Displaying similar documents to “Small-time behavior of beta coalescents”

On suprema of Lévy processes and application in risk theory

Renming Song, Zoran Vondraček (2008)

Annales de l'I.H.P. Probabilités et statistiques

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Let =− where is a general one-dimensional Lévy process and an independent subordinator. Consider the times when a new supremum of is reached by a jump of the subordinator . We give a necessary and sufficient condition in order for such times to be discrete. When this is the case and drifts to −∞, we decompose the absolute supremum of at these times, and derive a Pollaczek–Hinchin-type formula for the distribution function of the supremum.

Behavior near the extinction time in self-similar fragmentations I : the stable case

Christina Goldschmidt, Bénédicte Haas (2010)

Annales de l'I.H.P. Probabilités et statistiques

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The stable fragmentation with index of self-similarity ∈[−1/2, 0) is derived by looking at the masses of the subtrees formed by discarding the parts of a (1+)−1–stable continuum random tree below height , for ≥0. We give a detailed limiting description of the distribution of such a fragmentation, ((), ≥0), as it approaches its time of extinction, . In particular, we show that 1/ ((−)+) converges in distribution as →0 to a non-trivial limit. In order to prove this, we go...

Refracted Lévy processes

A. E. Kyprianou, R. L. Loeffen (2010)

Annales de l'I.H.P. Probabilités et statistiques

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Motivated by classical considerations from risk theory, we investigate boundary crossing problems for refracted Lévy processes. The latter is a Lévy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More formally, whenever it exists, a refracted Lévy process is described by the unique strong solution to the stochastic differential equation d =− { ...

Strong law of large numbers for fragmentation processes

S. C. Harris, R. Knobloch, A. E. Kyprianou (2010)

Annales de l'I.H.P. Probabilités et statistiques

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In the spirit of a classical result for Crump–Mode–Jagers processes, we prove a strong law of large numbers for fragmentation processes. Specifically, for self-similar fragmentation processes, including homogenous processes, we prove the almost sure convergence of an empirical measure associated with the stopping line corresponding to first fragments of size strictly smaller than for 1≥>0.