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

• 成果类型: Article
• 署名作者: Pitman, J; Yor, M
• 署名单位: University of California System; University of California Berkeley; Sorbonne Universite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2001
• 页码: 361-384
• 关键词: FUNCTIONALS; LAWS

摘要

The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian bridge (B-t(br), 0 less than or 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 same 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 formula 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 bridge (/B-t(br)/, 0 less than or equal to t less than or equal to 1) is given 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)/ 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.