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On a particular class of self-decomposable random variables : the durations of Bessel excursions straddling independent exponential times

100%

Bertoin J. , Fujita T. , Roynette B. , Yor M.

Probability and Mathematical Statistics

2006

tom Vol. 26, Fasc. 2

315--366

The distributional properties of the duration of a recurrent Bessel process straddling an independent exponential time are studied in detail. Although our study may be considered as a particular case of Winkel’s in [25], the infinite divisibility structure of these Bessel durations is particularly rich and we develop algebraic properties for a family of random variables arising from the Lévy measures of these durations.

On Paul Lévy’s arc sine law and Shiga-Watanabe’s time inversion result

100%

Graversen S. E. , Vuolle-Apiala J.

Probability and Mathematical Statistics

2000

tom Vol. 20, Fasc. 1

63--73

Let ((Xt), P) be a symmetric real-valued H-self-similar diffusion starting at 0. We characterize the distributions of At, the time spent on (0, ∞) before time t, and gt, the time of the last visit to 0 before t. This gives a simple new proof to well-known results in cluding P. Lévy’s arc sine law for Brownian motion and Brownian bridge and similar results for symmetrized Bessel processes. Our focus is more on simplicity of proofs than on novelty of results. Section 3 contains a generalization of T. Shiga’s and S. Watanabe’s theorem on time inversion for Bessel processes. We show that their result holds also for symmetrized Bessel processes.

A family of generalized gamma convoluted variables

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Roynette R. , Vallois P. , Yor M.

tom Vol. 29, Fasc. 2

181--204

This paper consists of three parts: in the first part, we describe a family of generalized gamma convoluted (abbreviated as GGC) variables. In the second part, we use this description to prove that several r.v.’s, related to the length of excursions away from 0 for a recurrent linear diffusion on R+, are GGC. Finally, in the third part, we apply our results to the case of Bessel processes with dimension d = 2(1 − α), where 0 < d < 2 or 0 < α< 1.

A warning about an independence property related to random Brownian scaling

75%

Fujita T. , Yor M.

Probability and Mathematical Statistics

2007

tom Vol. 27, Fasc. 1

105--108

In this note, which develops a part of our paper [2], we consider independence properties between Brownian motion, after Brownian scaling on a random interval (a, b), and the length (b—a) of the interval. We indicate three examples for which the Brownian scaled process is independent of the corresponding length. On the other hand, we discuss a case where this independence property does not hold and investigate further results for that example.