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A Central Limit Theorem for Contractive Stochastic Dynamical Systems

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Martin Benda
Martin Benda*
Affiliation:
Universität München
*
∗Postal address: Mathematisches Institut, Universität München, Theresienstrasse 39, D-80333, München, Germany
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Abstract

If (Fn)n∈ℕ is a sequence of independent and identically distributed random mappings from a second countable locally compact state space 𝕏 to 𝕏 which itself is independent of the 𝕏-valued initial variable X0, the discrete-time stochastic process (Xn)n≥0, defined by the recursion equation Xn = Fn(Xn−1) for n∈ℕ, has the Markov property. Since 𝕏 is Polish in particular, a complete metric d exists. The random mappings (Fn)n∈ℕ are assumed to satisfy ℙ-a.s. Conditions on the distribution of l(Fn) are given for the existence of an invariant distribution of X0 making the process (Xn)n≥0 stationary and ergodic. Our main result corrects a central limit theorem by Łoskot and Rudnicki (1995) and removes an error in its proof. Instead of trying to compare the sequence φ (Xn)n≥0 for some φ : 𝕏 → ℝ with a triangular scheme of independent random variables our proof is based on an approximation by a martingale difference scheme.

Keywords

Markov chaininvariant distributioncentral limit theoremiterated function system

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60F05: Central limit and other weak theorems
Type
Short Communications
Information
Journal of Applied Probability , Volume 35 , Issue 1 , March 1998 , pp. 200 - 205
Copyright
Copyright © Applied Probability Trust 1998 

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References

Billingsley, P. (1961). The Lindeberg–Levy theorem for martingales. Proc. Amer. Math. Soc. 12, 788–792.Google Scholar
Elton, J. (1990). A multiplicative ergodic theorem for Lipschitz maps. Stoch. Proc. Appl. 34, 39–47.Google Scholar
Loskot, K., and Rudnicki, R. (1995). Limit theorems for stochastically perturbed systems. J. Appl. Prob. 32, 459–469.Google Scholar