Shifting processes with cyclically exchangeable increments at random

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 0. It is seen that, under simple conditions, the weak limit as ε → 0 of the process conditioned on remaining above −ε 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, Lévy bridges, and the Brownian bridge. 1. Motivation: Weak convergence of conditioned Brownian bridge and the Vervaat transformation Excursion theory for Markov processes has proved to be an useful tool since its inception in [Itô72] (although some ideas date back to [Lév39]). This is true both in theoretical and applied investigations (see for example [GY93], [Ald97], [PY07], [Wat10], [LG10], [Wer10], [YY13]). Especially fruitful has been the application of excursion theory in the case of Brownian motion and other Lévy processes, where it lies at the foundation of the so called fluctuation theory aimed at studying their extremes (cf. [Ber96], [Kyp06], [Don07]). Brownian motion is undoubtedly one of the most tractable Lévy processes. It is therefore not a surprise that excursion theory takes a very explicit form for this process. In particular, we have the following interpretation of the normalized excursions above 0 of Brownian motion as Brownian motion conditioned to start at 0, end at 0 at a given time t (here we consider t = 1), and remain positive throughout (0, t). We recall that the Brownian bridge is a version of Brownian motion conditioned to start at 0 and end at 0 at time 1. Theorem 1 ([DIM77] and [Ver79]). Let X be a Brownian bridge from 0 to 0 of length 1. Then, the law of X conditioned to remain above −ε converges weakly as ε → 0 toward law of the normalized Brownian excursion. Furthermore, the weak limit can also be constructed as follows: if ρ is the unique instant at which X attains its minimum, then the weak limit has the same law as θρ(X)t = { m+Xρ+t 0 ≤ t ≤ 1− ρ m+Xt−(1−ρ) 1− ρ ≤ t ≤ 1 . 2000 Mathematics Subject Classification. 60G09, 60F17, 60G17, 60J65.