Advances in Decision Sciences

Advances in Decision Sciences / 2007 / Article
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Statistics and Applied Probability: A Tribute to Jeffrey J. Hunter

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Volume 2007 |Article ID 083852 | https://doi.org/10.1155/2007/83852

D. J. Daley, T. Rolski, R. Vesilo, "Long-Range Dependence in a Cox Process Directed by a Markov Renewal Process", Advances in Decision Sciences, vol. 2007, Article ID 083852, 15 pages, 2007. https://doi.org/10.1155/2007/83852

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

Academic Editor: Paul Cowpertwait
Received19 Jun 2007
Accepted08 Aug 2007
Published22 Nov 2007

Abstract

A Cox process NCox directed by a stationary random measure ξ has second moment var NCox(0,t]=E(ξ(0,t])+var ξ(0,t], where by stationarity E(ξ(0,t])=(const.)t=E(NCox(0,t]), so long-range dependence (LRD) properties of NCox coincide with LRD properties of the random measure ξ. When ξ(A)=AνJ(u)du is determined by a density that depends on rate parameters νi(i𝕏) and the current state J() of an 𝕏-valued stationary irreducible Markov renewal process (MRP) for some countable state space 𝕏 (so J(t) is a stationary semi-Markov process on 𝕏), the random measure is LRD if and only if each (and then by irreducibility, every) generic return time Yjj(jX) of the process for entries to state j has infinite second moment, for which a necessary and sufficient condition when 𝕏 is finite is that at least one generic holding time Xj in state j, with distribution function (DF)\ Hj, say, has infinite second moment (a simple example shows that this condition is not necessary when 𝕏 is countably infinite). Then, NCox has the same Hurst index as the MRP NMRP that counts the jumps of J(), while as t, for finite 𝕏, var NMRP(0,t]2λ20t𝒢(u)du, var NCox(0,t]20ti𝕏(νiν¯)2ϖii(t)du, where ν¯=iϖiνi=E[ξ(0,1]], ϖj=Pr{J(t)=j},1/λ=jpˇjμj, μj=E(Xj), {pˇj} is the stationary distribution for the embedded jump process of the MRP, j(t)=μi10min(u,t)[1Hj(u)]du, and 𝒢(t)0tmin(u,t)[1Gjj(u)]du/mjjiϖii(t) where Gjj is the DF and mjj the mean of the generic return time Yjj of the MRP between successive entries to the state j. These two variances are of similar order for t only when each i(t)/𝒢(t) converges to some [0,]-valued constant, say, γi, for t.

References

  1. D. J. Daley, “Long-range dependence in a Cox process directed by an alternating renewal process,” submitted for publication. View at: Google Scholar
  2. J. J. Hunter, “On the moments of Markov renewal processes,” Advances in Applied Probability, vol. 1, pp. 188–210, 1969. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  3. M. S. Sgibnev, “On the renewal theorem in the case of infinite variance,” Sibirskiĭ Matematicheskiĭ Zhurnal, vol. 22, no. 5, pp. 178–189, 224, 1981, Translation in Siberian Mathematical Journal, vol. 22, no. 5, pp. 787–796, 1981. View at: Google Scholar | MathSciNet
  4. D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes. Vol. I: Elementary Theory and Methods, Probability and Its Applications, Springer, New York, NY, USA, 2nd edition, 2003. View at: Zentralblatt MATH | MathSciNet
  5. E. Cinlar, Introduction to Stochastic Processes, Prentice-Hall, Englewood Cliffs, NJ, USA, 1975.
  6. V. G. Kulkarni, Modeling and Analysis of Stochastic Systems, Texts in Statistical Science Series, Chapman & Hall, London, UK, 1995. View at: Zentralblatt MATH | MathSciNet
  7. M. S. Sgibnev, “An infinite variance solidarity theorem for Markov renewal functions,” Journal of Applied Probability, vol. 33, no. 2, pp. 434–438, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

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.


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