Conditioned limit theorems for products of random matrices

• 成果类型: Article
• 署名作者: Grama, Ion; Le Page, Emile; Peigne, Marc
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-016-0719-z
• 发表日期: 2017
• 页码: 601-639
• 关键词: ordered random-walks; asymptotic-behavior; 1st-passage times; markov-chains; CONVERGENCE; distributions

摘要

Let g1, g2,... be i. i. d. random matrices in GL (d, R). For any n >= 1 consider the product G(n) = g(n) ... g(1) and the random process G(n)v = g(n) ... g(1)v in R-d starting at point v is an element of R-d \ {0}. It is well known that under appropriate assumptions, the sequence (log ||G(n)v||) (n >= 1) behaves like a sum of i. i. d. r. v.' s and satisfies standard classical properties such as the law of large numbers, the law of iterated logarithm and the central limit theorem. For any vector v with ||v|| > 1 denote by tau(v) the first time when the random process G(n)v enters the closed unit ball in R-d. We establish the asymptotic as n -> + infinity of the probability of the event {tau(v) > n} and find the limit law for the quantity 1/root n log ||G(n)v|| conditioned that tau(v) > n.