Advances in Applied Probability

Analyticity of iterates of random non-expansive maps

François Baccelli and Dohy Hong

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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.

Article information

Adv. in Appl. Probab. Volume 32, Number 1 (2000), 193-220.

First available in Project Euclid: 12 February 2002

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Zentralblatt MATH identifier

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 [See also 32B05, 32C05] 47H40: Random operators [See also 47B80, 60H25] 34D08: Characteristic and Lyapunov exponents 28A18

Contraction non-expansiveness analyticity vectorial recurrence relation Lyapunov exponents asymptotic mean value


Baccelli, François; Hong, Dohy. Analyticity of iterates of random non-expansive maps. Adv. in Appl. Probab. 32 (2000), no. 1, 193--220. doi:10.1239/aap/1013540030.

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