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Central limit theorems for iterated random Lipschitz mappings. (English) Zbl 1062.60017

Let \(M\) be a noncompact metric space such that every closed ball is compact, and let \(G\) be a semigroup of Lipschitz mappings of \(M\). Denote by \((Y_n)_{n\geq 1}\) a sequence of independent \(G\)-valued, identically distributed random variables, and by \(Z\) an \(M\)-valued random variable which is independent of \((Y_n)\). This paper considers the Markov chain \((Z_n)_{n\geq 0}\) with state space \(M\): \(Z_0=Z\) and \(Z_{n+1}=Y_{n+1}Z_n\) for all \(n\geq 0\). Let \(\xi\) be a real-valued function on \(G\times M\). This paper deals with the central and local central limit theorems of the random variables \((\xi(Y_n, Z_{n-1}))_{n\geq 1}\). The convergence rates are also presented. The main hypothesis is the contraction in mean for the action on \(M\) of the mappings \((Y_n)\). The main method used in the study is a quasi-compactness property of the transition probability of the Markov chain mentioned above, and a special perturbation theorem.

MSC:

60F05 Central limit and other weak theorems
60J05 Discrete-time Markov processes on general state spaces
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