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On the comparison of point processes

Published online by Cambridge University Press:  14 July 2016

Y. L. Deng
Y. L. Deng*
Affiliation:
Zhongshan University
*
∗ Postal address: Department of Mathematics, Zhongshan University, Guangzhou, China.
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Abstract

Several different orderings for the comparison of point processes have been introduced and their relationships discussed in Whitt [9], Daley [2] and Deng [4]. It is of some interest to know whether these orderings, in general, are preserved under various operations on point processes. Some results concerning limit operations were given in Deng [4].

In the present paper, we first further introduce some convex and concave orderings for counting processes, and survey the relationships among all orderings mentioned in [9], [4] and this paper. Then we focus our attention on the study of the conditions for the preservation of orderings under the operations of superposition, thinning, shift, and random change of time.

Keywords

COUNTING PROCESSESRENEWAL PROCESSESPOISSON PROCESSESORDERINGSUPERPOSITIONTHINNINGSHIFTSTOPPING TIMERANDOM CHANGE OF TIME

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 22 , Issue 2 , June 1985, pp. 300 - 313
Copyright
Copyright © Applied Probability Trust 1985 

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Footnotes

Work done while the author was on leave at The Department of Statistics (IAS), The Australian National University.

References

[1] Brémaud, P. (1981) Point Processes and Queues, Martingale Dynamics. Springer-Verlag. New York.Google Scholar
[2] Daley, D. J. (1981) Distances of random variables and point processes. Institute of Statistics Mimeo Series #1331, Department of Statistics, University of North Carolina, Chapel Hill.Google Scholar
[3] Dellacherie, C. and Meyer, P. A. (1980) Probabilités et Potentiel, Théorie des Martingales, 2ème edn. Hermann, Paris.Google Scholar
[4] Deng, Y. L. (1985) Comparison of inhomogeneous Poisson processes. Chinese Ann. Math. 6B, 83–96.Google Scholar
[5] Kamae, T., Krengel, U. and O'brien, G. L. (1977) Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899–912.Google Scholar
[6] Mertens, J. F. (1972) Théorie des processus stochastiques généraux applications aux surmartingales. Z. Wahrscheinlichkeitsth. 22, 45–68.Google Scholar
[7] Renyi, A. (1956) A characterization of the Poisson process (in Hungarian). Magy. Tud. Akad. Mat. Kut. Int. Közl. L, 519–527.Google Scholar
[8] Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. Wiley, New York.Google Scholar
[9] Whitt, W. (1981) Comparing counting processes and queues. Adv. Appl. Prob. 13, 207–220.Google Scholar