Abstract
We consider a Markov chain \({\{X_n\}_{n=0}^\infty}\) on \({\mathbb R^d}\) defined by the stochastic recursion X n = M n X n-1 + Q n , where (Q n , M n ) are i.i.d. random variables taking values in the affine group \({A(\mathbb R^d)=\mathbb R^d\rtimes {\rm GL}(\mathbb R^d)}\). Assume that M n takes values in the group of similarities of \({\mathbb R^d}\), and the Markov chain has a unique stationary measure ν, which has unbounded support. We denote by |M n | the expansion coefficient of M n and we assume \({\mathbb E [|M|^\alpha]=1}\) for some positive α. We show that the partial sums \({S_n=\sum_{k=0}^n X_k}\), properly normalized, converge to a normal law (α ≥ 2) or to an infinitely divisible law, which is stable in a natural sense (α < 2). These laws are fully nondegenerate, if ν is not supported on an affine hyperplane. Under an aperiodicity hypothesis, we prove also a local limit theorem for the sums S n . If α ≤ 2, proofs are based on the homogeneity at infinity of ν and on a detailed spectral analysis of a family of Fourier operators P v considered as perturbations of the transition operator P of the chain {X n }. The characteristic function of the limit law has a simple expression in terms of moments of ν (α > 2) or of the tails of ν and of stationary measure for an associated Markov operator (α ≤ 2). We extend the results to the situation where M n is a random generalized similarity.
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In memory of Andrzej Hulanicki.
This research project has been partially supported by Marie Curie Transfer of Knowledge Fellowship Harmonic Analysis, Nonlinear Analysis and Probability (contract number MTKD-CT-2004-013389). D. Buraczewski and E. Damek were also supported by MNiSW grant N201 012 31/1020.
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Buraczewski, D., Damek, E. & Guivarc’h, Y. Convergence to stable laws for a class of multidimensional stochastic recursions. Probab. Theory Relat. Fields 148, 333–402 (2010). https://doi.org/10.1007/s00440-009-0233-7
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DOI: https://doi.org/10.1007/s00440-009-0233-7
Mathematics Subject Classification (2000)
- 60F05
- 60J05
- 60B15