Spectral gap properties for linear random walks and Pareto's asymptotics for affine stochastic recursions

TL;DR

This paper investigates the spectral gap properties of linear and affine random walks on Euclidean space, deriving asymptotics of potential kernels and stationary measures, revealing their homogeneity at infinity under broad conditions.

Contribution

It establishes spectral gap results for convolution operators on homogeneous functions, providing precise asymptotics and homogeneity properties of potential kernels and stationary measures for affine stochastic recursions.

Findings

Spectral gap properties for convolution operators on homogeneous functions.

Asymptotic homogeneity of potential kernels and stationary measures.

Characterization of tail behavior and boundary properties of the measures.

Abstract

Let $V=\mathbb R^d$ be the Euclidean $d$-dimensional space, $\mu$ (resp $\lambda$) a probability measure on the linear (resp affine) group $G=G L (V)$ (resp $H= \Aff (V))$ and assume that $\mu$ is the projection of $\lambda$ on $G$. We study asymptotic properties of the iterated convolutions $\mu^n *\delta\_{v}$ (resp $\lambda^n*\delta\_{v})$ if $v\in V$, i.e asymptotics of the random walk on $V$ defined by $\mu$ (resp $\lambda$), if the subsemigroup $T\subset G$ (resp.\ $\Sigma \subset H$) generated by the support of $\mu$ (resp $\lambda$) is "large". We show spectral gap properties for the convolution operator defined by $\mu$ on spaces of homogeneous functions of degree $s\geq 0$ on $V$, which satisfy H{\"o}lder type conditions. As a consequence of our analysis we get precise asymptotics for the potential kernel $\Sigma\_{0}^{\infty} \mu^k * \delta\_{v}$, which imply its asymptotic homogeneity. Under natural conditions the $H$-space $V$ is a $\lambda$-boundary; then we use the above results and radial Fourier Analysis on $V\setminus \{0\}$ to show that the unique $\lambda$-stationary measure $\rho$ on $V$ is "homogeneous at infinity" with respect to dilations $v\rightarrow t v$ (for $t\textgreater{}0$), with a tail measure depending essentially of $\mu$ and $\Sigma$. Our proofs are based on the simplicity of the dominant Lyapunov exponent for certain products of Markov-dependent random matrices, on the use of renewal theorems for "tame" Markov walks, and on the dynamical properties of a conditional $\lambda$-boundary dual to $V$.