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Sufficient conditions for long-range count dependence of stationary point processes on the real line

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Rafał Kulik  and
Ryszard Szekli
Rafał Kulik*
Affiliation:
Wrocław University
Ryszard Szekli*
Affiliation:
Wrocław University
*
∗ Postal address: Mathematical Institute, Wrocław University, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
∗ Postal address: Mathematical Institute, Wrocław University, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
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Abstract

Daley and Vesilo (1997) introduced long-range count dependence (LRcD) for stationary point processes on the real line as a natural augmentation of the classical long-range dependence of the corresponding interpoint sequence. They studied LRcD for some renewal processes and some output processes of queueing systems, continuing the previous research on such processes of Daley (1968), (1975). Subsequently, Daley (1999) showed that a necessary and sufficient condition for a stationary renewal process to be LRcD is that under its Palm measure the generic lifetime distribution has infinite second moment. We show that point processes dominating, in a sense of stochastic ordering, LRcD point processes are LRcD, and as a corollary we obtain that for arbitrary stationary point processes with finite intensity a sufficient condition for LRcD is that under Palm measure the interpoint distances are positively dependent (associated) with infinite second moment. We give many examples of LRcD point processes, among them exchangeable, cluster, moving average, Wold, semi-Markov processes and some examples of LRcD point processes with finite second Palm moment of interpoint distances. These examples show that, in general, the condition of infiniteness of the second moment is not necessary for LRcD. It is an open question whether the infinite second Palm moment of interpoint distances suffices to make a stationary point process LRcD.

Keywords

long-range count dependencePalm measuresemi-Markov processes

MSC classification

Primary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 2 , June 2001 , pp. 570 - 581
Copyright
Copyright © Applied Probability Trust 2001 

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Footnotes

Work supported by Alexander von Humboldt Fellowship, and by the KBN Grant 2P03A04915.

References

Baccelli, F. and Brémaud, P. (1994). Elements of Queueing Theory. Palm–Martingale Calculus and Stochastic Recurrences. Springer, New York.Google Scholar
Bartlett, M. S. (1963). The spectral analysis of point processes. J. R. Statist. Soc. B 25, 264–296.Google Scholar
Beran, J. (1994). Statistics for Long-Memory Processes. Chapman and Hall, London.Google Scholar
Daley, D. J. (1968). The correlation structure of the output process of some single server queueing systems. Ann. Math. Statist. 39, 1007–1019.CrossRefGoogle Scholar
Daley, D. J. (1971). Weakly stationary point processes and random measures. J. R. Statist. Soc. B 33, 406–428.Google Scholar
Daley, D. J. (1975). Further second-order properties of certain single-server queueing systems. Stoch. Proc. Appl. 3, 185–191.Google Scholar
Daley, D. J. (1999). The Hurst Index of Long Range Dependent Renewal Processes. Ann. Prob. 27, 2035–2041.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.Google Scholar
Daley, D. J., and Vesilo, R. (2000). Long range dependence of inputs and outputs of some classical queues. In Analysis of Communication Networks: Call Centres, Traffic and Performance (Fields Inst. Commun. 28), eds McDonald, D. R. and Turner, S. R. E. American Mathematical Society, Providence, RI, pp. 179–186.Google Scholar
Daley, D. J., Rolski, T., and Vesilo, R. (2000). Long-range dependent point processes and their Palm–Khinchin distributions. Adv. Appl. Prob. 32, 1051–1063.CrossRefGoogle Scholar
Hewitt, E., and Savage, L. J. (1955). Symmetric measures on Cartesian products. Trans. Am. Math. Soc. 80, 470–501.Google Scholar
Karlin, S., and Rinott, Y. (1980). Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions. J. Multivar. Anal. 10, 467–498.CrossRefGoogle Scholar
Lowen, S. B., and Teich, M. C. (1991). Doubly stochastic Poisson point processes driven by fractal shot noise. Phys. Rev. 43, 4192–4215.CrossRefGoogle ScholarPubMed
Meester, L. E., and Shanthikumar, J. G. (1993). Regularity of stochastic processes: a theory based on directional convexity. Prob. Eng. Inf. Sci. 7, 343–360.Google Scholar
Müller, A., and Pflug, G. (2001). Asymptotic ruin probabilities for dependent claims. To appear in Insurance: Math. Econom.Google Scholar
Nelsen, R. B. (1997). Dependence and order in families of Archimedean copulas. J. Multivar. Anal. 60, 111–122.Google Scholar
Resnick, S., and Samorodnitsky, G. (1997). Performance decay in a single-server exponential queueing model with long range dependence. Operat. Res. 45, 235–243.Google Scholar
Scarsini, M. (1998). Multivariate convex orderings, dependence and stochastic equality. J. Appl. Prob. 35, 93–103.CrossRefGoogle Scholar
Shaked, M. (1977). A concept of positive dependence for exchangeable random variables. Ann. Statist. 5, 505–515.Google Scholar
Shaked, M., and Shanthikumar, J. G. (1997). Supermodular stochastic orders and positive dependence of random vectors. J. Multivar. Anal. 61, 86–101.Google Scholar
Szekli, R. (1995). Stochastic Ordering and Dependence in Applied Probability (Lecture Notes Statist. 97). Springer, Berlin.Google Scholar