The Annals of Applied Probability

Analytic expansions of max-plus Lyapunov exponents

François Baccelli and Dohy Hong

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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

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1019487510

Digital Object Identifier
doi:10.1214/aoap/1019487510

Mathematical Reviews number (MathSciNet)
MR1789980

Zentralblatt MATH identifier
1073.37526

Subjects
Primary: 41A58: Series expansions (e.g. Taylor, Lidstone series, but not Fourier series) 34D08: Characteristic and Lyapunov exponents 15A52 15A18: Eigenvalues, singular values, and eigenvectors
Secondary: 60K05: Renewal theory 60C05: Combinatorial probability 32D05: Domains of holomorphy 16A78 41A63: Multidimensional problems (should also be assigned at least one other classification number in this section)

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

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