Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-24T17:11:48.476Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Beran, J. (1994). Statistics for Long-Memory Processes. Chapman and Hall, London.Google Scholar
Daley, D. J. (1999). The Hurst index of long-range dependent renewal processes. Ann. Prob. 27, 2035–2041.CrossRefGoogle Scholar
Daley, D. J. (2000). The moment index of minima. Unpublished manuscript.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Daley, D. J. and Vesilo, R. (1997). Long range dependence of point processes, with queueing examples. Stoch. Proc. Appl. 70, 265–282.CrossRefGoogle Scholar
Feller, W. (1966). An Introduction to Probability Theory and its Applications, Vol. 2. John Wiley, New York.Google Scholar
Feller, W. (1968). An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edn. John Wiley, New York.Google Scholar
Kulik, R. and Szekli, R. (2000). Long-range dependent point processes and their Palm–Khinchin distribution. Unpublished manuscript.Google Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.CrossRefGoogle Scholar
Sgibnev, M. S. (1981). Renewal theorem in the case of infinite variance. Siberian Math. J. 22, 787–796.Google Scholar
Täcklind, S., (1944). Elementare Behandlung vom Erneuerungsproblem für den stationären Fall. Skand. Akt. 27, 1–15.Google Scholar
Teugels, J. L. (1968). Renewal theorems when the first or the second moment is infinite. Ann. Math. Statist. 39, 1210–1219.CrossRefGoogle Scholar