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

Let $V=\mathbb R^d$ be the Euclidean $d$-dimensional space, $μ$ (resp $λ$) a probability measure on the linear (resp affine) group $G=G L (V)$ (resp $H= \Aff (V))$ and assume that $μ$ is the projection of $λ$ on $G$. We study asymptotic properties of the iterated convolutions $μ^n *δ\_{v}$ (resp $λ^n*δ\_{v})$ if $v\in V$, i.e asymptotics of the random walk on $V$ defined by $μ$ (resp $λ$), if the subsemigroup $T\subset G$ (resp.\ $Σ\subset H$) generated by the support of $μ$ (resp $λ$) is "large". We show spectral gap properties for the convolution operator defined by $μ$ on spaces of homogeneous functions of degree $s\geq 0$ on $V$, which satisfy H{ö}lder type conditions. As a consequence of our analysis we get precise asymptotics for the potential kernel $Σ\_{0}^{\infty} μ^k * δ\_{v}$, which imply its asymptotic homogeneity. Under natural conditions the $H$-space $V$ is a $λ$-boundary; then we use the above results and radial Fourier Analysis on $V\setminus \{0\}$ to show that the unique $λ$-stationary measure $ρ$ 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 $μ$ and $Σ$. 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 $λ$-boundary dual to $V$.

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