Asymptotic behavior of ultimately contractive iterated random Lipschitz functions

• Autorzy: Alsmeyer G. , Hölker G.
• Języki publikacji: EN

Abstrakty

EN

Let (F_n)_n≥0 be a random sequence of i.i.d. global Lipschitz functions on a complete separable metric space (X; d) with Lipschit constants L₁; L₂; : : : For n ≥0, denote by M^x _n = F_n○ : : : ○ F₁(x) and ^M^x _n = F_n○ : : : ○ F₁(x) the associated sequences of forward and backward iterations, respectively. If E log⁺ L₁ < 0 (mean contraction) and E log⁺ d ( F₁(x₀); x₀) is finite for some x₀ЄX, then it is known (see [9]) that, for each x Є X, the Markov chain M^x _n converges weakly to its unique stationary distribution π, while ^M ^x_n is a.s. convergent to a random variable ^M∞ which does not depend on x and has distribution π. In [2], renewal theoretic methods have been successfully employed to provide convergence rate results for ^M ^x _n, which then also lead to corresponding assertions for M^x _n via M^x _n d= ^M ^x _n for all n and x, where d= means equality in law. Here our purpose is to demonstrate how these methods are extended to the more general situation where only ultimate contraction, i.e. an a.s. negative Lyapunov exponent lim_n→∞ n⁻¹ log l(F_n○ : : : ○ F₁) is assumed (here l(F) denotes the Lipschitz constant of F). This not only leads to an extension of the results from [2] but in fact also to improvements of the obtained convergence rate.

Słowa kluczowe

EN

random Lipschitz function; ultimately contractive; forward iterations; backward iterations; stationary distribution; Prokhorov metric; level ϒ ladder epochs; Lyapunov exponent

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2009
• Tom: Vol. 29, Fasc. 2
• Strony: 321--336
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Alsmeyer G.

• Institut für Mathematische Statistik, FB 10 Westfälische Wilhelms-Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany

autor

Hölker G.

• Institut für Mathematische Statistik, FB 10 Westfälische Wilhelms-Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany

Bibliografia

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