Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-23T11:55:04.994Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic Expansions in the Exponent: a Compound Poisson Approach

Part of: Limit theorems Distribution theory - Probability

Published online by Cambridge University Press:  01 July 2016

V. Čekanavičius
V. Čekanavičius*
Affiliation:
Vilnius University
*
∗ Postal address: Department of Mathematics, Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania.
Get access Rights & Permissions [Opens in a new window]

Abstract

The accuracy of the Normal or Poisson approximations can be significantly improved by adding part of an asymptotic expansion in the exponent. The signed-compound-Poisson measures obtained in this manner can be of the same structure as the Poisson distribution. For large deviations we prove that signed-compound-Poisson measures enlarge the zone of equivalence for tails.

Keywords

COMPOUND POISSONUNIFORM DISTANCESIGNED EXPONENTIAL MEASURES

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60E05: Distributions
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 2 , June 1997 , pp. 374 - 387
Copyright
Copyright © Applied Probability Trust 1997 

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

Arak, T. V. and Zaitsev, A. Yu. (1988) Uniform limit theorems for sums of independent random variables. Proc. Steklov Inst. Math. 174, 1–222.Google Scholar
Barbour, A. D., Chen, L. H. Y. and Loh, W.-L. (1992a) Compound Poisson approximation for nonnegative random variables via Stein's method. Ann. Prob. 20, 1843–1866.CrossRefGoogle Scholar
Barbour, A. D., Holst, L. and Janson, S. (1992b) Poisson Approximations. Oxford University Press, Oxford.Google Scholar
Cekanavicius, V. (1988) Approximation by generalized measures of Poisson type. Lithuanian. Math. J. 28, 284–288.CrossRefGoogle Scholar
Cuppens, R. (1975) Decomposition of Multivariate Probability. New York.CrossRefGoogle Scholar
Dhaene, J. and De Pril, N. (1994) On a class of approximate computation methods in the individual risk model. Insurance: Math. Econ. 14, 181–196.Google Scholar
Feller, W. (1968) Introduction to Probability Theory and its Applications. 3rd edn. Wiley, New York.Google Scholar
Franken, P. (1964) Approximation der Verteilungen von Summen unabhängiger nichtnegativen ganzzahliger Zufalgrösen durch Poissonsche Verteilungen. Math. Nachr. 27, 303–340.CrossRefGoogle Scholar
Hipp, C. (1986) Improved approximations for the aggregate claims distribution in the individual model. ASTIN Bull. 16, 89–100.Google Scholar
Ibragimov, I. A. and Linnik, Yu. V. (1971) Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.Google Scholar
Kornya, P. (1983) Distribution of aggregate claims in the individual risk theory model. Soc. Act. Trans. 35, 823–858.Google Scholar
Kruopis, J. (1986) Precision of approximations of the generalized Binomial distribution by convolutions of Poisson measures. Lithuanian Math. J. 26, 37–49.CrossRefGoogle Scholar
Le Cam, L. (1965) On the distribution of sums of independent random variables. In Bernoulli, Bayes, Laplace. Springer, Berlin. pp. 179–202.Google Scholar
Meshalkin, L. D. (1961) On the approximation of distributions of sums by infinitely divisible laws. (In Russian.) Teor. Veroyatnost. i Primenen. 6, 257–275.Google Scholar
Michel, R. (1988) An improved error bound for the compound Poisson approximation of a nearly homogeneous portfolio. ASTIN Bull. 17, 165–169.Google Scholar
Petrov, V. V. (1975) Sums of Independent Random Variables. Springer, Berlin.Google Scholar
Presman, E. L. (1983) Approximation of binomial distributions by infinitely divisible ones. Theory Prob. Appl. 28, 393–403.Google Scholar
Quine, M. P. (1994) Probability approximations for divisible discrete distributions. Austral. J. Statist. 36, 339–349.Google Scholar
Rachev, S. T. and Rüschendorf, L. (1990) Approximation of sums by compound Poisson distributions with respect to stop-loss distances. Adv. Appl. Prob. 22, 350–374.CrossRefGoogle Scholar
Roos, M. (1994) Stein-Chen method for compound Poisson approximation: the coupling approach. In Probability Theory and Mathematical Statistics. ed. Grigelionis, B. et al. VSP, Utrecht. pp. 645–660.Google Scholar
Serfling, R. J. (1975) A general Poisson approximation theorem. Ann. Prob. 3, 726–731.Google Scholar
ŠIaulys, J. and Cekanavicius, V. (1988) Approximation of distributions of integer-valued additive functions by discrete charges. I. Lithuanian Math. J. 28, 392–401.CrossRefGoogle Scholar
ŠIaulys, J. and Cekanavicius, V. (1989) Approximation of distributions of integer-valued additive functions by discrete charges. II. Lithuanian Math. J. 29, 80–95.CrossRefGoogle Scholar
Wang, Y. H. (1991) A compound Poisson convergence theorem. Ann. Prob. 19, 452–455.Google Scholar