Moderate deviations and local limit theorems for the coefficients of random walks on the general linear group (Q2689893)

Consider the random walk \(G_n=g_1\cdots g_n\), where \(g_1,\ldots,g_n\) are independent and identically distributed elements with law \(\mu\) of the general linear group GL(\(V\)), with \(V=\mathbb{R}^d\). The authors investigate limiting results for the coefficients \(\langle f,G_nv\rangle\) as \(n\to\infty\), where \(v\in V\) and \(f\in V^*\), the dual vector space of \(V\). In particular, they obtain Cramer-type moderate deviation expansions in the central limit theorem for these coefficients, for both the upper and lower tail probabilities. They further obtain moderate deviations in a local limit theorem for these coefficients. These results are established under exponential moment, strong irreducibility and proximality conditions on \(\mu\).