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

摘要

Let V = 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 A on G. We study asymptotic properties of the iterated convolutions mu(n) * delta(upsilon), (resp lambda(n) * delta(upsilon)) if upsilon epsilon V, i.e. asymptotics of the random walk on V defined by mu (resp. lambda), if the subsemigroup T subset of G (resp. Sigma subset of H) generated by the support of mu (resp. lambda) is "large." We show spectral gap properties for the convolution operator defined by it on spaces of homogeneous functions of degree s >= 0 on V, which satisfy Holder type conditions. As a consequence of our analysis we get precise asymptotics for the potential kernel Sigma(infinity)(0) mu(k) * delta(upsilon), 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 {0} to show that the unique lambda-stationary measure rho on V is "homogeneous at infinity" with respect to dilations upsilon -> t upsilon (for t > 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.