Heavy tail properties of stationary solutions of multidimensional stochastic recursions

Abstract

We consider the following recurrence relation with random i.i.d. coefficients $(a_{n}, b_{n})$: $$ x_{n+1}=a_{n+1} x_{n}+b {n+1} %%\leqno (0)$$ 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, Random difference equations and renewal theory for products of random matrices, under positivity or density assumptions, and the results recently developed in On the tail of the stationary distribution of a random coefficient AR(q) model, in a special framework.