Products of random matrices: Dimension and growth in norm

Suppose that $X_1,\dot{\mathrm{.}}\mathrm{.},X_n,\dot{\mathrm{.}}\mathrm{.}$ are i.i.d. rotationally invariant $N$-by-$N$ matrices. Let ${\mathrm{\Pi }}_n=X_n\dot{\mathrm{.}}\mathrm{.}{X}_1$. It is known that $n^{-1}\log \mathrm{\mid }{\mathrm{\Pi }}_n\mathrm{\mid }$ 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.