Long-Range Dependence in a Cox Process Directed by a Markov Renewal ProcessReport as inadecuate




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Journal of Applied Mathematics and Decision Sciences - Volume 2007 2007, Article ID 83852, 15 pages

Research Article

Centre for Mathematics and its Applications, The Australian National University, ACT 0200, Australia

Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2-4, Wrocław 50384, Poland

Department of Electronics, Macquarie University, North Ryde, NSW 2109, Australia

Received 19 June 2007; Accepted 8 August 2007

Academic Editor: Paul Cowpertwait

Copyright © 2007 D. J. Daley et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A Cox process NCox directed by a stationary random measure ξ has secondmoment var NCox0,t=Eξ0,t+var ξ0,t, where bystationarity Eξ0,t=const.t=ENCox0,t, so long-range dependence LRD properties ofNCox coincide with LRD properties of the random measure ξ.When ξA=∫AνJudu is determined by a density that dependson rate parameters νii∈𝕏 and the current state J⋅of an 𝕏-valued stationary irreducible Markov renewal process MRP forsome countable state space 𝕏 so Jt is a stationary semi-Markovprocess on 𝕏, the random measure is LRD if and only if each and thenby irreducibility, every generic return time Yjjj∈X of theprocess for entries to state j has infinite second moment, for which anecessary and sufficient condition when 𝕏 is finite is that at leastone generic holding time Xj in state j, with distribution function DF\Hj, say, has infinite second moment a simple example shows that thiscondition is not necessary when 𝕏 is countably infinite.Then, NCox has the same Hurst index as the MRP NMRP that counts the jumpsof J⋅, while as t→∞, for finite 𝕏,var NMRP0,t∼2λ2∫0t𝒢udu,var NCox0,t∼2∫0t∑i∈𝕏νi−ν¯2ϖiℋitdu,whereν¯=∑iϖiνi=Eξ0,1,ϖj=Pr{Jt=j},1-λ=∑jpˇjμj,μj=EXj,{pˇj}is the stationary distribution for the embedded jump processof the MRP, ℋjt=μi−1∫0∞minu,t1−Hjudu, and𝒢t∼∫0tminu,t1−Gjjudu-mjj∼∑iϖiℋitwhere Gjj is theDF and mjj the mean of the generic return time Yjj of the MRPbetween successiveentries to the state j. These two variances are of similar orderfor t→∞ only when each ℋit-𝒢t converges to some0,∞-valued constant, say, γi, for t→∞.





Author: D. J. Daley, T. Rolski, and R. Vesilo

Source: https://www.hindawi.com/



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