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

Abstract

Let $V={\mathbb{R}}^d$ be the Euclidean $d$-dimensional space, $\mu$ (resp $\lambda$) a probability measure on the linear (resp affine) group $G=GL(V)$ (resp $H= \Aff (V))$ and assume that $\mu$ is the projection of $\lambda$ on $G$. We study asymptotic properties of the iterated convolutions ${\mu }^n\ast \delta \_v$ (resp ${\lambda }^n\ast \delta \_v)$ 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 $s\ge 0$ on $V$, which satisfy H{ö}lder type conditions. As a consequence of our analysis we get precise asymptotics for the potential kernel $\Sigma \_0^{\infty }{\mu }^k\ast \delta \_v$, 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 $v\to tv$ (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 $V$.

References in corpus (1)

• On multidimensional Mandelbrot's cascades

Cited by in corpus (3)

• Conditioned limit theorems for products of random matrices
• On Kesten's Multivariate Choquet-Deny Lemma
• Stable laws and spectral gap properties for affine random walks