Products of random matrices: Dimension and growth in norm

Abstract

Suppose that $X_1,\dot{.}.,X_n,\dot{.}.$ are i.i.d. rotationally invariant $N$-by-$N$ matrices. Let ${\Pi }_n=X_n\dot{.}.X_1$. It is known that $n^{-1}\log |{\Pi }_n|$ converges to a nonrandom limit. We prove that under certain additional assumptions on matrices $X_i$ the speed of convergence to this limit does not decrease when the size of matrices, $N$, grows.

Published in at http://dx.doi.org/10.1214/09-AAP658 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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