Analytic expansions of max-plus Lyapunov exponents

• 成果类型: Article
• 署名作者: Baccelli, F; Hong, D
• 署名单位: Universite PSL; Ecole Normale Superieure (ENS)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• 发表日期: 2000
• 页码: 779-827
• 关键词: random matrices; PRODUCTS

摘要

We give an explicit analytic series expansion of the (max, plus)Lyapunov exponent gamma (p) of a sequence of independent and identically distributed random matrices, generated via a Bernoulli scheme depending on a small parameter p. A key assumption is that one of the matrices has a unique normalized eigenvector. This allows us to obtain a representation of this exponent as the mean value of a certain random variable. We then use a discrete analogue of the so-called light-traffic perturbation formulas to derive the expansion. We show that it is analytic under a simple condition on p. This also provides a closed form expression for all derivatives of gamma (p) at p = 0 and approximations of gamma (p) of any order, together with an error estimate for finite order Taylor approximations. Several extensions of this are discussed, including expansions of multinomial schemes depending on small parameters (p(1),...,p(m)) and expansions for exponents associated with iterates of a class of random operators which includes the class of nonexpansive and homogeneous operators. Several examples pertaining to computer and communication sciences are investigated: timed event graphs, resource sharing models and heap models.