Asymptotics of stationary solutions of multivariate stochastic recursions with heavy tailed inputs and related limit theorems

Abstract

Let ${\Phi }_n$ be an i.i.d. sequence of Lipschitz mappings of ${\mathbb{R}}^d$. We study the Markov chain $\{X_n^x{\}}_{n=0}^{\infty }$ on ${\mathbb{R}}^d$ defined by the recursion $X_n^x={\Phi }_n(X_{n-1}^x)$, $n\in \mathbb{N}$, $X_0^x=x\in {\mathbb{R}}^d$. We assume that ${\Phi }_n(x)=\Phi (A_{n}x,B_n(x))$ for a fixed continuous function $\Phi :{\mathbb{R}}^d\times {\mathbb{R}}^d\to {\mathbb{R}}^d$, commuting with dilations and i.i.d random pairs $(A_n,B_n)$, where $A_n\in End({\mathbb{R}}^d)$ and $B_n$ is a continuous mapping of ${\mathbb{R}}^d$. Moreover, $B_n$ is $\alpha$-regularly varying and $A_n$ has a faster decay at infinity than $B_n$. We prove that the stationary measure $\nu$ of the Markov chain $\{X_n^x\}$ is $\alpha$-regularly varying. Using this result we show that, if $\alpha <2$, the partial sums $S_n^x={\sum }_{k=1}^n{X}_k^x$, appropriately normalized, converge to an $\alpha$-stable random variable. In particular, we obtain new results concerning the random coefficient autoregressive process $X_n=A_n{X}_{n-1}+B_n$.

arXiv comment23 pages, 0 figures. Accepted for publication in Stochastic Processes and their Applications