Elements related to the largest complete excursion of a reflected BM stopped at a fixed time. Application to local score

We calculate the density function of $(U^1\left(t\right),{\theta }^1\left(t\right))$, where $U^1\left(t\right)$ is the maximum over $[0,g\left(t\right)]$ of a reflected Brownian motion $U$, where $g\left(t\right)$ stands for the last zero of $U$ before $t$, ${\theta }^1\left(t\right)=f^1\left(t\right)-g^1\left(t\right)$, $f^1\left(t\right)$ is the hitting time of the level $U^1\left(t\right)$, and $g^1\left(t\right)$ is the left-hand point of the interval straddling $f^1\left(t\right)$. We also calculate explicitly the marginal density functions of $U^1\left(t\right)$ and ${\theta }^1\left(t\right)$. Let $U_n^1$ and ${\theta }_n^1$ be the analogs of $U^1\left(t\right)$ and ${\theta }^1\left(t\right)$ respectively where the underlying process $\left(U_n\right)$ is the Lindley process, i.e. the difference between a centered real random walk and its minimum. We prove that $(\frac{U_n^1}{\sqrt{n}},\frac{{\theta }_n^1}{n})$ converges weakly to $(U^1\left(1\right),{\theta }^1\left(1\right))$ as $n\to \infty$.