

**Convergence of Convolution Powers of a Probability Measure on $d\times d$ Stochastic Matrices and a Cyclicity Condition**

Edgardo Cureg, University of South Florida, USA

Arunava Mukherjea, University of Texas -- Pan American, Edinburg, USA

SUMMARY. This paper is concerned with a very interesting cyclicity condition introduced by Chakraborty and Rao (1998). The results show that if $\mu$ is a probability measure on $3\times 3$ stochastic matrices, and the minimal rank $r$ of the matrices in the closed semigroup $S$ generated by $S_\mu,$ the support of $\mu,$ is $2,$ then the sequence $(\mu^n)$ of convolution powers of $\mu$ does {\em not} converge weakly if and only if $S_\mu$ is cyclic. Here we extend this result to any $d>3.$ Moreover, we show that when the minimal rank $r$ above is not $2,$ this result does not always hold.

*AMS (2000) subject classification. *Primary 60B10, 43A05; secondary 15A52.

*Key words and phrases. *Convolution sequence, weak convergence, probability
measure, semigroups, stochastic matrices, cyclic support.