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

Sebastian Mentemeier, Ewa Damek, Mariusz Mirek, Jacek Zienkiewicz
published 2011-05-05

Given a sequence $(M_{n},Q_{n})_{n\ge 1}$ of i.i.d.\ random variables with generic copy $(M,Q) \in GL(d, \R) \times \R^d$, 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 $$ \lim_{n \to \infty}(\E{\norm{M_1 ... M_n}^\kappa})^{\frac{1}{n}} = 1 .$$ 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 on the tail behaviour of $R$ by Alsmeyer and the second author.

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