Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T19:38:56.774Z Has data issue: false hasContentIssue false
  • English
  • Français

Poisson process via martingale and related characteristics

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Jacek Wesołowski
Jacek Wesołowski*
Affiliation:
Warsaw University of Technology
*
∗Postal address: Mathematical Institute, Warsaw University of Technology, Plac Politechniki 1, 00–661 Warsaw, Poland. Email address: wesolo@alpha.im.pw.edu.pl.
Get access Rights & Permissions [Opens in a new window]

Abstract

The classical martingale characterizations of the Poisson process were obtained for point process or purely discontinuous martingale i.e. under additional assumptions on properties of trajectories. Here our aim is to search for related characterizations without relying on properties of trajectories. Except for a new martingale characterization, results based on conditional moments jointly involving the past and the nearest future are presented.

Keywords

Poisson processmartingalesconditional variancecharacterizationsconditioning given the futureconditioning given the past

MSC classification

Primary: 60E99: None of the above, but in this section
Type
Short Communications
Information
Journal of Applied Probability , Volume 36 , Issue 3 , September 1999 , pp. 919 - 926
Copyright
Copyright © Applied Probability Trust 1999 

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

Beśka, M., Kłopotowski, A. and Słomiński, L. (1982). Limit theorems for random sums of dependent d-dimensional random vectors. Z. Wahrscheinlichkeitsth. 61, 43–57.Google Scholar
Brémaud, P. (1981). Point Processes and Queues. Springer, New York.Google Scholar
Bryc, W. (1985). Some remarks on random vectors with nice enough behaviour of conditional moments. Bull. Pol. Acad. Sci. Math. 33, 677–684.Google Scholar
Bryc, W. (1987). A characterization of the Poisson process by conditional moments. Stochastics 20, 17–26.CrossRefGoogle Scholar
Bryc, W. (1995). The Normal Distribution. Characterizations with Applications. Springer, New York.Google Scholar
Bryc, W. (1998). Stationary random fields with linear regression. Preprint. 1–12.Google Scholar
Çinlar, E., and Jagers, P. (1973). Two mean values which characterize the Poisson process. J. Appl. Prob. 10, 678–681.Google Scholar
Elliott, R. J. (1982). Stochastic Calculus and Applications. Springer, New York.Google Scholar
Fang, Z. (1991). Characterization of nonhomogeneous Poisson process via moment conditions. Statist. Prob. Lett. 12, 83–90.CrossRefGoogle Scholar
Gupta, P. L., and Gupta, R. C. (1986). A characterization of the Poisson process. J. Appl. Prob. 23, 233–235.Google Scholar
Holmes, P. L. (1974). Another characterization of the Poisson process. Sankhyā A 36, 449–450.Google Scholar
Ivanoff, B. G. (1985). Poisson convergence for point processes on the plane. J. Austral. Math. Soc. (Ser. A) 39, 253–269.CrossRefGoogle Scholar
Ivanoff, B. G., and Merzbach, E. (1993). A martingale characterization of the set-indexed Poisson Process. Preprint. Theor. Prob. 9, 903–913.Google Scholar
Li, S. H., Huang, W. J., and Huang, M. N. L. (1994). A characterization of the Poisson process as a renewal process by conditional moments. Ann. Inst. Statist. Math. 46, 351–360.Google Scholar
Merzbach, E., and Nualart, D. (1986). A characterization of the spatial Poisson process and changing time. Ann. Prob. 14, 1380–1390.Google Scholar
Plucińska, A. (1983). On a stochastic process determined by the conditional expectation and the conditional variance. Stochastics 10, 115–129.CrossRefGoogle Scholar
Prakasa Rao, B. L. S. (1998). Characterization and identifiability for stochastic processes. In Handbook of Statistics, eds Rao, C. R. and Shanbhag, D. N. To appear.Google Scholar
Watanabe, S. (1964). On discontinuous additive functionals and Lévy measures of a Markov process. Japan J. Math. 34, 53–70.Google Scholar
Wesołowski, J. (1984). A characterization of a Gaussian process based on properties of conditional moments. Demonstratio Math. 17, 795–808.Google Scholar
Wesołowski, J. (1988). A remark on a characterization of the Poisson process. Demonstratio Math. 21, 555–557.Google Scholar
Wesołowski, J. (1989). A characterization of the Gamma process by conditional moments. Metrika 36, 299–309.Google Scholar
Wesołowski, J. (1990a). A martingale characterization of the Poisson process. Bull. Pol. Acad. Sci. Math. 38, 49–53.Google Scholar
Wesołowski, J. (1990b). A martingale characterization of the Wiener process. Statist. Prob. Lett. 10, 213–215 ((1994) Erratum 19, 167).Google Scholar
Wesołowski, J. (1993). Stochastic processes with linear conditional expectation and quadratic conditional variance. Prob. Math. Statist. 14, 33–44.Google Scholar
Wise, G. L. (1992). A counterexample to a martingale characterization of a Wiener process. Statist. Prob. Lett. 15, 337–338.CrossRefGoogle Scholar