The Annals of Probability

Kerstan's method for compound Poisson approximation

Bero Roos

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We consider the approximation of the distribution of the sum of independent but not necessarily identically distributed random variables by a compound Poisson distribution and also by a finite signed measure of higher accuracy. Using Kerstan's method, some new bounds for the total variation distance are presented. Recently, several authors had difficulties applying Stein's method to the problem given. For instance, Barbour, Chen and Loh used this method in the case of random variables on the nonnegative integers. Under additional assumptions, they obtained some bounds for the total variation distance containing an undesirable log term. In the present paper, we shall show that Kerstan's approach works without such restrictions and yields bounds without log terms.

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Ann. Probab., Volume 31, Number 4 (2003), 1754-1771.

First available in Project Euclid: 12 November 2003

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Zentralblatt MATH identifier

Primary: 62E17: Approximations to distributions (nonasymptotic)
Secondary: 60F05: Central limit and other weak theorems 60G50: Sums of independent random variables; random walks

Compound Poisson approximation discrete self-decomposable distributions discrete unimodal distributions finite signed measure Kerstan's method sums of independent random variables total variation distance


Roos, Bero. Kerstan's method for compound Poisson approximation. Ann. Probab. 31 (2003), no. 4, 1754--1771. doi:10.1214/aop/1068646365.

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  • Barbour, A. D. (1988). Stein's method and Poisson process convergence. J. Appl. Probab. 25A 175--184.
  • Barbour, A. D., Chen, L. H. Y. and Loh, W.-L. (1992). Compound Poisson approximation for nonnegative random variables via Stein's method. Ann. Probab. 20 1843--1866.
  • Barbour, A. D. and Chryssaphinou, O. (2001). Compound Poisson approximation: A user's guide. Ann. Appl. Probab. 11 964--1002.
  • Barbour, A. D. and Hall, P. (1984). On the rate of Poisson convergence. Math. Proc. Cambridge Philos. Soc. 95 473--480.
  • Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Clarendon, Oxford.
  • Barbour, A. D. and Utev, S. (1998). Solving the Stein equation in compound Poisson approximation. Adv. in Appl. Probab. 30 449--475.
  • Barbour, A. D. and Utev, S. (1999). Compound Poisson approximation in total variation. Stochastic Process. Appl. 82 89--125.
  • Barbour, A. D. and Xia, A. (1999). Poisson perturbations. ESAIM Probab. Statist. 3 131--150.
  • Barbour, A. D. and Xia, A. (2000). Estimating Stein's constants for compound Poisson approximation. Bernoulli 6 581--590.
  • Čekanavičius, V. (1997). Approximation of the generalized Poisson binomial distribution: Asymptotic expansions. Liet. Mat. Rink. 37 1--17. [English translation (1997) Lithuanian Math. J. 37 1--12.]
  • Čekanavičius, V. (1998). Estimates in total variation for convolutions of compound distributions. J. London Math. Soc. (2) 58 748--760.
  • Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.
  • Deheuvels, P. and Pfeifer, D. (1986). A semigroup approach to Poisson approximation. Ann. Probab. 14 663--676.
  • Doeblin, W. (1939). Sur les sommes d'un grand nombre de variables aléatoires indépendantes. Bull. Sci. Math. 63 23--32 and 35--64.
  • Gerber, H. U. (1979). An Introduction to Mathematical Risk Theory. Huebner Foundation, Philadelphia.
  • Hipp, C. (1985). Approximation of aggregate claims distributions by compound Poisson distributions. Insurance Math. Econom. 4 227--232. [Correction (1987) Insurance Math. Econom. 6 165.]
  • Hipp, C. and Michel, R. (1990). Risikotheorie: Stochastische Modelle und Statistische Verfahren. Verlag Versicherungswirtschaft, Karlsruhe.
  • Kerstan, J. (1964). Verallgemeinerung eines Satzes von Prochorow und Le Cam. Z. Wahrsch. Verw. Gebiete 2 173--179.
  • Khintchine, A. Ya. (1933). Asymptotische Gesetze der Wahrscheinlichkeitsrechnung. Springer, Berlin.
  • Le Cam, L. (1960). An approximation theorem for the Poisson binomial distribution. Pacific J. Math. 10 1181--1197.
  • Le Cam, L. (1965). On the distribution of sums of independent random variables. In Bernoulli, Bayes, Laplace (J. Neyman and L. Le Cam, eds.) 179--202. Springer, New York.
  • Michel, R. (1987). An improved error bound for the compound Poisson approximation of a nearly homogeneous portfolio. Astin Bull. 17 165--169.
  • Prohorov, Y. V. (1953). Asymptotic behaviour of the binomial distribution. Uspekhi Mat. Nauk 8 135--142. [English translation (1961) Sel. Transl. Math. Statist. Probab. 1 87--95.]
  • Roos, B. (1996). Metrische Poisson-Approximation (in German). Ph.D. dissertation, Fachbereich Mathematik, Univ. Oldenburg.
  • Roos, B. (1999a). On the rate of multivariate Poisson convergence. J. Multivariate Anal. 69 120--134.
  • Roos, B. (1999b). Asymptotics and sharp bounds in the Poisson approximation to the Poisson binomial distribution. Bernoulli 5 1021--1034.
  • Roos, B. (2000). Binomial approximation to the Poisson binomial distribution: The Krawtchouk expansion. Theory Probab. Appl. 45 258--272.
  • Roos, B. (2001). Sharp constants in the Poisson approximation. Statist. Probab. Lett. 52 155--168.
  • Steutel, F. W. and van Harn, K. (1979). Discrete analogues of self-decomposability and stability. Ann. Probab. 7 893--899.
  • Vellaisamy, P. and Chaudhuri, B. (1999). On compound Poisson approximation for sums of random variables. Statist. Probab. Lett. 41 179--189.
  • Witte, H.-J. (1990). A unification of some approaches to Poisson approximation. J. Appl. Probab. 27 611--621.
  • Zaitsev, A. Yu. (1983). On the accuracy of approximation of distributions of sums of independent random variables---which are nonzero with a small probability---by means of accompanying laws. Theory Probab. Appl. 28 657--669.
  • Zaitsev, A. Yu. (1989). Multidimensional version of the second uniform limit theorem of Kolmogorov. Theory Probab. Appl. 34 108--128.