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Régularité du plus grand exposant caractéristique des produits de matrices aléatoires indépendantes et applications. (Regularity of the largest characteristic exponent of products of independent random matrices and applications). (French) Zbl 0679.60010
Regularity properties (Hölder, \(C^{\infty})\) for the first characteristic exponent of a product of independent random matrices are given. Let \(\Omega =X^{{\mathbb{N}}}\) be a product space with a product probability P. Consider a map \((\lambda,\omega)\to g^{\lambda}(\omega)\) from \({\mathbb{R}}\times \Omega\) into the group GL(d,\({\mathbb{R}})\) of \(d\times d\) invertible matrices. By \(\mu_{\lambda}\) we denote the distribution law of \(g^{\lambda}\). Suppose that \(\{g_ k^{\lambda}(\cdot)\), \(k\geq 1\}\) is a sequence of independent random matrices having the common law \(\mu_{\lambda}\) and \(\gamma\) (\(\lambda)\) the first characteristic exponent of that sequence, i.e. \[ \gamma (\lambda)=\lim_{n\to \infty}n^{-1}\log \| g_ n^{\lambda}g^{\lambda}_{n-1}...g_ 1^{\lambda}\| \quad P\quad p.s. \] Regularity properties of the map \(\lambda\) \(\to \gamma (\lambda)\) are investigated. Theorem 1 gives conditions under which \(\gamma\) (\(\lambda)\) is a Hölder function. One of the assumption requires that for \(\lambda\),\(\mu\in T\) (a compact subset of an open set \(U\subset {\mathbb{R}})\) \[ \| g^{\lambda}(\omega)- g^{\mu}(\omega)\| \leq C(\omega)| \lambda -\mu |^{\epsilon_ 1(T)}\quad P\quad p.s. \] and \[ \int C^{\epsilon_ 1(T)}(\omega)dP(\omega)<\infty. \] Theorem 2 gives conditions under which \(\lambda\) \(\to \gamma (\lambda)\) is C on U. The properties for the integrated density in the one-dimensional Anderson model are obtained in Theorem 2.
Reviewer: D.Szynal

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60F99 Limit theorems in probability theory
60G50 Sums of independent random variables; random walks
60K99 Special processes
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