Local times in a Brownian excursion

Abstract

Let $\{B(t), t \geq 0\}$ be a standard Brownian motion in $\mathbb{R}$. Let $T$ be the first return time to 0 after hitting 1, and $\{L(T,x), x \in \mathbb{R}\}$ be the local time process at time $T$ and level $x$. The distribution of $L(T,x)$ for each $x \in \mathbb{R}$ is determined. This is applied to the estimation of a $L^1$ integral on $\mathbb{R}$.