Shifting processes with cyclically exchangeable increments at random

TL;DR

This paper introduces a path transformation for cyclically exchangeable increment processes that conditions their minimum within a specified interval, with implications for processes like Lévy and Brownian bridges, and explores their weak limits.

Contribution

It presents a novel path transformation technique for cyclically exchangeable increment processes and analyzes its effects on various stochastic processes, including Lévy and Brownian bridges.

Findings

Weak limit of conditioned process exists as epsilon approaches zero

Transformation relates to Vervaat transformation of the process

Applicable to Lévy bridges and Brownian bridges

Abstract

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.