Heavy tail properties of stationary solutions of multidimensional stochastic recursions

Abstract

We consider the following recurrence relation with random i.i.d. coefficients $(a_n,b_n)$: $$x_{n+1}=a_{n+1}{x}_n+b_{n+1}$$ where $a_n\in GL(d,\mathbb{R}),b_n\in {\mathbb{R}}^d$. Under natural conditions on $(a_n,b_n)$ this equation has a unique stationary solution, and its support is non-compact. We show that, in general, its law has a heavy tail behavior and we study the corresponding directions. This provides a natural construction of laws with heavy tails in great generality. Our main result extends to the general case the results previously obtained by H. Kesten in [16] under positivity or density assumptions, and the results recently developed in [17] in a special framework.

Published at http://dx.doi.org/10.1214/074921706000000121 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)

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