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 U n ∗ and θ n ∗ be the analogs of U ∗ ( t ) and θ ∗ ( t ) respectively where the underlying process ( U n ) is the Lindley process, i.e. the difference between a centered real random walk and its minimum. We prove that ( U n ∗ n , θ n ∗ n ) converges weakly to ( U ∗ ( 1 ) , θ ∗ ( 1 ) ) as n → ∞ .