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

Abstract

The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian bridge $(B^{\mathrm{br}}_t, 0 \le t \le 1)$ is described. The height $M^{\mathrm{br} +}_j$of the $j$th highest maximum over a positive excursion of the bridge has the same distribution as $M^{\mathrm{br} +}_1 /j$, where the distribution of $M^{\mathrm{br} +}_1 = \sup_{0 \le t \le 1} B^{\mathrm{br}}_t$ is given by Lévy’s formula $P(M^{\mathrm{br} +}_1 > x) = e^{−2x^{2}}$. The probability density of the height $M^{\mathrm{br}}_j$ of the $j$th highest maximum of excursions of the reflecting Brownian bridge $(|B^{\mathrm{br}}_t|, 0 \le t \le 1)$ is given by a modification of the known $\theta$-function series for the density of $M^{\mathrm{br}}_1 = \sup_{0 \le t \le 1} |B^{\mathrm{br}}_t|$. 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.