Skip to main content Accessibility help

On the comparison of point processes

  • Y. L. Deng (a1)


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.


Corresponding author

Postal address: Department of Mathematics, Zhongshan University, Guangzhou, China.


Hide All

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



Hide All
[1] Brémaud, P. (1981) Point Processes and Queues, Martingale Dynamics. Springer-Verlag. New York.
[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.
[3] Dellacherie, C. and Meyer, P. A. (1980) Probabilités et Potentiel, Théorie des Martingales, 2ème edn. Hermann, Paris.
[4] Deng, Y. L. (1985) Comparison of inhomogeneous Poisson processes. Chinese Ann. Math. 6B, 8396.
[5] Kamae, T., Krengel, U. and O'brien, G. L. (1977) Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899912.
[6] Mertens, J. F. (1972) Théorie des processus stochastiques généraux applications aux surmartingales. Z. Wahrscheinlichkeitsth. 22, 4568.
[7] Renyi, A. (1956) A characterization of the Poisson process (in Hungarian). Magy. Tud. Akad. Mat. Kut. Int. Közl. L, 519527.
[8] Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. Wiley, New York.
[9] Whitt, W. (1981) Comparing counting processes and queues. Adv. Appl. Prob. 13, 207220.


On the comparison of point processes

  • Y. L. Deng (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed