ON THE DISTRIBUTION OF RANKED HEIGHTS OF EXCURSIONS OF A BROWNIAN BRIDGE1 By Jim Pitman and Marc Yor

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 Mbr+ j of the jth highest maximum over a positive excursion of the bridge has the same distribution as Mbr+ 1 /j, where the distribution of Mbr+ 1 = sup0≤t≤1 Bbr t is given by Lévy’s formula P Mbr+ 1 > x = e−2x 2 . The probability density of the height Mbr j of the jth highest maximum of excursions of the reflecting Brownian bridge Bbr t 0 ≤ t ≤ 1 is given by a modification of the known θ-function series for the density of Mbr 1 = sup0≤t≤1 Bbr t . 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.