Limit theorems for a strongly irreducible product of independent random matrices under optimal moment assumptions

Let be a probability distribution over the linear semi-group End (E) for E a finite dimensional vector space over a locally compact field. We assume that is proximal, strongly irreducible and that ^*n\0\=0 for all integers n. We consider the random sequence ₙ: = ₀ ₍-₁ for (ₖ) ₊ ₀ independents of distribution law. We define the logarithmic singular gap as sqz = (₁₂), where ₁ and ₂ are the two largest singular values. We show that (sqz (ₙ) ) ₍ escapes to infinity linearly and satisfies exponential large deviations estimates below its escape rate. With the same assumptions, we also show that the image of a generic line by ₙ as well as its eigenspace of maximal eigenvalue both converge to the same random line l_ at an exponential speed. If we moreover assume that the push-forward distribution N () is Lᵖ for N: g (\|g\|\|g^-1\|) and for some p 1, then we show that |w (l_) | is Lᵖ for all unitary linear form w and the logarithm of each coefficient of ₙ is almost surely equivalent to the logarithm of the norm. To prove these results, we do not rely on any classical results for random products of invertible matrices with L¹ moment assumption. Instead we describe an effective way to group the i. i. d factors into i. i. d random words that are aligned in the Cartan projection. We moreover have an explicit control over the moments.