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Heavy tail properties of stationary solutions of multidimensional stochastic recursions

Yves Guivarc'h
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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.

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BibTeXRIS

Yves Guivarc'h. 2006-08-10. Heavy tail properties of stationary solutions of multidimensional stochastic recursions. https://doi.org/10.1214/074921706000000121

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