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 $κ>0$ 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 $κ$. As a by-product, we show that the unique stationary solution $R$ of the RDE is regularly varying with index $κ$, and give a precise description of its tail measure. This extends the prior work http://arxiv.org/abs/1009.1728v3 .
Recurrence of multidimensional affine recursions in the critical case
Laboratoire d’Analyse et de Mathématiques Appliquées UMR 8050, Université Gustave Eiffel, Université Paris-Est Créteil, CNRS（应用分析与数学实验室 UMR 8050，巴黎东克雷泰伊大学，古斯塔夫·埃菲尔大学，法国国家科学研究中心）；Institut Denis Poisson UMR 7013, Université de Tours, Université d’Orléans, CNRS（丹尼斯·普瓦松研究所 UMR 7013，图尔大学，奥尔良大学，法国国家科学研究中心）
On the tails of the invariant measure for multidimensional affine stochastic recursions in the critical case
Univ Bretagne Sud, CNRS UMR 6205, LMBA（南布列塔尼大学）；Universität Hildesheim, Institut für Mathematik und Angewandte Informatik（希尔德斯海姆大学）；State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
Maxima of asymptotically Gaussian random fields and moderate deviation approximations to boundary crossing probabilities of sums of random variables with multidimensional indices
