  Results of penalization of a one-dimensional Brownian motion $(X_t) $, by its
one-sided maximum $\dis (S_t=\sup_{0 \leq u \leq t}X_u)$, 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 $t\to \infty$, or penalizations by functions of $(S_t,X_t)$,
and also the study of the speed of convergence, as $t\to \infty$, of the
penalized distributions at time $t$.
