Convergence to stable laws for multidimensional stochastic recursions: the case of regular matrices

Abstract

Given a sequence $(M_n,Q_n{)}_{n\ge 1}$ of i.i.d.\ random variables with generic copy $(M,Q)\in GL(d,\mathbb{R})\times {\mathbb{R}}^d$, we consider the random difference equation (RDE) $$R_n=M_n{R}_{n-1}+Q_n,$$ $n\ge 1$, and assume the existence of $\kappa >0$ such that $$ \lim_{n \to \infty}(\E{\norm{M_1 ... M_n}^κ})^{\frac{1}{n}} = 1 .$$ We prove, under suitable assumptions, that the sequence $S_n=R_1+...+R_n$, appropriately normalized, converges in law to a multidimensional stable distribution with index $\kappa$. As a by-product, we show that the unique stationary solution $R$ of the RDE is regularly varying with index $\kappa$, and give a precise description of its tail measure. This extends the prior work http://arxiv.org/abs/1009.1728v3 .

15 pages

References in corpus (2)

• The tail of the stationary distribution of a random coefficient AR(q) model
• On multidimensional Mandelbrot's cascades

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