Convergence to stable laws for a class of multidimensional stochastic recursions

Abstract

We consider a Markov chain $\{X_n\}_{n=0}^\8$ 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 $H={\mathbb{R}}^d⋊\mathrm{GL}({\mathbb{R}}^d)$. Assume that $M_n$ takes values in the similarity group of ${\mathbb{R}}^d$, and the Markov chain has a unique stationary measure $\nu$, 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 $\nu$ 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 $\nu$ 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 $\nu$ ($\a > 2$) or of the tails of $\nu$ 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.