## The Annals of Applied Probability

### Analytic expansions of max-plus Lyapunov exponents

#### Abstract

We give an explicit analytic series expansion of the (max, plus)-Lyapunov exponent $\gamma(p)$ of a sequence of independent and identically distributed randommatrices, 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 formexpression 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 $(p_1,\dots, p_m)$ 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.

#### Article information

Source
Ann. Appl. Probab. Volume 10, Number 3 (2000), 779-827.

Dates
First available in Project Euclid: 22 April 2002

https://projecteuclid.org/euclid.aoap/1019487510

Digital Object Identifier
doi:10.1214/aoap/1019487510

Mathematical Reviews number (MathSciNet)
MR1789980

Zentralblatt MATH identifier
1073.37526

#### Citation

Baccelli, François; Hong, Dohy. Analytic expansions of max-plus Lyapunov exponents. Ann. Appl. Probab. 10 (2000), no. 3, 779--827. doi:10.1214/aoap/1019487510. https://projecteuclid.org/euclid.aoap/1019487510

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• ENS, DMI-LIENS 45 rue d'Ulm 75230 Paris Cedex 05 France E-mail: francois.baccelli@ens.fr dohy.hong@ens.fr