The classical Furstenberg Theorem states that the norm of a product of random i.i.d. matrices (under very mild assumptions) grows exponentially: almost surely one has
\lim_n (1/n) \log |A_n…A_1| = \lambda > 0.
My talk will be devoted to our recent work with Anton Gorodetski, where we have considered an analogous setting with A_j being independent, but no longer identically distributed. In such a setting it is natural to expect an (accordingly modified) version of the Furstenberg Theorem to hold. And indeed, we show that (again, under some mild assumptions on the distributions of A_j) there exists a deterministic sequence L_n such that

\liminf (1/n) L_n >0

 and almost surely

\lim (1/n) [\log |A_n…A_1| - L_n] = 0.

Moreover, there is an analogue of the Large Deviations Theorem.

The difficulty here is that in the nonstationary setting one cannot use the usual tools of the dynamical systems theory (stationary measure, ergodic theorem, etc.). I will discuss the proof of above results, as well as the general intuition of survival in the nonstationary world.