Sankhya: The Indian Journal of Statistics

2007, Volume 69, Pt. 2, 304--313

Cyclicity and Weak Convergence for Convolution of Measures on Non-negative Matrices

Santanu Chakraborty, University of Texas -- Pan American, Edinburg, USA

SUMMARY. Let $\mu$ be a probability measure on Borel subsets of $d\times d$ non-negative matrices. Let $S$ be the closed (multiplicative) semigroup generated by the support of $\mu,\, S_{\mu}$. Let the minimal rank of the matrices in $S$ be $2$. Then, we obtain necessary and sufficient conditions for weak convergence of $\mu^{n}$ in terms of cyclicity of $S_{\mu}$ -- a concept first introduced in the work of Chakraborty and Rao (1998, {\it Sankhy\=a A}).

AMS (2000) subject classification. Primary 60B05, 60B10; secondary 15A51, 15A52.

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

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