BERRY-ESSEEN BOUNDS AND MODERATE DEVIATIONS FOR THE RANDOM WALK ON $GL_d (R)$

Overview & Synopsis

Let $(g_n)$ be a sequence of independent and identically distributed random real $d \times d$ matrices with law $\mu$. Consider the random walk $G_n := g_n... g_1$ on the general linear group $GL_d (R)$ and the Markov chain $ X^x_n := G_n^x/|G_n^x|$, where $x$ is a starting point on the unit sphere $S^{d−1}$. Denote respectively by $\|G_n\|$ and $\rho(G_n)$ the operator norm and the spectral radius of $G_n$. For $\log \|G_n\|$ and $\log \rho(Gn)$, we prove moderate deviation principles under exponential moment and strong irreducibility conditions on $\mu$; we also establish moderate deviation expansions in the normal range $[0, o(n^{1/6})]$ and Berry-Esseen bounds under the additional proximality condition on $\mu$. Similar results are found for the couples $(X^x_n, \log\|G_n\|)$ and $(X^x_n, \log \rho(G_n))$ with target functions.