Abstract

The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian bridge (B br t ; 0 t 1) is described. The height M br+ j of the jth highest maximum over a positive excursion of the bridge has the same distribution as M br+ 1 =j, where the distribution of M br+ 1 = sup 0t1 B br t is given by L'evy's formula P (M br+ 1 ? x) = e \Gamma2x 2 . The probability density of the height M br j of the jth highest maximum of excursions of the reflecting Brownian bridge (jB br t j; 0 t 1) is given by a modification of the known `-function series for the density of M br 1 = sup 0t1 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. Keywords: Brownian bridge, Brownian excursion, Brownian scaling, local time, selfsimilar recurrent Markov process, Bessel p...

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