Heavy tail properties of stationary solutions of multidimensional stochastic recursions

Details Heavy tail properties of stationary solutions of multidimensional stochastic recursions Heavy tail properties of stationary solutions of multidimensional stochastic recursions

2006-08-10

arxiv.org/abs/math/0608239v1

IMS Lecture Notes--Monograph Series 2006, Vol. 48, 85-99

Mathematics

Probability

Published at http://dx.doi.org/10.1214/074921706000000121 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/

Scientific paper

10.1214/074921706000000121

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.

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