On excursions inside an excursion

May-Ru Chen and Ju-Yi Yen

Stochastic Processes and their Applications, 2021, vol. 138, issue C, 96-116

Abstract: The distribution of ranked heights of excursions of a Brownian bridge is given by Pitman and Yor (2001). In this work, we consider excursions of a Brownian excursion above a random level x, where x is the value of the excursion at an independent uniform time on [0,1]. We study the maximum heights of these excursions as Pitman and Yor did for excursions of a Brownian bridge. In particular, the probability functions and the moments of the sum of the jth highest maximum over an excursion above x and the absolute value of the kth lowest minimum over an excursion below x, including j=1 and k=2, are computed in this paper.

Keywords: Brownian excursion theory; Brownian bridge; Vervaat transform (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1016/j.spa.2021.04.010

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