Heavy Tail Properties of Stationary Solutions of Multidimensional Stochastic Recursions

We consider the following recurrence relation with random i.i.d. coefficients (an,bn):$x_{n+1}=a_{n+1}x_{n}+b_{n+1}$where$a_{n}\in GL(d,{\Bbb R}),b_{n}\in {\Bbb R}^{d}$. Under natural conditions on (an,bn) this equation has a unique stationary solution, and its support is non-compact. We show that,...

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Published in:Lecture notes-monograph series 2006-01, Vol.48, p.85-99
Main Author: Yves Guivarc'h
Format: Article
Language:English
Subjects:
Geometry
Infinity
Markov chains
Mathematical theorems
Matrices
Mellin transforms
Radon
Random Processes
Random walk
Recursion
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description We consider the following recurrence relation with random i.i.d. coefficients (an,bn):$x_{n+1}=a_{n+1}x_{n}+b_{n+1}$where$a_{n}\in GL(d,{\Bbb R}),b_{n}\in {\Bbb R}^{d}$. Under natural conditions on (an,bn) 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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subjects Geometry
Infinity
Markov chains
Mathematical theorems
Matrices
Mellin transforms
Radon
Random Processes
Random walk
Recursion
title Heavy Tail Properties of Stationary Solutions of Multidimensional Stochastic Recursions
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