^{1}Centre for Mathematics and its Applications, The Australian National University, ACT 0200, Australia ^{2}Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, Wrocław 50384, Poland ^{3}Department of Electronics, Macquarie University, North Ryde, NSW 2109, Australia

A Cox process NCox directed by a stationary random measure ξ has second
moment varNCox(0,t]=E(ξ(0,t])+varξ(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,
varNMRP(0,t]∼2λ2∫0t𝒢(u)du,
varNCox(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→∞.