Related papers: Some central limit theorems for random walks assoc…

The spherical functions of the noncompact Grassmann manifolds $G_{p,q}(\mathbb F)=G/K$ over the (skew-)fields $\mathbb F=\mathbb R, \mathbb C, \mathbb H$ with rank $q\ge1$ and dimension parameter $p>q$ can be described as Heckman-Opdam…

We consider compact Grassmann manifolds $G/K$ over the real, complex or quaternionic numbers whose spherical functions are Heckman-Opdam polynomials of type $BC$. From an explicit integral representation of these polynomials we deduce a…

The Heckman-Opdam hypergeometric functions of type BC extend classical Jacobi functions in one variable and include the spherical functions of non-compact Grassmann manifolds over the real, complex or quaternionic numbers. There are various…

Let $F_{BC}(\lambda,k;t)$ be the Heckman-Opdam hypergeometric function of type BC with multiplicities $k=(k_1,k_2,k_3)$ and weighted half sum $\rho(k)$ of positive roots. We prove that $F_{BC}(\lambda+\rho(k),k;t)$ converges for…

We show that almost any one-dimensional projection of a suitably scaled random walk on a hypercube, inscribed in a hypersphere, converges weakly to an Ornstein-Uhlenbeck process as the dimension of the sphere tends to infinity. We also…

We study the Heckman-Opdam hypergeometric functions associated to a root system of type $BC$ and a multiplicity function which is allowed to assume some non-positive values (a standard multiplicity function). For such functions, we obtain…

We show the existence of a trace process at infinity for random walks on hyperbolic groups of conformal dimension < 2 and relate it to the existence of a reflecting random walk. To do so, we employ the theory of Dirichlet forms which…

In the eighties, A. Connes and E. J. Woods made a connection between hyperfinite von Neumann algebras and Poisson boundaries of time dependent random walks. The present paper explains this connection and gives a detailed proof of two…

We consider the sum of the coordinates of a simple random walk on the K-dimensional hypercube, and prove a double asymptotic of this process, as both the time parameter n and the space parameter K tend to infinity. Depending on the…

We establish a central limit theorem, a local limit theorem, and a law of large numbers for a natural random walk on a symmetric space $M$ of non-compact type and rank one. This class of spaces, which includes the complex and quaternionic…

We derive a functional central limit theorem for the excursion of a random walk conditioned on sweeping a prescribed geometric area. We assume that the increments of the random walk are integer-valued, centered, with a third moment equal to…

Spherically symmetric random walks in arbitrary dimension $D$ can be described in terms of Gegenbauer (ultraspherical) polynomials. For example, Legendre polynomials can be used to represent the special case of two-dimensional spherically…

We survey some geometrical properties of trajectories of $d$-dimensional random walks via the application of functional limit theorems. We focus on the functional law of large numbers and functional central limit theorem (Donsker's…

We introduce a new method for proving central limit theorems for random walk on nilpotent groups. The method is illustrated in a local central limit theorem on the Heisenberg group, weakening the necessary conditions on the driving measure.…

We study random walks on the isometry group of a Gromov hyperbolic space or Teichm\"uller space. We prove that the translation lengths of random isometries satisfy a central limit theorem if and only if the random walk has finite second…

It is known that every infinite index quasi-convex subgroup $H$ of a non-elementary hyperbolic group $G$ is a free factor in a larger quasi-convex subgroup of $G$. We give a probabilistic generalization of this result. That is, we show that…

In this paper we study asymptotic properties of symmetric and non-degenerate random walks on transient hyperbolic groups. We prove a central limit theorem and a law of iterated logarithm for the drift of a random walk, extending previous…

We present a unified approach to a couple of central limit theorems for radial random walks on hyperbolic spaces and time-homogeneous Markov chains on the positive half line whose transition probabilities are defined in terms of the Jacobi…

In this paper we present explicit product formulas for a continuous two-parameter family of Heckman-Opdam hypergeometric functions of type BC on Weyl chambers $C_q\subset \mathbb R^q$ of type $B$. These formulas are related to continuous…

We prove central limit theorems for the random walks on either the mapping class group of a closed, connected, orientable, hyperbolic surface, or on $\text{Out}(F_N)$, each time under a finite second moment condition on the measure (either…