An extension of Vervaat's transformation and its consequences (Q5918056)

\textit{W. Vervaat} [Ann. Probab. 7, 143-149 (1979; Zbl 0392.60058)] pointed out the following connection between the standard Brownian bridge \(b=(b_t, 0\leq t \leq 1)\) and the normalized Brownian excursion. If \(m_0\) is the a.s. unique instant at which \(b\) reaches its overall minimum, then the process obtained from \(b\) by exchanging the pre-\(m_0\) and the post-\(m_0\) paths, is a normalized Brownian excursion which is independent of \(m_0\). It is also well-known that \(m_0\) has the uniform distribution. The purpose of the paper under review is to extend Vervaat's result by presenting a related path-construction of a Brownian bridge conditioned to spend a given amount \(s\geq 0\) of time below \(0\) (for \(s=0\), this conditioned Brownian bridge can be identified as a Brownian excursion). Some applications of this construction are discussed.