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

The distribution of the sequence of ranked maximum and minimum values attai ned during excursions of a standard Brownian bridge (B-t(br), 0 less than o r equal to t less than or equal to 1) is described. The height M-j(br+) of the jth highest maximum over a positive excursion of the bridge has the sam e distribution as M-1(br+)/j, where the distribution of Mb(1)(br+) = Sup(0 less than or equal tot less than or equal to1)B(t)(br) is given by Levy's f ormula P(M-1(br+) > x) = e(-2x2). The probability density of the height M-j (br+) of the jth highest maximum of excursions of the reflecting Brownian b ridge (/B-t(br)/, 0 less than or equal to t less than or equal to 1) is giv en by a modification of the known theta -function series for the density of M-1(br) = sup(0 less than or equal tot less than or equal to1) /B-t(br)/ T hese results are obtained from a more general description of the distributi on of ranked values of a homogeneous functional of excursions of the standa rdized bridge of a self-similar recurrent Markov process.