Limit theorems for products of positive random matrices

• 成果类型: Article
• 署名作者: Hennion, H
• 署名单位: Universite de Rennes
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1023481103
• 发表日期: 1997
• 页码: 1545-1587
• 关键词: convergence; exponent

摘要

Let S be the set of q x q matrices with positive entries, such that each column and each row contains a strictly positive element, and denote by S degrees the subset of these matrices, all entries of which are strictly positive. Consider a random ergodic sequence (X-n)(n greater than or equal to 1) in S. The aim of this paper is to describe the asymptotic behavior of the random products X-(n) = X-n... X-1, n greater than or equal to 1, under the main hypothesis P(U)(n greater than or equal to 1)[X-(n) is an element of S degrees]) > 0. We first study the behavior in direction of row and column vectors of X-(n). Then, adding a moment condition, we prove a law of large numbers for the entries and lengths of these Vectors and also for the spectral radius of X-(n). Under the mixing hypotheses that are usual in the case of sums of real random variables, we get a central limit theorem for the previous quantities. The variance of the Gaussian limit law is strictly positive except when (X-(n))(n greater than or equal to 1) is tight. This tightness property is fully studied when the X-n, n greater than or equal to 1, are independent.