## 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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### References:

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