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

D. J. Daley*, T. Rolski, Rein Vesilo

*Corresponding author for this work

Research output: Contribution to journalArticle

12 Citations (Scopus)


For a stationary long-range dependent point process N(·) with Palm distribution P0, the Hurst index H ≡ sup{h:lim supt→∞ t-2h var N(0, t] = ∞} is related to the moment index κ ≡ sup{k:E0(Tk) < ∞} 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) < ∞.

Original languageEnglish
Pages (from-to)1051-1063
Number of pages13
JournalAdvances in Applied Probability
Issue number4
Publication statusPublished - Dec 2000

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