Journal of Applied Mathematics and Decision Sciences
Volume 2007 (2007), Article ID 83852, 15 pages
Research Article

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

D. J. Daley,1 T. Rolski,2 and R. Vesilo3

1Centre for Mathematics and its Applications, The Australian National University, ACT 0200, Australia
2Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, Wrocław 50384, Poland
3Department 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.


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.