## Abstract

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) = ∫_{A}ν_{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] ∼ 2λ^{2} ∫_{0}^{t}G(u)du, var N_{Cox}(0,t] ∼ 2_{0}^{t} ∑_{i∈}(ν_{i} - ν̄)^{2}ω_{i}ℋ_{i}(t) du, where ν̄ = ∑_{i}Σ_{i}ν_{i} = E[ξ(0,1]], ω_{j} = Pr{J(t) = j}, 1/λ = ∑_{j}p_{j}μ_{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} ∼ ∑_{i}Σ_{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 → ∞.

Original language | English |
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Article number | 83852 |

Pages (from-to) | 1-15 |

Number of pages | 15 |

Journal | Journal of Applied Mathematics and Decision Sciences |

Volume | 2007 |

DOIs | |

Publication status | Published - 2007 |