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On Growth-Collapse Processes with Stationary Structure and Their Shot-Noise Counterparts

Part of: Stochastic processes Mathematical economics Special processes

Published online by Cambridge University Press:  14 July 2016

Offer Kella
Offer Kella*
Affiliation:
The Hebrew University of Jerusalem
*
∗Postal address: Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel. Email address: [email protected]
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Abstract

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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.

Keywords

Growth-collapse processrisk processshot-noise process

MSC classification

Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) 60G10: Stationary processes 91B70: Stochastic models
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 2 , June 2009 , pp. 363 - 371
Copyright
Copyright © Applied Probability Trust 2009 

Footnotes

Supported by grant number 964/06 from the Israel Science Foundation and the Vigevani Chair in Statistics.

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