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Branching random walks on binary search trees: convergence of the occupation measure

Eric Fekete (2010)

ESAIM: Probability and Statistics

We consider branching random walks with binary search trees as underlying trees. We show that the occupation measure of the branching random walk, up to some scaling factors, converges weakly to a deterministic measure. The limit depends on the stable law whose domain of attraction contains the law of the increments. The existence of such stable law is our fundamental hypothesis. As a consequence, using a one-to-one correspondence between binary trees and plane trees, we give a description of the...

Brownian particles with electrostatic repulsion on the circle : Dyson’s model for unitary random matrices revisited

Emmanuel Cépa, Dominique Lépingle (2001)

ESAIM: Probability and Statistics

The brownian motion model introduced by Dyson [7] for the eigenvalues of unitary random matrices N × N is interpreted as a system of N interacting brownian particles on the circle with electrostatic inter-particles repulsion. The aim of this paper is to define the finite particle system in a general setting including collisions between particles. Then, we study the behaviour of this system when the number of particles N goes to infinity (through the empirical measure process). We prove that a limiting...

Brownian particles with electrostatic repulsion on the circle: Dyson's model for unitary random matrices revisited

Emmanuel Cépa, Dominique Lépingle (2010)

ESAIM: Probability and Statistics

The Brownian motion model introduced by Dyson [7] for the eigenvalues of unitary random matrices N x N is interpreted as a system of N interacting Brownian particles on the circle with electrostatic inter-particles repulsion. The aim of this paper is to define the finite particle system in a general setting including collisions between particles. Then, we study the behaviour of this system when the number of particles N goes to infinity (through the empirical measure process). We prove...

Central and non-central limit theorems for weighted power variations of fractional brownian motion

Ivan Nourdin, David Nualart, Ciprian A. Tudor (2010)

Annales de l'I.H.P. Probabilités et statistiques

In this paper, we prove some central and non-central limit theorems for renormalized weighted power variations of order q≥2 of the fractional brownian motion with Hurst parameter H∈(0, 1), where q is an integer. The central limit holds for 1/2q<H≤1−1/2q, the limit being a conditionally gaussian distribution. If H<1/2q we show the convergence in L2 to a limit which only depends on the fractional brownian motion, and if H>1−1/2q we show the convergence in L2 to a stochastic integral...

Central Limit Theorem for Diffusion Processes in an Anisotropic Random Environment

Ernest Nieznaj (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

We prove the central limit theorem for symmetric diffusion processes with non-zero drift in a random environment. The case of zero drift has been investigated in e.g. [18], [7]. In addition we show that the covariance matrix of the limiting Gaussian random vector corresponding to the diffusion with drift converges, as the drift vanishes, to the covariance of the homogenized diffusion with zero drift.

Central limit theorem for hitting times of functionals of Markov jump processes

Christian Paroissin, Bernard Ycart (2004)

ESAIM: Probability and Statistics

A sample of i.i.d. continuous time Markov chains being defined, the sum over each component of a real function of the state is considered. For this functional, a central limit theorem for the first hitting time of a prescribed level is proved. The result extends the classical central limit theorem for order statistics. Various reliability models are presented as examples of applications.

Central limit theorem for hitting times of functionals of Markov jump processes

Christian Paroissin, Bernard Ycart (2010)

ESAIM: Probability and Statistics

A sample of i.i.d. continuous time Markov chains being defined, the sum over each component of a real function of the state is considered. For this functional, a central limit theorem for the first hitting time of a prescribed level is proved. The result extends the classical central limit theorem for order statistics. Various reliability models are presented as examples of applications.

Central limit theorem for random measures generated by stationary processes of compact sets

Zbyněk Pawlas (2003)

Kybernetika

Random measures derived from a stationary process of compact subsets of the Euclidean space are introduced and the corresponding central limit theorem is formulated. The result does not require the Poisson assumption on the process. Approximate confidence intervals for the intensity of the corresponding random measure are constructed in the case of fibre processes.

Central limit theorem for sampled sums of dependent random variables

Nadine Guillotin-Plantard, Clémentine Prieur (2010)

ESAIM: Probability and Statistics

We prove a central limit theorem for linear triangular arrays under weak dependence conditions. Our result is then applied to dependent random variables sampled by a ℤ -valued transient random walk. This extends the results obtained by [N. Guillotin-Plantard and D. Schneider, Stoch. Dynamics3 (2003) 477–497]. An application to parametric estimation by random sampling is also provided.

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