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One-dimensional distributions of subordinators with upper truncated Lévy measure, and applications

Part of: Limit theorems Distribution theory - Probability Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Shai Covo
Shai Covo*
Affiliation:
Bar Ilan University
*
∗ Postal address: Department of Mathematics, Bar Ilan University, 52900 Ramat-Gan, Israel. Email address: [email protected]
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Abstract

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Given a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with continuous Lévy measure ν, we derive a formula for the distribution function Fs (x; t) at time t of the associated subordinator whose Lévy measure is the restriction of ν to (0,s]. It will be expressed in terms of ν and the marginal distribution function F (⋅; t) of the original process. A generalization concerning an arbitrary truncation of ν will follow. Under certain conditions, an analogous formula will be obtained for the nth derivative, ∂nFs (x; t) ∂ xn. The requirement that ν is continuous is shown to have no intrinsic meaning. A number of interesting results involving the size ordered jumps of subordinators will be derived. An appropriate approximation for the small jumps of a gamma process will be considered, leading to a revisiting of the generalized Dickman distribution.

Keywords

SubordinatorLévy processcompound Poisson processtruncated Lévy measureordered jumpsgamma processsmall jumpsweak convergenceDickman distribution

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60E07: Infinitely divisible distributions; stable distributions 60F17: Functional limit theorems; invariance principles
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 2 , June 2009 , pp. 367 - 392
Copyright
Copyright © Applied Probability Trust 2009 

Footnotes

This paper is part of the author's PhD thesis, prepared at Bar Ilan University under the supervision of Professor E. Merzbach. This work was supported by the Doctoral Fellowship of Excellence, Bar Ilan University.

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