The famous Oseledets theorem states that if gn is a station-
ary sequence of m × m matrices, then with probability 1 there is a (random) basis in R m such that for any vector x the asymptotic behaviour of ||gn . . . g1 x|| is the same as that for one of the vectors from this basis. The fact that the sequence is stationary is crucial for the existence of such a basis. I shall consider the product of non-identically distributed independent matrices and will explain under what conditions one can prove the existence of distinct Lyapunov exponents as well as the Oseledetss dichotomy (or rather multihotomy) of the space.