Extreme lengths in Brownian and Bessel excursions

Abstract

We establish some strong limit theorems for the longest excursion lengths of a Bessel process of dimension d ∈ (0,2). In the special case d=1, we recover and improve some well-known results for Wiener processes, and solve an open problem raised. The proof relies on exact distributions evaluated by Pitman and Yor and on a careful analysis of the Bessel sample paths.