## Speed of stochastic locally contractive systems.(English)Zbl 1046.60067

The author considers the autoregressive model on $${\mathbb R}^d$$ defined by $$Y_n^y = a_n Y_{n-1}^y +B_n$$ where $$\{(a_n,B_n)\}$$ is a sequence of i.i.d. random variables in $${\mathbb R}_+^* \times {\mathbb R}^d$$ and $$Y_0^y = y$$. In this case $$Y_n^y-Y_n^x= e^{S_n} (y-x)$$ where $$S_n=\sum_{i=1}^n \log a_i$$. If $$E(\log a_1)<0$$, then $$| Y_n^y-Y_n^x|$$ goes to zero with an exponential speed and the iterated function system $$Y_n^y$$ is strongly contractive. In the critical case $$E(\log a_1)=0$$ has $$Y_n^y$$ a local contraction property, i.e. if $$Y_n^y$$ is in a compact set, then $$| Y_n^y-Y_n^x| \to 0$$ a.s. The speed of this convergence is determined and the results are extended to some higher-dimensional situations and to a Markov chain on a nilpotent Lie group induced by a random walk on a solvable Lie group of $${\mathcal N}{\mathcal A}$$ type.

### MSC:

 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 60G50 Sums of independent random variables; random walks
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### References:

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