Long-Range dependence in a cox process Directed by a Markov Renewal process

A Cox process N_Cox directed by a stationary random measure ξ has second moment var & N_Cox(0,t] = E(ξ(0,t]) + var ξ(0,t], where bystationarity E(ξ(0,t]) = (const.) t = E N_Cox(0,t]), so long-range dependence (LRD) properties of N_Cox coincide with LRD properties of the random measure ξ. When ξ(A) = ∫_Aν_J(u) du is determined by a density that depends on rate parameters ν_i (i ∈ χ) and the current state J(·) of an χ-valued stationary irreducible Markov renewal process (MRP) for some countable state space χ (so J(t) is a stationary semi-Markov process on χ), the random measure is LRD if and only if each (and then by irreducibility, every) generic return time Y_jj (j ∈ X) of the process for entries to state j has infinite second moment, for which a necessary and sufficient condition when χ is finite is that at least one generic holding time X_j in state j, with distribution function (DF) H_j, say, has infinite second moment (a simple example shows that this condition is not necessary when χ is countably infinite). Then, N_Cox has the same Hurst index as the MRP N_MRP MRP that counts the jumps of J(·), while as t → ∞, for finite χ, var N_MRP (0,t] ∼ 2λ² ∫₀^tG(u)du, var N_Cox(0,t] ∼ 2₀^t ∑_i∈(ν_i - ν̄)²ω_iℋ_i(t) du, where ν̄ = ∑_iΣ_iν_i = E[ξ(0,1]], ω_j = Pr{J(t) = j}, 1/λ = ∑_jp_jμ_j, μ_j = E(X_j), {p_j is the stationary distribution for the embedded jump process of the MRP, ℋ_j(t) = μ_i^-1 ∫₀^∞ min (u,t) [1 - H_j (u)]du, and G(t) ∼ ∫₀^t min (u,t)[1 - G_jj(u)]du/ m_jj ∼ ∑_iΣ_i ℋ_i (t) where G_jj is the DF and m_jj the mean of the generic return time Y_jj of the MRP between successive entries to the state j. These two variances are of similar order for t → ∞ only when each ℋ_i(t)/G(t) converges to some [0, ∞]-valued constant, say, γ_i, for t → ∞.