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

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 Yj j ( 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] ∼ 2λ ∫ t 0 (u)du, var NCox(0, t] ∼ 2 ∫ t 0 ∑ i∈X(νi − ν)2 i i(t)du, where ν = ∑ i iνi = E[ξ(0,1]], j = Pr{J(t) = j},1/λ = ∑ j p̌ jμ j , μj = E(Xj), { p̌ j} is the stationary distribution for the embedded jump process of the MRP, j(t) = μ−1 i ∫ ∞ 0 min(u, t)[1 − Hj(u)]du, and (t) ∼ ∫ t 0min(u, t)[1 − Gj j(u)]du/mj j ∼ ∑ i i i(t) where Gj j is the DF and mj j the mean of the generic return time Yj j of the MRP between successive entries to the state j. These two variances are of similar order for t →∞ only when each i(t)/ (t) converges to some [0,∞]-valued constant, say, γi, for t→∞.