Convergence to stable laws for a class of multidimensional stochastic recursions

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.