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Analyticity of iterates of random non-expansive maps

Part of: Stability theory Analytic continuation Probability theory on algebraic and topological structures Nonlinear operators and their properties Real functions: Miscellaneous topics Genetics and population dynamics

Published online by Cambridge University Press:  01 July 2016

François Baccelli  and
Dohy Hong
François Baccelli*
Affiliation:
ENS Paris
Dohy Hong*
Affiliation:
ENS Paris
*
∗ Postal address: ENS, DMI-LIENS, 45 rue d'Ulm, 75230 Paris Cedex 05, France.
∗∗ Email address: [email protected]
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Abstract

This paper focuses on the analyticity of the limiting behavior of a class of dynamical systems defined by iteration of non-expansive random operators. The analyticity is understood with respect to the parameters which govern the law of the operators. The proofs are based on contraction with respect to certain projective semi-norms. Several examples are considered, including Lyapunov exponents associated with products of random matrices both in the conventional algebra, and in the (max, +) semi-field, and Lyapunov exponents associated with non-linear dynamical systems arising in stochastic control. For the class of reducible operators (defined in the paper), we also address the issue of analyticity of the expectation of functionals of the limiting behavior, and connect this with contraction properties with respect to the supremum norm. We give several applications to queueing theory.

Keywords

Contractionnon-expansivenessanalyticityvectorial recurrence relationLyapunov exponentsasymptotic mean value

MSC classification

Primary: 47H09: Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc. 32D05: Domains of holomorphy 60B99: None of the above, but in this section 34D05: Asymptotic properties
Secondary: 26E05: Real-analytic functions 47H40: Random operators 34D08: Characteristic and Lyapunov exponents
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 1 , March 2000 , pp. 193 - 220
Copyright
Copyright © Applied Probability Trust 2000 

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Footnotes

The work of FB was supported in part by the TMR grant ALAPEDES (‘The Algebraic Approach to Performance Evaluation of Discrete Event System’, RB-FMRX-CT-96-0074).

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