Large deviation expansions for the coefficients of random walks on the general linear group

Abstract

Consider ${({\mathit{g}}_{\mathit{n}})}_{\mathit{n}\ge 1}$ a sequence of independent and identically distributed random matrices and the left random walk ${\mathit{G}}_{\mathit{n}}:={\mathit{g}}_{\mathit{n}}\dots {\mathit{g}}_1$, $\mathit{n}\ge 1$ on the general linear group $GL(\mathit{d},\mathbb{R})$. Under suitable conditions we establish a Bahadur–Rao–Petrov type large deviation expansion for the coefficient $⟨\mathit{f},{\mathit{G}}_{\mathit{n}}\mathit{v}⟩$ of the product ${\mathit{G}}_{\mathit{n}}$, where $\mathit{v}\in {\mathbb{R}}^{\mathit{d}}$ and $\mathit{f}\in {({\mathbb{R}}^{\mathit{d}})}^{\ast }$. In particular, we obtain an explicit rate function in the large deviation principle, thus improving significantly the known large deviation bounds. A local limit theorem with large deviations for the coefficients and large deviation expansions under the change of probability measure are also proved.