Limiting laws for long Brownian Bridges perturbed by their one-sided maximum, III

Results of penalization of a one-dimensional Brownian motion $(X_t)$, by its one-sided maximum $\text{dis}(S_t={\sup }_{0\le u\le t}{X}_u)$, which were recently obtained by the authors are improved with the consideration-in the present paper- of the asymptotic behaviour of the likewise penalized Brownian bridges of length $t$, as $t\to \mathrm{\infty }$, or penalizations by functions of $(S_t,X_t)$, and also the study of the speed of convergence, as $t\to \mathrm{\infty }$, of the penalized distributions at time $t$.