Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-22T06:53:33.423Z Has data issue: false hasContentIssue false
  • English
  • Français

On the moment determinacy of the distributions of compound geometric sums

Part of: Distribution theory - Probability Integral transforms, operational calculus

Published online by Cambridge University Press:  14 July 2016

Gwo Dong Lin  and
Jordan Stoyanov
Gwo Dong Lin*
Affiliation:
Academia Sinica, Taiwan
Jordan Stoyanov*
Affiliation:
University of Newcastle upon Tyne
*
∗ Postal address: Institute of Statistical Science, Academia Sinica, Taipei 11529, Taiwan, Republic of China. Email address: gdlin@stat.sinica.edu.tw
∗∗ Postal address: School of Mathematics and Statistics, University of Newcastle upon Tyne, Newcastle upon Tyne NE1 7RU, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

We deal with compound geometric sums of independent positive random variables and study the moment problem for the distributions of such sums (the Stieltjes moment problem). We find conditions under which the distributions are uniquely determined by their moments. We also treat related topics, including the Hamburger moment problem involving random variables on the whole real line. Some conjectures are outlined.

Keywords

Compound geometric summomentsdeterminate and indeterminate moment problemsCarleman criterionKrein criterionLin condition

MSC classification

Primary: 60E05: Distributions 44A60: Moment problems
Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 3 , September 2002 , pp. 545 - 554
Copyright
Copyright © Applied Probability Trust 2002 

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

Berg, C. (1985). On the preservation of determinacy under convolution. Proc. Amer. Math. Soc. 93, 351–357.Google Scholar
Cai, J., and Kalashnikov, V. (2000). NWU property of a class of random sums. J. Appl. Prob. 37, 283–289.Google Scholar
Devinatz, A. (1959). On a theorem of Lévy–Raikov. Ann. Math. Statist. 30, 583–586.Google Scholar
Diaconis, P., and Ylvisaker, D. (1985). Quantifying prior opinion (with discussion). In Bayesian Statistics 2 (Proc. 2nd Valencia Internat. Meeting, 6–10 September 1983), eds Bernardo, J. M., DeGroot, M. H., Lindley, D. V. and Smith, A. F. M., North-Holland, Amsterdam, pp. 133–156.Google Scholar
Embrechts, P., Klüppelberg, C., and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Gnedenko, B. V., and Korolev, V. (1996). Random Summation: Limit Theorems and Applications. CRC Press, Boca Raton, FL.Google Scholar
Gut, A. (1988). Stopped Random Walks: Limit Theorems and Applications (Appl Prob. 5). Springer, New York.Google Scholar
Hu, C.-Y., and Lin, G. D. (2001). On the geometric compounding model with applications. Prob. Math. Statist. 21, 135–147.Google Scholar
Kalashnikov, V. (1997). Geometric Sums: Bounds for Rare Events with Applications (Math. Appl. 413). Kluwer, Dordrecht.Google Scholar
Lin, G. D. (1997). On the moment problems. Statist. Prob. Lett. 35, 85–90.Google Scholar
Lin, G. D. and Hu, C.-Y. (2000). A note on the ℒ-class of life distributions. Sankhyā A 62, 267–272.Google Scholar
Milne, R. K., and Yeo, G. F. (1989). Random sum characterizations. Math. Scientist 14, 120–126.Google Scholar
Pakes, A. G., Hung, W.-L., and Wu, J.-W. (2001). Criteria for the unique determination of probability distributions by moments. Austral. N. Z. J. Statist. 43, 101–111.Google Scholar
Rolski, T., Schmidli, H., Schmidt, V., and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, New York.Google Scholar
Slud, E. (1993). The moment problem for polynomial forms in normal random variables. Ann. Prob. 21, 2200–2214.CrossRefGoogle Scholar
Stoyanov, J. (1997). Counterexamples in Probability, 2nd edn. John Wiley, Chichester.Google Scholar
Stoyanov, J. (2000). Krein condition in probabilistic moment problems. Bernoulli 6, 939–949.Google Scholar
Szekli, R. (1988). A note on preservation of self-decomposability under geometric compounding. Statist. Prob. Lett. 6, 231–236.Google Scholar