Stable laws and spectral gap properties for affine random walks

Abstract

We consider a general multidimensional affine recursion with corresponding Markov operator $P$ and a unique $P$-stationary measure. We show spectral gap properties on Hölder spaces for the corresponding Fourier operators and we deduce convergence to stable laws for the Birkhoff sums along the recursion. The parameters of the stable laws are expressed in terms of basic quantities depending essentially on the matricial multiplicative part of $P$. Spectral gap properties of $P$ and homogeneity at infinity of the $P$-stationary measure play an important role in the proofs.

31 pages. Accepted by AIHP

References in corpus (4)

• Heavy tail properties of stationary solutions of multidimensional stochastic recursions
• Branching structure for an (L-1) random walk in random environment and its applications
• Convergence to stable laws for multidimensional stochastic recursions: the case of regular matrices
• Spectral gap properties for linear random walks and Pareto's asymptotics for affine stochastic recursions