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- W1594513799 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." @default.
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- W1594513799 date "2001-02-01" @default.
- W1594513799 modified "2023-09-25" @default.
- W1594513799 title "On the distribution of ranked heights of excursions of a Brownian bridge" @default.
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- W1594513799 doi "https://doi.org/10.1214/aop/1008956334" @default.
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