Functional central limit theorems for vicious walkers

Details Functional central limit theorems for vicious walkers Functional central limit theorems for vicious walkers

2002-03-28

arxiv.org/abs/math/0203286v6

Stoch. Stoch. Rep. 75 (2003) 369-390

Mathematics

Probability

AMS-LaTeX, 20 pages, 2 figures, v6: minor corrections made for publication

Scientific paper

10.1080/10451120310001633711

We consider the diffusion scaling limit of the vicious walker model that is a system of nonintersecting random walks. We prove a functional central limit theorem for the model and derive two types of nonintersecting Brownian motions, in which the nonintersecting condition is imposed in a finite time interval $(0,T]$ for the first type and in an infinite time interval $(0,\infty)$ for the second type, respectively. The limit process of the first type is a temporally inhomogeneous diffusion, and that of the second type is a temporally homogeneous diffusion that is identified with a Dyson's model of Brownian motions studied in the random matrix theory. We show that these two types of processes are related to each other by a multi-dimensional generalization of Imhof's relation, whose original form relates the Brownian meander and the three-dimensional Bessel process. We also study the vicious walkers with wall restriction and prove a functional central limit theorem in the diffusion scaling limit.

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