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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 γ(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 γ(p) at p=0 and approximations of γ(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 (p1,,pm) 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.

Citation

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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

Vol.10 • No. 3 • August 2000
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