  We propose a path transformation which applied to a cyclically exchangeable
increment process conditions its minimum to belong to a given interval.
  This path transformation is then applied to processes with start and end at
zero. It is seen that, under simple conditions, the weak limit as epsilon tends
to zero of the process conditioned on remaining above minus epsilon exists and
has the law of the Vervaat transformation of the process.
  We examine the consequences of this path transformation on processes with
exchangeable increments, L\'evy bridges, and the Brownian bridge.
