Precise large deviation asymptotics for products of random matrices
Résumé
Let $(g_{n})_{n\geq 1}$ be a sequence of independent identically distributed $d\times d$ real random matrices
with Lyapunov exponent $\lambda$. For any starting point $x$ on the unit sphere in $\mathbb R^d$,
we deal with the norm $ | G_n x | $, where $G_{n}:=g_{n} \ldots g_{1}$. The goal of this paper is to establish precise asymptotics for large deviation probabilities $\mathbb P(\log | G_n x | \geq n(q+l))$, where $q>\lambda$ is fixed and $l$ is vanishing as $n\to \infty$. We study both invertible matrices and positive matrices and give analogous results for the couple $(X_n^x,\log | G_n x |)$ with target functions, where $X_n^x= G_n x /| G_n x |$. As applications we improve previous results on the large deviation principle for the matrix norm $\|G_n\|$ and obtain a precise local limit theorem with large deviations.
Domaines
Probabilités [math.PR]
Origine : Fichiers produits par l'(les) auteur(s)
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