Statistics and Applied Probability: A Tribute to Jeffrey J. HunterView this Special Issue
Research Article | Open Access
D. J. Daley, T. Rolski, R. Vesilo, "Long-Range Dependence in a Cox Process Directed by a Markov Renewal Process", Advances in Decision Sciences, vol. 2007, Article ID 083852, 15 pages, 2007. https://doi.org/10.1155/2007/83852
Long-Range Dependence in a Cox Process Directed by a Markov Renewal Process
A Cox process directed by a stationary random measure has second moment , where by stationarity , so long-range dependence (LRD) properties of coincide with LRD properties of the random measure . When is determined by a density that depends on rate parameters and the current state of an -valued stationary irreducible Markov renewal process (MRP) for some countable state space (so is a stationary semi-Markov process on ), the random measure is LRD if and only if each (and then by irreducibility, every) generic return time of the process for entries to state has infinite second moment, for which a necessary and sufficient condition when is finite is that at least one generic holding time in state , with distribution function (DF)\ , say, has infinite second moment (a simple example shows that this condition is not necessary when is countably infinite). Then, has the same Hurst index as the MRP that counts the jumps of , while as , for finite , , , where , , , is the stationary distribution for the embedded jump process of the MRP, , and where is the DF and the mean of the generic return time of the MRP between successive entries to the state . These two variances are of similar order for only when each converges to some -valued constant, say, , for .
- D. J. Daley, “Long-range dependence in a Cox process directed by an alternating renewal process,” submitted for publication.
- J. J. Hunter, “On the moments of Markov renewal processes,” Advances in Applied Probability, vol. 1, pp. 188–210, 1969.
- M. S. Sgibnev, “On the renewal theorem in the case of infinite variance,” Sibirskiĭ Matematicheskiĭ Zhurnal, vol. 22, no. 5, pp. 178–189, 224, 1981, Translation in Siberian Mathematical Journal, vol. 22, no. 5, pp. 787–796, 1981.
- D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes. Vol. I: Elementary Theory and Methods, Probability and Its Applications, Springer, New York, NY, USA, 2nd edition, 2003.
- E. Cinlar, Introduction to Stochastic Processes, Prentice-Hall, Englewood Cliffs, NJ, USA, 1975.
- V. G. Kulkarni, Modeling and Analysis of Stochastic Systems, Texts in Statistical Science Series, Chapman & Hall, London, UK, 1995.
- M. S. Sgibnev, “An infinite variance solidarity theorem for Markov renewal functions,” Journal of Applied Probability, vol. 33, no. 2, pp. 434–438, 1996.
Copyright © 2007 D. J. Daley et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.