On the distribution of ranked heights of excursions of a Brownian bridge

The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian bridge (Bbr t ; 0 t 1) is described. The height M j of the jth highest maximum over a positive excursion of the bridge has the same distribution as M 1 =j, where the distribution of M 1 = sup0 t 1B br t is given by L evy's formula P (M br+ 1 > x) = e 2x . The probability density of the height M j of the jth highest maximum of excursions of the re ecting Brownian bridge (jB t j; 0 t 1) is given by a modi cation of the known -function series for the density of Mbr 1 = sup0 t 1 jB br t j. These results are obtained from a more general description of the distribution of ranked values of a homogeneous functional of excursions of the standardized bridge of a self-similar recurrent Markov process.