Title: Path decompositions of a Brownian bridge related to the ratio of its maximum and amplitude
Author: Jim Pitman and Marc Yor
Date: August 1998 
Abstract:
We give two new proofs of Csaki's formula for the law of the ratio 1-Q of
the maximum relative to the amplitude (i.e. the maximum minus minimum)
for a standard Brownian bridge. 
The second of these proofs is based on an absolute continuity relation
between the law of the Brownian bridge restricted to the event (Q < v)
and the law of a process obtained by a Brownian scaling operation 
after back-to back joining of two independent three-dimensional Bessel 
processes, each started at v and run until it first hits 1.
Variants of this construction and some properties of the joint
law of Q and the amplitude are described.