Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-22T12:00:20.001Z Has data issue: false hasContentIssue false
  • English
  • Français

UNIFORM ORDER STATISTICS PROPERTY AND [ell ]∞-SPHERICAL DENSITIES

Published online by Cambridge University Press:  01 July 2004

Moshe Shaked ,
Fabio Spizzichino  and
Florentina Suter
Moshe Shaked
Affiliation:
Department of Mathematics, University of Arizona, Tucson, AZ 85721, E-mail: [email protected]
Fabio Spizzichino
Affiliation:
Department of Mathematics, University “La Sapienza”, Piazzale Aldo Moro, 5, 00185 Rome, Italy
Florentina Suter
Affiliation:
Department of Mathematics, University of Bucharest, Academiei 14, 010014 Bucharest, Romania
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article, we observe that processes with the uniform order statistics property (UOSP) can be characterized by the condition that their first n epoch times have a joint [ell ]∞≤-spherical density, n ≥ 1. Some related results, and some further properties of [ell ]∞≤-spherical densities, are also given. We also extend some of the results regarding the UOSP to the more general (not necessarily uniform) order statistics property. Finally, we develop a theory of discrete-time discrete-state processes with the UOSP, where the need to consider multiple jumps, at a single time point, arises.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 18 , Issue 3 , July 2004 , pp. 275 - 297
Copyright
© 2004 Cambridge University Press

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

REFERENCES

Benczur, A. (1968). On sequences of equivalent events and the compound Poisson process. Studia Scientiarum Mathematicarum Hungarica 3: 451–458.Google Scholar
Berman, S.M. (1980). Stationarity, isotropy and sphericity in [ell ]p. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 54: 21–23.Google Scholar
Crump, K.S. (1975). On point processes having an order statistic structure. Sankhya A 37: 396–404.Google Scholar
Deffner, A. & Haeusler, E. (1985). A characterization of order statistic point processes that are mixed Poisson processes and mixed sample processes simultaneously. Journal of Applied Probability 22: 314–323.Google Scholar
Diaconis, P. & Freedman, D. (1987). A dozen de Finetti-style results in search of a theory. Annales de L'Institut Henri Poincaré: Probabilités et Statistiques 23 (Suppl. 2): 397–423.Google Scholar
Feller, W. (1968). An introduction to probability theory and its applications, Vol. I, 3rd ed. New York: Wiley.
Grandell, J. (1997). Mixed Poisson processes. London: Chapman & Hall.
Hayakawa, Y. (2000). A new characterization property of mixed Poisson processes via Berman's Theorem. Journal of Applied Probability 37: 261–268.Google Scholar
Huang, W.-J. & Shoung, J.-M. (1994). On a study of some properties of point processes. Sankhya A 56: 67–76.Google Scholar
Iglesias, Z.P., Matús, F., Pereira, C.A.B., & Tanaka, N.I. (2003). On finite sequences conditionally uniform given minima and maxima. Technical Report, PUC, Santiago, Chile.
Johnson, N.L., Kotz, S., & Balakrishnan, N. (1997). Discrete multivariate distributions. New York: Wiley.
Johnson, N.L., Kotz, S., & Kemp, A.W. (1992). Univariate discrete distributions, 2nd ed. New York: Wiley.
Kamps, U. (1995). A concept of generalized order statistics. Stuttgart: Teubner.
Neuts, M.F. & Resnick, S.I. (1971). On the times of births in a linear birthprocess. Journal of the Australian Mathematical Society 12: 473–475.Google Scholar
Puri, P. (1982). On the characterization of point processes with the order statistic property without moment condition. Journal of Applied Probability 19: 39–51.Google Scholar
Shaked, M., Spizzichino, F., & Suter, F. (2002). Nonhomogeneous birth processes and [ell ]∞-spherical densities, with applications in reliability theory. Probability in the Engineering and Informational Sciences 16: 271–288.Google Scholar
Spizzichino, F. (2001). Subjective probability models for lifetimes. London: Chapman & Hall/CRC.