On the tails of the invariant measure for multidimensional affine stochastic recursions in the critical case

Abstract: We study the behavior at infinity of the invariant Radon measure for the multidimensional affine stochastic recursion $V_n = A_n V_{n-1} + B_n,$ where $(A_n)<em>{n \geq 1}$ are positive random matrices, $(B_n)</em>{n \geq 1}$ are random vectors with nonnegative entries, and $(A_n, B_n)_{n \geq 1}$ are independent and identically distributed. In the critical regime where the top Lyapunov exponent of the random matrix products $A_n \cdots A_1$ is zero, Brofferio, Peigné and Pham [6] recently established the existence and uniqueness, up to multiplication by a constant, of an invariant Radon measure with infinite total mass. They proved that the tail behavior of this measure when applied to radial sets is governed by a slowly varying function. Our goal is to show that this slowly varying function is actually bounded. Moreover, we investigate directional tail behavior.