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∗(t),θ∗(t)), where U∗(t) is the maximum over [0,g(t)] of a reflected Brownian motion U, where g(t) stands for the last zero of U before t, θ∗(t)=f∗(t)−g∗(t), f∗(t) is the hitting time of the level U∗(t), and g∗(t) is the left-hand point of the interval straddling f∗(t). We also calculate explicitly the marginal density functions of U∗(t) and θ∗(t). Let Un∗ and θn∗ be the analogs of U∗(t) and θ∗(t) respectively where the underlying process (Un) is the Lindley process, i.e. the difference between a centered real random walk and its minimum. We prove that (Un∗n,θn∗n) converges weakly to (U∗(1),θ∗(1)) as n→∞.