Advances in Decision Sciences
Volume 2007 (2007), Article ID 83852, 15 pages
doi:10.1155/2007/83852

Long-range dependence in a cox process directed by a Markov renewal process

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(i∈𝕏) 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(j∈X) 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]∼2λ2∫0t𝒢(u)du, var NCox(0,t]∼2∫0t∑i∈𝕏(νi−ν¯)2ϖiℋi(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, ℋj(t)=μi−1∫0∞min(u,t)[1−Hj(u)]du, and 𝒢(t)∼∫0tmin(u,t)[1−Gjj(u)]du/mjj∼∑iϖiℋi(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 ℋi(t)/𝒢(t) converges to some [0,∞]-valued constant, say, γi, for t→∞.