Asymptotic Distribution of Brownian Excursions into an Interval

TL;DR

This paper derives the asymptotic distribution of Brownian excursions into an interval as time approaches infinity, conditioned on the occurrence of such excursions, extending earlier results in the field.

Contribution

It provides a new asymptotic distribution result for Brownian excursions into an interval, building on previous work and focusing on the behavior as time tends to infinity.

Findings

Asymptotic distribution derived for Brownian excursions into an interval

Conditional distribution given the existence of an excursion

Extends earlier results in stochastic process theory

Abstract

In this paper, following earlier results in [2] we derive the asymptotic distribution as $t \to \infty$, of the excursion of Brownian motion straddling $t$, into an interval $(a,b)$, conditional on the event that there is such an excursion.