On excursions inside an excursion.

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 j th highest maximum over an excursion above x and the absolute value of the k th lowest minimum over an excursion below x , including j = 1 and k = 2 , are computed in this paper. [ABSTRACT FROM AUTHOR]