The law of the maximum of a Bessel bridge. (English) Zbl 0943.60084

Pitman, Jim; Yor, Marc

Electron. J. Probab. 4, Paper No. 15, 34 p. (1999).

Let \(M_\delta\) be the maximum of a standard Bessel bridge of dimension \(\delta\). A series formula for \(P(M_\delta\leq a)\) due to Gikhman and Kiefer for \(\delta= 1,2,\dots\) is shown to be valid for all real \(\delta>0\). Various other characterizations of the distribution of \(M_\delta\) are given, including formulae for its Mellin transform, which is an entire function. The asymptotic distribution of \(M_\delta\) is described both as \(\delta\to \infty\) and as \(\delta\downarrow \infty\).

Reviewer: O.Brockhaus (London)

MSC:

60J65	Brownian motion
60J60	Diffusion processes
33C10	Bessel and Airy functions, cylinder functions, \({}_0F_1\)

Keywords:

Brownian brigde; Brownian excursion; Brownian scaling; local time; Bessel process; zeros of Bessel functions; Riemann zeta function