Limit theorems for a strongly irreducible product of independent random matrices under optimal moment assumptions (2402.05751v2)

Abstract: Let $\nu$ be a probability distribution over the semi-group of square matrices of size $d\ge2$. We assume that $\nu$ is proximal, strongly irreducible and that $\nu^{*n}{0}=0$ for all $n$. We consider the sequence $\overline\gamma_n:=\gamma_0\cdots\gamma_{n-1}$ for $(\gamma_k){k}$ independent of distribution law $\nu$. Let $s_1\ge s_2\ge\dots s_d$ be the singular values given by the Cartan projection. We show that $(\log\frac{s_1}{s_2}(\overline{\gamma}_n)){n}$, escapes to infinity linearly and satisfies exponential large deviations inequalities below its escape rate. This is an extension the simplicity of the top Lyapunov exponent result, proven by Guivarc'h and Raugi under a first moment assumption. We also show that the image of a generic line by $\overline{\gamma}n$ as well as its eigenspace of maximal eigenvalue both converge to the same random line $l^\infty$ at an exponential speed. If we moreover assume that $\nu$ is supported on the group of invertible matrices and that the push-forward distribution $N*\nu$ is $\mathrm{L}^p$ for $N:g\mapsto\log|g||g^{-1}|$ and for some $p>0$, then we show that $-\log d(l^\infty,H)$ is uniformly $\mathrm{L}^p$ for all proper subspace $H\subset\mathbb{R}^d$. For $p=1$, we moreover show that the rescaled logarithm of each coefficient of $\overline{\gamma}_n$ almost surely converges to the top Lyapunov exponent. This is an extension of results by Benoist and Quint which were themselves improvements of historic results by Guivarc'h, Lepage and Raugi, first proven under exponential moment assumptions. To prove these results, we do not rely on the existence of the stationary measure nor on the existence of the Lyapunov exponent. 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 decomposition. We moreover have an explicit control over the moments.