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

August, 1999
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Jim Pitman and Marc Yor
Annals of Probability vol. 29, pages 362-384 (2001)

The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian bridge is described. The height of the $j$th highest maximum $M_j$ over a positive excursion of the bridge has the same distribution as $M_1/j$, where the distribution of $M_1$ is given by L\'evy's formula $P( M_1 > x ) = e^{-2x^2}$. The probability density of the height of the $j$th highest maximum of excursions of the reflecting Brownian bridge is given by a modification of the known $\theta$-function series for the density of the maximum absolute value of the bridge. 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.

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