Random walks in Euclidean space

Abstract

Abstract. Fix a probability measure on the space of isometries of Euclidean spacebe a sequence of random points such that Y l+1 is the image of Y l under a random isometry of the previously fixed probability law, which is independent of Y l . We prove a Local Limit Theorem for Y l under necessary non-degeneracy conditions. Moreover, under more restrictive but still general conditions we give a quantitative estimate which describes the behavior of the law of Y l on scales e −cl 1/4 < r < l

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

“…Using the result of [3] we give an affirmative answer (see [10]). We note that the same question has been recently solved in full generality in [51] i.e without using the spectral gap property in SU (d). Our method here is of general interest in the larger context of Markov operators with a Hilbert space spectral condition, i.e Q strongly mixing in the sense of [45].…”

A horospherical ratio ergodic theorem for actions of free groups

“…His argument was corrected and the results extended by Guivarc'h [Gu76] (see also [Vo04]). An important advance in the case when X equals the isometry group of Euclidean n-space was very recently obtained by Varju [Va12]. Breuillard has obtained positive results when X is the Heisenberg group [Br05] and the averages are random walk averages.…”

Large deviation expansions for the coefficients of random walks on the general linear group

“…In this subsection we formulate the precise local limit theorems with large deviations for the coefficients f, G n v . For sums of independent real-valued random variables, local limit theorems with large and moderate deviations can be found for instance in Gnedenko [28], Sheep [54], Stone [55], Borovkov and Borovkov [7], Breuillard [10], Varju [56]. For products of random matrices, such types of local limit theorems for the vector norm |G n v| have been recently established in [6,57,58].…”