A Cox processNCoxdirected by a stationary random measureξhas secondmomentvar?NCox(0,t]=E(ξ(0,t])+var?ξ(0,t], where bystationarityE(ξ(0,t])=(const.)t=E(NCox(0,t]), so long-range dependence (LRD) properties ofNCoxcoincide with LRD properties of the random measureξ.Whenξ(A)=∫AνJ(u)duis determined by a density that dependson rate parametersνi(i∈??)and the current stateJ(?)of an??-valued stationary irreducible Markov renewal process (MRP) forsome countable state space??(soJ(t)is a stationary semi-Markovprocess on??), the random measure is LRD if and only if each (and thenby irreducibility, every) generic return timeYjj(j∈X)of theprocess for entries to statejhas infinite second moment, for which anecessary and sufficient condition when??is finite is that at leastone generic holding timeXjin statej, with distribution function (DF)Hj, say, has infinite second moment (a simple example shows that thiscondition is not necessary when??is countably infinite).Then,NCoxhas the same Hurst index as the MRPNMRPthat counts the jumpsofJ(?), while ast→∞, for finite??,var?NMRP(0,t]～2λ2∫0t??(u)du,var?NCox(0,t]～2∫0t∑i∈??(νi?νˉ)2?i?i(t)du,whereνˉ=∑i?iνi=E[ξ(0,1]],?j=Pr{J(t)=j},1/λ=∑jpˇjμj,μj=E(Xj),{pˇj}is the stationary distribution for the embedded jump processof the MRP,?j(t)=μi?1∫0∞min(u,t)[1?Hj(u)]du, and??(t)～∫0tmin(u,t)[1?Gjj(u)]du/mjj～∑i?i?i(t)whereGjjis theDF andmjjthe mean of the generic return timeYjjof the MRPbetween successiveentries to the statej. These two variances are of similar orderfort→∞only when each?i(t)/??(t)converges to some[0,∞]-valued constant, say,γi, fort→∞.
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