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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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