Products of random matrices: Dimension and growth in norm

Suppose that $X_1,\...,X_n,\...$ are i.i.d. rotationally invariant $N$-by-$N$ matrices. Let $Π_n=X_n\... X_1$. It is known that $n^{-1}\log |Π_n|$ 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.

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