Long-range dependent point processes and their Palm-Khinchin distributions

For a stationary long-range dependent point process N(·) with Palm distribution P⁰, the Hurst index H ≡ sup{h:lim sup_t→∞ t^-2h var N(0, t] = ∞} is related to the moment index κ ≡ sup{k:E⁰(T^k) < ∞} of a generic stationary interval T between points (E⁰ denotes expectation with respect to P⁰) by 2H + κ ≥ 3, it being known that equality holds for a stationary renewal process. Thus, a stationary point process for which κ < 2 is necessarily long-range dependent with Hurst index greater than 1/2 . An extended example of a Wold process shows that a stationary point process can be both long-range count dependent and long-range interval dependent and have finite mean square interval length, i.e., E⁰(T²) < ∞.