Stable laws and spectral gap properties for affine random walks

TL;DR

This paper studies the spectral gap and convergence to stable laws for multidimensional affine random walks, linking the spectral properties of the associated operators to the behavior of the sums of the process.

Contribution

It establishes spectral gap properties for Fourier operators of affine recursions and connects these to the convergence of Birkhoff sums to stable laws, highlighting the role of the stationary measure.

Findings

Spectral gap properties are proven for Fourier operators on Hölder spaces.

Convergence to stable laws for Birkhoff sums is demonstrated.

Parameters of the stable laws depend on the multiplicative part of the recursion.

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 H\"older 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.