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

Abstract

Let $(g_n{)}_{n\ge 1}$ be a sequence of independent and identically distributed elements of the general linear group $GL(d,\mathbb{R})$. Consider the random walk $G_n:=g_n\dots g_1$. Under suitable conditions, we establish Bahadur-Rao-Petrov type large deviation expansion for the coefficients $⟨f,G_{n}v⟩$, where $f\in ({\mathbb{R}}^d{)}^{\ast }$ and $v\in {\mathbb{R}}^d$. In particular, our result implies the large deviation principle with an explicit rate function, thus improving significantly the large deviation bounds established earlier. Moreover, we establish Bahadur-Rao-Petrov type large deviation expansion for the coefficients $⟨f,G_{n}v⟩$ under the changed measure. Toward this end we prove the Hölder regularity of the stationary measure corresponding to the Markov chain $G_{n}v/|G_{n}v|$ under the changed measure, which is of independent interest. In addition, we also prove local limit theorems with large deviations for the coefficients of $G_n$.

References in corpus (1)

• Random products of matrices: a dynamical point of view