Let An=εn⋯ε1, where (εn)n≥1 is a sequence of independent random matrices taking values in GLd(R), d≥2, with common distribution μ. In this paper,
under standard assumptions on μ (strong irreducibility and proximality), we
prove Berry-Esseen type theorems for log(∥An∥) when μ has a
polynomial moment. More precisely, we get the rate ((logn)/n)q/2−1
when μ has a moment of order q∈]2,3] and the rate 1/n when
μ has a moment of order 4, which significantly improves earlier results
in this setting