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

Part of: Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

D. J. Daley ,
T. Rolski  and
Rein Vesilo
D. J. Daley*
Affiliation:
The Australian National University
T. Rolski*
Affiliation:
Wrocław University
Rein Vesilo*
Affiliation:
Macquarie University
*
∗ Postal address: School of Mathematical Sciences, The Australian National University, Canberra ACT 0200, Australia. Email address: daryl@maths.anu.edu.au
∗∗ Postal address: Mathematics Institute, Wrocław University, Pl. Grunwaldski 2-4, 50 384 Wrocław, Poland.
∗∗∗ Postal address: Department of Electronics, Macquarie University, Sydney, NSW 2109, Australia.
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Abstract

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(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 ½. 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) < ∞.

Keywords

Ergodic point processlong-range dependencePalm-Khinchin distributionHurst indexmoment indexWold processdistribution tails

MSC classification

Primary: 60G55: Point processes
Secondary: 62M10: Time series, auto-correlation, regression, etc.
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 4 , December 2000 , pp. 1051 - 1063
Copyright
Copyright © Applied Probability Trust 2000 

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