Tail estimates for the Brownian excursion area and other Brownian areas

TL;DR

This paper derives asymptotic tail estimates for various Brownian areas, including excursion, bridge, and meander areas, using a saddle point method to invert Laplace transforms and compute detailed asymptotic expansions.

Contribution

It introduces a systematic technique to obtain asymptotic tail estimates for multiple Brownian areas through Laplace transform inversion and saddle point analysis.

Findings

Derived asymptotic tail estimates for several Brownian areas.

Computed the first four terms of the asymptotic expansion.

Provided asymptotics for distribution functions and moments.

Abstract

Several Brownian areas are considered in this paper: the Brownian excursion area, the Brownian bridge area, the Brownian motion area, the Brownian meander area, the Brownian double meander area, the positive part of Brownian bridge area, the positive part of Brownian motion area. We are interested in the asymptotics of the right tail of their density function. Inverting a double Laplace transform, we can derive, in a mechanical way, all terms of an asymptotic expansion. We illustrate our technique with the computation of the first four terms. We also obtain asymptotics for the right tail of the distribution function and for the moments. Our main tool is the two-dimensional saddle point method.