Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T04:36:44.640Z Has data issue: false hasContentIssue false
  • English
  • Français

On a generalization of the Rényi–Srivastava characterization of the Poisson law

Part of: Probability theory and stochastic processes Mathematical modeling, applications of mathematics Distribution theory - Probability Combinatorial probability Combinatorics

Published online by Cambridge University Press:  25 February 2021

Jean-Renaud Pycke
Jean-Renaud Pycke*
Affiliation:
University of Évry and University of Paris
*
*Postal address: University of Évry, LaMME, CNRSUMR 8071, 23 bvd de France, 91 037 Évry Cedex, France; University of Paris, Laboratory I3SP (URP 3625) Sports Faculty, France. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a new method of proof for a result of D. Pierre-Loti-Viaud and P. Boulongne which can be seen as a generalization of a characterization of Poisson law due to Rényi and Srivastava. We also provide explicit formulas, in terms of Bell polynomials, for the moments of the compound distributions occurring in the extended collective model in non-life insurance.

Keywords

Poisson lawcompound variablesBell polynomialsnon-life insurance

MSC classification

Primary: 60E05: Distributions 60E10: Characteristic functions; other transforms 60C05: Combinatorial probability
Secondary: 05A17: Partitions of integers 60-08: Computational methods (not classified at a more specific level) 97M30: Financial and insurance mathematics
Type
Research Papers
Information
Journal of Applied Probability , Volume 58 , Issue 1 , March 2021 , pp. 68 - 82
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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

Comtet, L. (1974). Advanced Combinatorics. Reidel.CrossRefGoogle Scholar
Evgrafov, M. A. (1966). Analytic Functions. Sanders.Google Scholar
Feller, W. (1950). An Introduction to Probability Theory and its Applications, Vol. 2, 3rd edn. Wiley.Google Scholar
Kendall, M. G. and Stuart, A. (1994). The Advanced Theory of Statistics, Vol. 1, 6th edn.Google Scholar
Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012). Loss Models: From Data to Decisions, 4th edn. Wiley.Google Scholar
Johnson, N. L., Kemp, A. W. and Kotz, S. (2005). Univariate Discrete Distributions. Wiley-Interscience.CrossRefGoogle Scholar
Johnson, N. L., Kemp, A. W. and Kotz, S. (1997). Multivariate Discrete Distributions. Wiley-Interscience.Google Scholar
Partrat, C. and Besson, J.-L. (2005). Assurance non-vie: Modélisation, simulation. Economica.Google Scholar
Pierre Loti-Viaud, D. and Boulongne, P. (2014). Mathématiques et assurance: Premiers éléments. Ellipses.Google Scholar
Rényi, A. (1964). On two mathematical models of the traffic on a divided highway. J. Appl. Prob. 1, 311–320.CrossRefGoogle Scholar
Roman, S. (1984). The Umbral Calculus. Academic Press.Google Scholar
Srivastava, R. C. (1971). On a characterization of the Poisson process. J. Appl. Prob. 8, 615–616.CrossRefGoogle Scholar
Sundt, B. and Vernic, R. (2009). Recursions for Convolutions and Compound Distributions with Insurance Applications. Springer Science & Business Media.Google Scholar
Widder, D. W. (1946). The Laplace Transform. Princeton Mathematical Series.Google Scholar