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On the distance between the distributions of random sums

Part of: Distribution theory Distribution theory - Probability Limit theorems

Published online by Cambridge University Press:  14 July 2016

Bero Roos  and
Dietmar Pfeifer
Bero Roos*
Affiliation:
Universität Hamburg
Dietmar Pfeifer*
Affiliation:
Universität Oldenburg
*
∗ Postal address: Fachbereich Mathematik, SPST, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany. Email address: [email protected]
∗∗ Postal address: Fachbereich Mathematik, Universität Oldenburg, 26111 Oldenburg, Germany.
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Abstract

In this paper, we consider the total variation distance between the distributions of two random sums SM and SN with different random summation indices M and N. We derive upper bounds, some of which are sharp. Further, bounds with so-called magic factors are possible. Better results are possible when M and N are stochastically or stop-loss ordered. It turns out that the solution of this approximation problem strongly depends on how many of the first moments of M and N coincide. As approximations, we therefore choose suitable finite signed measures, which coincide with the distribution of the approximating random sum SN if M and N have the same first moments.

Keywords

Binomial distributiondiscrete Taylor formulafinite signed measurePoisson approximationrandom sumsharpness resultsstochastic orderstop-loss ordertotal variation distance

MSC classification

Primary: 60E05: Distributions
Secondary: 62E17: Approximations to distributions (nonasymptotic) 60E15: Inequalities; stochastic orderings 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 1 , March 2003 , pp. 87 - 106
Copyright
Copyright © Applied Probability Trust 2003 

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References

Barbour, A. D., Holst, L., and Janson, S. (1992). Poisson Approximation. Oxford University Press.Google Scholar
Denuit, M. and Lefèvre, C. (1997). Some new classes of stochastic order relations among arithmetic random variables, with applications in actuarial sciences. Insurance Math. Econom. 20, 197–213.Google Scholar
Denuit, M., and Van Bellegem, S. (2001). On the stop-loss and total variation distances between random sums. Statist. Prob. Lett. 53, 153–165.Google Scholar
Denuit, M., Lefèvre, C., and Mesfioui, M. (1999). On s-convex stochastic extrema for arithmetic risks. Insurance Math. Econom. 25, 143–155.CrossRefGoogle Scholar
Denuit, M., Lefèvre, C., and Utev, S. (1999). Stochastic orderings of convex/concave-type on an arbitrary grid. Math. Operat. Res. 24, 835–846.Google Scholar
Doeblin, W. (1938). Exposé de la theorie des chaînes simples constantes de Markov à un nombre fini d’états. Rev. Math. Union Interbalkanique 2, 77–105.Google Scholar
Finkelstein, M., Tucker, H. G., and Veeh, J. A. (1990). The limit distribution of the number of rare mutants. J. Appl. Prob. 27, 239–250.Google Scholar
Gnedenko, B. V. and Korolev, V. Yu. (1996). Random Summation. Limit Theorems and Applications. CRC Press, Boca Raton, FL.Google Scholar
Gut, A. (1988). Stopped Random Walks. Limit Theorems and Applications. Springer, New York.Google Scholar
Heilmann, W.-R. (1988). Fundamentals of Risk Theory. Versicherungswirtschaft, Karlsruhe.Google Scholar
Kalashnikov, V. (1997). Geometric Sums: Bounds for Rare Events with Applications: Risk Analysis, Reliability, Queuing. Kluwer, Dordrecht.Google Scholar
Krajka, A., and Rychlik, Z. (1987). On the difference between the distribution functions of two random sums of independent random variables. Math. Nachr. 134, 317–322.Google Scholar
Kruglov, V. M. and Korolev, V. Yu. (1990). Limit Theorems for Random Sums. Izdatel' stvo Moskovskogo Universiteta, Moskva (in Russian).Google Scholar
Lindeberg, J. W. (1922). Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung. Math. Z. 15, 211–225.Google Scholar
Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley, New York.Google Scholar
Logunov, P. L. (1990). Estimates for the convergence rate to the Poisson distribution for random sums of independent indicators. Theory Prob. Appl. 35, 587–590.Google Scholar
Mitrinović, D. S. (1970). Analytic Inequalities. Springer, Berlin.CrossRefGoogle Scholar
Müller, A., and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, New York.Google Scholar
Rahimov, I. (1995). Random Sums and Branching Stochastic Processes. Springer, New York.CrossRefGoogle Scholar
Roos, B. (2000). Binomial approximation to the Poisson binomial distribution: the Krawtchouk expansion. Theory Prob. Appl. 45, 258–272.Google Scholar
Roos, B. (2001). Improvements in the Poisson approximation of mixed Poisson distributions. To appear in J. Statist. Planning Infer.Google Scholar
Shaked, M., and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, Boston, MA.Google Scholar
Vellaisamy, P., and Chaudhuri, B. (1996). Poisson and compound Poisson approximations for random sums of random variables. J. Appl. Prob. 33, 127–137.Google Scholar
Yannaros, N. (1991). Poisson approximation for random sums of Bernoulli random variables. Statist. Prob. Lett. 11, 161–165.Google Scholar