Heavy tail properties of stationary solutions of multidimensional stochastic recursions

We consider the following recurrence relation with random i.i.d. coefficients $(a_n,b_n)$: $$x_{n+1}=a_{n+1} x_n+b_{n+1}$$ where $a_n\in GL(d,\mathbb{R}),b_n\in \mathbb{R}^d$. Under natural conditions on $(a_n,b_n)$ this equation has a unique stationary solution, and its support is non-compact. We show that, in general, its law has a heavy tail behavior and we study the corresponding directions. This provides a natural construction of laws with heavy tails in great generality. Our main result extends to the general case the results previously obtained by H. Kesten in [16] under positivity or density assumptions, and the results recently developed in [17] in a special framework.Comment: Published at http://dx.doi.org/10.1214/074921706000000121 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

A spectral gap property for random walks under unitary representations

Let $G$ be a locally compact group and $\mu$ a probability measure on $G,$ which is not assumed to be absolutely continuous with respect to Haar measure. Given a unitary representation $(\pi, \cal H)$ of $G,$ we study spectral properties of the operator $\pi(\mu)$ acting on $\cal H.$ Assume that $\mu$ is adapted and that the trivial representation $1_G$ is not weakly contained in the tensor product $\pi\otimes \bar\pi.$ We show that $\pi(\mu)$ has a spectral gap, that is, for the spectral radius $r_{\rm spec}(\pi(\mu))$ of $\pi(\mu),$ we have $r_{\rm spec}(\pi(\mu))<1.$ This provides a common generalization of several previously known results. Another consequence is that, if $G$ has Kazhdan's Property (T), then $r_{\rm spec}(\pi(\mu))<1$ for every unitary representation $\pi$ of $G$ without finite dimensional subrepresentations. Moreover, we give new examples of so-called identity excluding groups.Comment: 19 page

Group-theoretic compactification of Bruhat-Tits buildings

Let GF denote the rational points of a semisimple group G over a non-archimedean local field F, with Bruhat-Tits building X. This paper contains five main results. We prove a convergence theorem for sequences of parahoric subgroups of GF in the Chabauty topology, which enables to compactify the vertices of X. We obtain a structure theorem showing that the Bruhat-Tits buildings of the Levi factors all lie in the boundary of the compactification. Then we obtain an identification theorem with the polyhedral compactification (previously defined in analogy with the case of symmetric spaces). We finally prove two parametrization theorems extending the BruhatTits dictionary between maximal compact subgroups and vertices of X: one is about Zariski connected amenable subgroups, and the other is about subgroups with distal adjoint action

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, $\mu$ (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 $V$

Spectral gap properties and limit theorems for some random walks and dynamical systems

Papers from the Special Semester held at the Centre Interfacultaire Bernoulli, École Polytechnique Fédérale de Lausanne, Lausanne, January–June 2013International audienceWe give a description of some limit theorems and the corresponding proofs for various transfer operators. Our examples are closely related with random walks on homogeneous spaces. The results are obtained using spectral gap methods in Hölder spaces or Hilbert spaces. We describe also their geometrical setting and the basic corresponding properties. In particular we focus on precise large deviations for products of random matrices, Fréchet's law for affine random walks and local limit theorems for Euclidean motion groups or nilmanifolds

Semigroup actions on tori and stationary measures on projective spaces

International audienceLet $\Gamma$ be a sub-semigroup of $G=GL(d,\mathbb R),$ $d>1.$ We assume that the action of $\Gamma$ on $\R^d$ is strongly irreducible and that $\Gamma$ contains a proximal and expanding element. We describe contraction properties of the dynamics of $\Gamma$ on $\R^d$ at infinity. This amounts to the consideration of the action of $\Gamma$ on some compact homogeneous spaces of $G,$ which are extensions of the projective space \pr^{d-1}. In the case where $\Gamma$ is a sub-semigroup of $GL(d,\R)\cap M(d,\Z)$ and $\Gamma$ has the above properties, we deduce that the $\Gamma$-orbits on \T^d=\R^d\slash\Z^d are finite or dense