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

Pitman, Jim ; Yor, Marc

Ann. Probab., Tome 29 (2001) no. 1, p. 361-384 / Harvested from Project Euclid

Résumé

The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian bridge $(B^{br}_t, 0 \le t \le 1)$ is described. The height $M^{br +}_j$of the $j$th highest maximum over a positive excursion of the bridge has the same distribution as $M^{br +}_1 /j$, where the distribution of $M^{br +}_1 = \sup_{0 x) = e^{2x^{2}}$. The probability density of the height $M^{br}_j$ of the$j$th highest maximum of excursions of the reflecting Brownian bridge $(|B^{br}_t|, 0 \le t \le 1)$ is given by a modification of the known $\theta$-function series for the density of $M^{br}_1 = \sup_{0 < t 1} |B^{br}_t$. 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.

Détail

Publié le : 2001-02-14
Classification:  Brownian bridge,  Brownian excursion,  Brownian scaling,  local time,  self-similar recurrent Markov process,  Bessel process,  60J65