Convergence to stable laws for a class of multidimensional stochastic recursions

TL;DR

This paper studies the convergence behavior of multidimensional stochastic recursions, showing that normalized sums of the process converge to stable laws or normal distributions depending on the tail index, with detailed spectral analysis.

Contribution

It establishes convergence to stable or normal laws for multidimensional recursions with similarity transformations, extending classical limit theorems to a broader class of stochastic recursions.

Findings

Normalized sums converge to stable laws or normal distribution.

The limit laws are explicitly characterized in terms of the stationary measure.

A local limit theorem is proved under natural conditions.

Abstract

We 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 $\nu$, 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.