Hostname: page-component-5c6d5d7d68-txr5j Total loading time: 0 Render date: 2024-08-15T11:24:26.219Z Has data issue: false hasContentIssue false
  • English
  • Français

Compound Poisson approximations for sums of discrete nonlattice variables

Part of: Distribution theory - Probability Limit theorems

Published online by Cambridge University Press:  01 July 2016

V. Čekanavičius  and
Y. H. Wang
V. Čekanavičius*
Affiliation:
Vilnius University
Y. H. Wang*
Affiliation:
Tunghai University
*
∗ Postal address: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Email address: vydas.cekanavicius@maf.vu.lt
∗∗ Postal address: Department of Statistics, Tunghai University, No. 181, Section 3, Taichung-Kan Road, Taichung, Taiwan 407-04, Republic of China.
Get access Rights & Permissions [Opens in a new window]

Abstract

Sums of independent random variables concentrated on the same finite discrete, not necessarily lattice, set of points are approximated by compound Poisson distributions and signed compound Poisson measures. Such approximations can be more accurate than the normal distribution. Short asymptotic expansions are constructed.

Keywords

Compound Poisson distributionsigned compound Poisson measureuniform distance

MSC classification

Primary: 60F99: None of the above, but in this section
Secondary: 60F05: Central limit and other weak theorems 60E05: Distributions
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 1 , March 2003 , pp. 228 - 250
Copyright
Copyright © Applied Probability Trust 2003 

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

Aleskevicienė, A. and Statulevicius, A. (1997). Inversion formulas in the case of a discontinuous limit law. Theory Prob. Appl. 42, 1–16.Google Scholar
Arak, T. V. (1980). Approximation of n-fold convolutions of distributions, having a nonnegative characteristic function, with accompanying laws. Theory Prob. Appl. 25, 221–243.Google Scholar
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. and Čekanavicius, V. (2002). Total variation asymptotics for sums of independent integer random variables. Ann. Prob. 30, 509–545.Google Scholar
Barbour, A. D. and Chryssaphinou, O. (2001). Compound Poisson approximation: a user's guide. Ann. Appl. Prob. 11, 964–1002.Google Scholar
Barbour, A. D. and Hall, P. (1984). On the rate of Poisson convergence. Math. Proc. Camb. Phil. Soc. 95, 473–480.CrossRefGoogle Scholar
Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximations. Oxford University Press.Google Scholar
Bikelis, A. (1996). Asymptotic expansions for distributions of statistics. In Proc. 36th Conf. Lithuanian Math. Soc. (Vilnius, 22–23 June 1995), eds Kudžma, R. and Mackevičius, V., Vilnius University Press, pp. 5–28 (in Russian).Google Scholar
Booth, J. G., Hall, P. and Wood, A. T. A. (1994). On the validity of Edgeworth and saddlepoint approximations. J. Multivariate Anal. 51, 121–138.Google Scholar
Borovkov, K. and Pfeifer, D. (1996). Pseudo-Poisson approximation for Markov chains. Stoch. Process Appl. 61, 163–180.Google Scholar
Čekanavicius, V., (1997). Approximation of the generalized Poisson binomial distribution: asymptotic expansions. Lithuanian Math. J. 37, 1–12.CrossRefGoogle Scholar
Čekanavicius, V., (1998). Estimates in total variation for convolutions of compound distributions. J. London Math. Soc. 58, 748–760.Google Scholar
Čekanavicius, V. and Kruopis, J. (2000). Signed Poisson approximation: a possible alternative to normal and Poisson laws. Bernoulli 6, 591–606.CrossRefGoogle Scholar
Cuppens, R. (1975). Decomposition of Multivariate Probability (Prob. Math. Statist. 29). Academic Press, New York.Google Scholar
Deheuvels, P. and Pfeifer, D. (1986). A semigroup approach to Poisson approximation. Ann. Prob. 14, 663–676.Google Scholar
Dhaene, J. and De Pril, N. (1994). On a class of approximative computation methods in the individual risk model. Insurance Math. Econom. 14, 181–196.Google Scholar
Doeblin, W. (1939). Sur les sommes d'un grand nombre de variables aleatoires independantes. Bull. Sci. Math. 63, 23–32, 35–64.Google Scholar
Franken, P. (1964). Approximation der Verteilungen von Summen unabhängiger nichtnegativer ganzzahliger Zufallsgrößen durch Poissonsche Verteilungen. Math. Nachr. 27, 303–340.Google Scholar
Gani, J. (1982). On the probability generating function of the sum of Markov–Bernoulli random variables. In Essays in Statistical Science (J. Appl. Prob. Spec. Vol. 19A), eds Gani, J. and Hannan, E. J., Applied Probability Trust, Sheffield, pp. 321–326.Google Scholar
Hipp, C. (1986). Improved approximations for the aggregate claims distribution in the individual model. ASTIN Bull. 16, 89–100.Google Scholar
Khintchine, A. (1933). Asymptotische Gesetze der Wahrsheinlichkeitsrechnung. Springer, Berlin.CrossRefGoogle Scholar
Kornya, P. (1983). Distribution of aggregate claims in the individual risk theory model. Soc. Actuaries Trans. 35, 823–858.Google Scholar
Kruopis, J. (1986). Approximations for distributions of sums of lattice random variables I. Lithuanian Math. J. 26, 234–244.Google Scholar
Le Cam, L. (1960). An approximation theorem for the Poisson binomial distribution. Pacific J. Math. 10, 1181–1197.Google 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
Michel, R. (1988). An improved error bound for the compound Poisson approximation of a nearly homogeneous portfolio. ASTIN Bull. 17, 165–169.Google Scholar
Panjer, H. H. and Willmot, G. E. (1983). Compound Poisson models in actuarial risk theory. J. Econometrics 23, 67–76.Google Scholar
Petrov, V. V. (1995). Limit Theorems of Probability Theory. Oxford University Press.Google Scholar
Presman, E. L. (1983). Approximation of binomial distributions by infinitely divisible ones. Theory Prob. Appl. 28, 393–403.Google Scholar
Prohorov, Yu. V. (1952). Some improvements of Liapunov's theorem. Izv. Akad. Nauk SSSR Ser. Mat. 16, 281–292 (in Russian).Google Scholar
Roos, B. (1999). Asymptotics and sharp bounds in the Poisson approximation to the Poisson-binomial distribution. Bernoulli 5, 1021–1034.CrossRefGoogle Scholar
Roos, B. (2001). Sharp constants in the Poisson approximation. Statist. Prob. Lett. 52, 155–168.Google Scholar
Steele, J. M. (1994). Le Cam's inequality and Poisson approximations. Amer. Math. Monthly 101, 48–54.CrossRefGoogle Scholar
Wang, Y. H. (1981). On the limit of the Markov binomial distribution. J. Appl. Prob. 18, 937–942.Google Scholar
Wang, Y. H. (1986). Coupling methods in approximation. Canad. J. Statist. 14, 69–74.Google Scholar
Wang, Y. H. (1992). Approximating kth order two-state Markov chains. J. Appl. Prob. 29, 861–868.Google Scholar
Wang, Y. H. (1993). On the number of successes in independent trials. Statist. Sinica 3, 295–312.Google Scholar
Zaitsev, A. Yu. (1984). On the accuracy of approximation of distributions of sums of independent random variables which are nonzero with a small probability by accompanying laws. Theory Prob. Appl. 28, 657–669.Google Scholar
Zaitsev, A. Yu. (1989). A multidimensional variant of Kolmogorov's second uniform limit theorem. Theory Prob. Appl. 34, 108–128.Google Scholar
Zaitsev, A. Yu. (1992). On the approximation of convolutions of multidimensional symmetric distributions by accompanying laws. J. Soviet. Math. 62, 1859–1872.Google Scholar