Convergence to stable laws for a class of multidimensional stochastic recursions

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dc.creatorBuraczewski, Dariusz
dc.creatorDamek, Ewa
dc.creatorGuivarc'h, Yves
dc.date2008-09-25
dc.date2008-11-10
dc.date.accessioned2026-07-07T10:16:46Z
dc.date.available2026-07-07T10:16:46Z
dc.descriptionWe consider a Markov chain $\{X_n\}_{n=0}^\8$ on $\R^d$ 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 $H=\R^d\rtimes {\rm GL}(\R^d)$. Assume that $M_n$ takes values in the similarity group of $\R^d$, 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 $\E |M|^\a=1$ for some positive $\a$. We show that the partial sums $S_n=\sum_{k=0}^n X_k$, properly normalized, converge to a normal law ($\a\ge 2$) or to an infinitely divisible law, which is stable in a natural sense ($\a<2$). These laws are fully nondegenerate, if $ν$ is not supported on an affine hyperplane. Under a natural hypothesis, we prove also a local limit theorem for the sums $S_n$. If $\a\le 2$, 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 $ν$ ($\a > 2$) or of the tails of $ν$ and of stationary measure for an associated Markov operator ($\a\le 2$). We extend the results to the situation where $M_n$ is a random generalized similarity.
dc.identifierhttps://arxiv.org/abs/0809.4349
dc.identifierhttp://arxiv.org/abs/0809.4349
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/173624
dc.subjectProbability
dc.subject60J05, 60F05
dc.titleConvergence to stable laws for a class of multidimensional stochastic recursions
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