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Hitting Times and the Running Maximum of Markovian Growth-Collapse Processes

Part of: Special processes Limit theorems Markov processes

Published online by Cambridge University Press:  14 July 2016

Andreas Löpker  and
Wolfgang Stadje
Andreas Löpker*
Affiliation:
Eindhoven University of Technology and EURANDOM
Wolfgang Stadje*
Affiliation:
University of Osnabrück
*
∗Current address: Department of Economics and Social Sciences, Helmut Schmidt University, PO Box 700822, 22008 Hamburg, Germany. Email address: lopker@hsu-hh.de
∗∗Postal address: Department of Mathematics and Computer Science, University of Osnabrück, 49069 Osnabrück, Germany. Email address: wolfgang@mathematik.uos.de
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Abstract

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We consider the level hitting times τy = inf{t ≥ 0 | Xt = y} and the running maximum process Mt = sup{Xs | 0 ≤ s ≤ t} of a growth-collapse process (Xt)t≥0, defined as a [0, ∞)-valued Markov process that grows linearly between random ‘collapse’ times at which downward jumps with state-dependent distributions occur. We show how the moments and the Laplace transform of τy can be determined in terms of the extended generator of Xt and give a power series expansion of the reciprocal of Ee−sτy. We prove asymptotic results for τy and Mt: for example, if m(y) = Eτy is of rapid variation then Mt / m-1(t) →w 1 as t → ∞, where m-1 is the inverse function of m, while if m(y) is of regular variation with index a ∈ (0, ∞) and Xt is ergodic, then Mt / m-1(t) converges weakly to a Fréchet distribution with exponent a. In several special cases we provide explicit formulae.

Keywords

Growth-collapse processpiecewise deterministic Markov processhitting timerunning maximumasymptotic behaviorregular variationseparable jump measure

MSC classification

Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) 60J25: Continuous-time Markov processes on general state spaces 60J75: Jump processes 60F05: Central limit and other weak theorems
Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 2 , June 2011 , pp. 295 - 312
Copyright
Copyright © Applied Probability Trust 2011 

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