On growth-collapse processes with stationary structure and their shot-noise counterparts
Offer Kella
Source: J. Appl. Probab.
Volume 46, Number 2
(2009), 363-371.
Abstract
In this paper we generalize existing results for the steady-state distribution of growth-collapse processes. We begin
with a stationary setup with some relatively general growth process and observe that, under certain expected
conditions, point- and time-stationary versions of the processes exist as well as a limiting distribution for these
processes which is independent of initial conditions and necessarily has the marginal distribution of the stationary
version. We then specialize to the cases where an independent and identically distributed (i.i.d.)\ structure holds and
where the growth process is a nondecreasing Lévy process, and in particular linear, and the times between collapses
form an i.i.d.\ sequence. Known results can be seen as special cases, for example, when the inter-collapse times form a
Poisson process or when the collapse ratio is deterministic. Finally, we comment on the relation between these
processes and shot-noise type processes, and observe that, under certain conditions, the steady-state distribution of
one may be directly inferred from the other.
Primary Subjects: 60K30
Secondary Subjects: 60G10, 91B70
Keywords: Growth-collapse process; risk process; shot-noise process
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jap/1245676093
Digital Object Identifier: doi:10.1239/jap/1245676093
Zentralblatt MATH identifier:
05578812
Mathematical Reviews number (MathSciNet):
MR2535819
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