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Theory of Probability and Mathematical Statistics

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The cumulant representation of the Lundberg root in the case of semicontinuous processes

Author: D. V. Gusak
Translated by: Oleg Klesov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 82 (2010).
Journal: Theor. Probability and Math. Statist. 82 (2011), 1-10
MSC (2010): Primary 60G50; Secondary 60K10
Published electronically: August 2, 2011
MathSciNet review: 2790479
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Abstract | References | Similar Articles | Additional Information

Abstract: For the case of homogeneous processes $ \xi (t)$, $ \xi (0)=0$, $ t\geq 0$, with independent increments and negative jumps, it is proved in A. V. Skorokhod, Random Processes with Independent Increments, Nauka, Moscow, 1964 that the functional

$\displaystyle \tau ^+(x)=\inf\left\{t\geq 0\colon \xi (t)>x\right\}, \qquad x\geq 0, $

is a nondecreasing process with independent increments with respect to $ x$, and its moment generating function is expressed via the cumulant that satisfies the corresponding Lundberg equation. The corresponding representations of this cumulant are specified and its Lévy characteristics (namely, $ \gamma $ and Lévy's integral measure $ N(x)$) are evaluated by using some of the results of the author's work of 2007 for the processes under consideration.

References [Enhancements On Off] (What's this?)

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Additional Information

D. V. Gusak
Affiliation: Institute for Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivs’ka Street 3, Kyiv–4 252601, Ukraine

Keywords: Compound semicontinuous Poisson processes, semicontinuous homogeneous processes with independent increments, Lundberg’s root
Received by editor(s): October 19, 2009
Published electronically: August 2, 2011
Article copyright: © Copyright 2011 American Mathematical Society