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

机译:线性随机游走的谱隙特性和仿射随机递归的帕累托渐近性

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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.
机译:令V = Rd为欧几里得d维空间,mu(reslam。lambda)是线性(resaff。仿射)组G = GL(V)(respon。H = Aff(V))的概率度量。是A在G上的投影。我们研究迭代卷积mu(n)* delta(upsilon),(resp lambda(n)* delta(upsilon))的渐近性质,如果upsilonεV是随机的,则其渐近渐近如果由mu(分别为lambda)的支持所生成的G的亚半群T子集(H的分别为Sigma子集)为“大”,则由mu(分别为lambda)定义的V。我们展示了在V上满足solder类型条件的s> = 0的齐次函数的空间上由它定义的卷积算子的谱隙性质。作为我们分析的结果,我们为潜在内核Sigma(infinity)(0)mu(k)* delta(upsilon)给出了精确的渐近线,这暗示了它的渐近同质性。在自然条件下,H空间V为Lambda边界;然后我们使用上述结果和对V {0}的径向傅立叶分析,得出关于膨胀upsilon-> t upsilon(对于t> 0),V上唯一的λ平稳测度rho是“无穷大的均匀”。尾部量度基本上取决于mu和Sigma。我们的证明是基于对某些依赖于马尔科夫的随机矩阵的乘积的主导Lyapunov指数的简单性,对“ tame”马尔可夫游动的更新定理的使用以及对V的条件Lambda边界对偶的动力学性质。

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