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August 2000 Analytic expansions of max-plus Lyapunov exponents
François Baccelli, Dohy Hong
Ann. Appl. Probab. 10(3): 779-827 (August 2000). DOI: 10.1214/aoap/1019487510

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.

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François Baccelli. Dohy Hong. "Analytic expansions of max-plus Lyapunov exponents." Ann. Appl. Probab. 10 (3) 779 - 827, August 2000. https://doi.org/10.1214/aoap/1019487510

Information

Published: August 2000
First available in Project Euclid: 22 April 2002

zbMATH: 1073.37526
MathSciNet: MR1789980
Digital Object Identifier: 10.1214/aoap/1019487510

Subjects:
Primary: 15A18‎ , 15A52 , 34D08 , 41A58
Secondary: 16A78 , 32D05 , 41A63 , 60C05 , 60K05

Keywords: (max, plus) semiring , analyticity , Lyapunov exponents , network modeling , renovating events , stationary state variables , stochastic Petri nets. , strong coupling , Taylor series , vectorial recurrence relation

Rights: Copyright © 2000 Institute of Mathematical Statistics

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Vol.10 • No. 3 • August 2000
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