×

zbMATH — the first resource for mathematics

A Charlier-Parseval approach to Poisson approximation and its applications. (English) Zbl 1209.60019
Summary: We propose a new approach to Poisson approximation. The basic idea is very simple and based on properties of Charlier polynomials and the Parseval identity. Such an approach quickly leads to new effective bounds for several Poisson approximation problems. We also give a selected survey on diverse Poisson approximation results.

MSC:
60F05 Central limit and other weak theorems
60E05 Probability distributions: general theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] D. Aldous, Probability Approximations via the Poisson Clumping Heuristic, Springer-Verlag, New York, 1989. · Zbl 0679.60013
[2] T.V. Arak and A.Yu. Zaĭtsev, Uniform Limit Theorems for Sums of Independent Random Variables, English transl.: Proc. Steklov Inst. Math. 1988, no 1, Amer. Math. Soc., Providence, RI, 1988.
[3] A.D. Barbour, Asymptotic expansions in the Poisson limit theorem, Ann. Probab., 15:748–766, 1987. · Zbl 0622.60049
[4] A.D. Barbour, Topics in Poisson approximation, in D.N. Shanbhag and C.R. Rao (Eds.), Stochastic Processes: Theory and Methods, Handbook of Statistics, Vol. 19, North-Holland, Amsterdam, 2001, pp. 79–115. · Zbl 0981.60021
[5] A.D. Barbour, V. Čekanavičius, and A. Xia, On Stein’s method and perturbations, ALEA Lat. Am. J. Probab. Math. Stat., 3:31–53, 2007. · Zbl 1121.62016
[6] A.D. Barbour and L.H.-Y. Chen, Stein’s Method and Applications, Singapore Univ. Press,World Scientific Publishing Co., Singapore, 2005. · Zbl 1094.62017
[7] A.D. Barbour and O. Chryssaphinou, Compound Poisson approximation: A user’s guide, Ann. Appl. Probab., 11:964–1002, 2001. · Zbl 1018.60051
[8] A.D. Barbour and P. Hall, On the rate of Poisson convergence, Math. Proc. Cambridge Philos. Soc., 95:473–480, 1984. · Zbl 0544.60029
[9] A.D. Barbour, L. Holst, and S. Janson, Poisson Approximation, Oxford Science Publications, Clarendon Press, Oxford, 1992.
[10] A.D. Barbour and J.L. Jensen, Local and tail approximations near the Poisson limit, Scand. J. Stat., 16:75–87, 1989. · Zbl 0674.60022
[11] A.D. Barbour and A. Xia, Poisson perturbations, ESAIM Probab. Stat., 3:131–150, 1999. · Zbl 0949.62015
[12] A.D. Barbour and A. Xia, On Stein’s factors for Poisson approximation in Wasserstein distance, Bernoulli, 12:943–954, 2006. · Zbl 1328.62076
[13] R.P. Boas Jr., Representation of probability distributions by Charlier series, Ann. Math. Stat., 20:376–392, 1949. · Zbl 0041.24801
[14] I.S. Borisov and I.S. Vorozheĭkin, Accuracy of approximation in the Poisson theorem in terms of the {\(\chi\)}2-distance, Sib. Math. J., 49:5–17, 2008.
[15] K.A. Borovkov, On the problem of improving Poisson approximation, Theory Probab. Appl., 33:343–347, 1989. · Zbl 0666.60024
[16] K.A. Borovkov and D. Pfeifer, On improvements of the order of approximation in the Poisson limit theorem, J. Appl. Probab., 33:146–155, 1996. · Zbl 0852.60025
[17] L. von Bortkiewicz, Das Gesetz der Kleinen Zahlen, B.G. Teubner, Leipzig, 1898.
[18] V. Čekanavičius, Asymptotic expansions in the exponent: A compound Poisson approach, Adv. Appl. Probab., 29:374–387, 1997. · Zbl 0895.60029
[19] V. Čekanavičius, On local estimates and the Stein method, Bernoulli, 10:665–683, 2004. · Zbl 1055.62017
[20] V. Čekanavičius and J. Kruopis, Signed Poisson approximation: A possible alternative to normal and Poisson laws, Bernoulli, 6:591–606, 2000. · Zbl 0976.60035
[21] C.V.L. Charlier, Die zweite Form des Fehlergesetzes, Ark. Mat. Astron. Fys., 2(15):1–8, 1905. · JFM 36.0305.03
[22] S. Chatterjee, P. Diaconis, and E. Meckes, Exchangeable pairs and Poisson approximation, Probab. Surv., 2:64–106, 2005. · Zbl 1189.60072
[23] L.H.Y. Chen, Poisson approximation for dependent trials, Ann. Probab., 3:534–545, 1975. · Zbl 0335.60016
[24] L. Comtet, Advanced Combinatorics, revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974. · Zbl 0283.05001
[25] D.J. Daley, A note on bounds for the supremum metric for discrete random variables, Math. Nachr., 99:95–98, 1980. · Zbl 0466.60028
[26] D.J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes. Vol. II. General Theory and Structure, 2nd edition, Springer, New York, 2008. · Zbl 1159.60003
[27] A. de Moivre, The Doctrine of Chances, 3rd edition, W. Pearson, London, 1756; available at www.ibiblio.org/chance . · Zbl 0153.30801
[28] P. Deheuvels, A. Karr, D. Pfeifer, and R. R. Serfling, Poisson approximations in selected metrics by coupling and semigroup methods with applications, J. Stat. Plann. Inference, 20:1–22, 1988. · Zbl 0656.60035
[29] P. Deheuvels and D. Pfeifer, Operator semigroups and Poisson convergence in selected metrics, Semigroup Forum, 34:203–224, 1986. Errata: Semigroup Forum, 35:251. · Zbl 0606.60034
[30] P. Deheuvels and D. Pfeifer, A semigroup approach to Poisson approximation, Ann. Probab., 14:663–676, 1986. · Zbl 0597.60019
[31] P. Deheuvels and D. Pfeifer, On a relationship between Uspensky’s theorem and Poisson approximation, Ann. Inst. Stat. Math., 40:671–681, 1988. · Zbl 0675.60027
[32] P. Deheuvels, D. Pfeifer, and M.L. Puri, A new semigroup technique in Poisson approximation, Semigroup Forum, 38:189–201, 1989. · Zbl 0674.60026
[33] W. Ehm, Binomial approximation to the Poisson binomial distribution, Stat. Probab. Lett., 11:7–16, 1991. · Zbl 0724.60021
[34] A.A. Fedotov, P. Harremoës, and F. Topsøe, Refinements of Pinsker’s inequality, IEEE Trans. Inform. Theory, 49:1491–1498, 2003. · Zbl 1063.94017
[35] P. Franken, Approximation des Verteilungen von Summen unabhangiger nichtnegativer ganzzahler Zufallsgrossen durch Poissonsche verteilungen, Math. Nachr., 23:303–340, 1964. · Zbl 0192.25204
[36] L. Goldstein and G. Reinert, Distributional transformations, orthogonal polynomials, and Stein characterizations, J. Theor. Probab., 18:237–260, 2005. · Zbl 1072.62002
[37] I.J. Good, Some statistical applications of Poisson’s work (with comments by Persi Diaconis and Eduardo Engel, Herbert Solomon, C.C. Heyde, and Nozer D. Singpurwalla, and with a reply by the author), Stat. Sci., 1:157–180, 1986. · Zbl 0611.60001
[38] F.A. Haight, Handbook of the Poisson Distribution, John Wiley & Sons, Inc., New York, London, Sydney. · Zbl 0152.37706
[39] H. Herrmann, Variationsabstand zwischen der Verteilung einer Summe unabhängiger nichtnegativer ganzzahliger Zufallsgrössen und Poissonschen Verteilungen, Math. Nachr., 29:265–289, 1965. · Zbl 0147.17203
[40] C. Hipp, Approximation of aggregate claims distributions by compound Poisson distributions, Insur. Math. Econ., 4:227–232, 1985. Correction note: Insur. Math. Econ., 6:165, 1987. · Zbl 0598.62139
[41] C. Hipp, Improved approximations for the aggregate claims distribution in the individual model, Astin Bull., 16:89–100, 1986.
[42] J.L. Hodges Jr. and L. Le Cam, The Poisson approximation to the Poisson binomial distribution, Ann. Math. Stat., 31:737–740, 1960. · Zbl 0119.14704
[43] H.-K. Hwang, Asymptotics of Poisson approximation to random discrete distributions: an analytic approach, Adv. Appl. Probab., 31:448–491, 1999. · Zbl 0945.60001
[44] H.-K. Hwang and V. Zacharovas, Uniform asymptotics of Poisson approximation to the Poisson-binomial distribution, Theory Probab. Appl., 2010 (in press). · Zbl 1230.60021
[45] M. Jacob, Sullo sviluppo di una curva di frequenze in serie di Charlier tipo B, G. Ist. Ital. Attuari, 4:221–234, 1933.
[46] S. Janson, Coupling and Poisson approximation, Acta Appl. Math., 34:7–15, 1994. · Zbl 0802.60021
[47] C. Jordan, Sur la probabilité des épreuves répétées, le théorème de Bernoulli et son inversion, Bull. Soc. Math. Fr., 54:101–137, 1926.
[48] J.E. Kennedy and M.P. Quine, The total variation distance between the binomial and Poisson distributions, Ann. Probab., 17:396–400, 1989. · Zbl 0664.60027
[49] J. Kerstan, Verallgemeinerung eines Satzes von Prochorow und Le Cam, Z. Wahrscheinlichkeitstheor. Verw. Geb., 2:173–179, 1964. · Zbl 0123.35403
[50] A.N. Kolmogorov, Two uniform limit theorems for sums of independent random variables, Theory Probab. Appl., 1:384–394, 1956. · Zbl 0079.34502
[51] I. Kontoyiannis, P. Harremöes, and O. Johnson, Entropy and the law of small numbers, IEEE Trans. Inform. Theory, 51:466–472, 2005. · Zbl 1297.94016
[52] P.S. Kornya, Distribution of aggregate claims in the individual risk theory model, Trans. Soc. Actuaries, 35:823–858, 1983.
[53] Y. Kruopis, The accuracy of approximation of the generalized binomial distribution by convolutions of Poisson measures, Lith. Math. J., 26:37–49, 1986. · Zbl 0631.60019
[54] L. Le Cam, An approximation theorem for the Poisson binomial distribution, Pac. J. Math., 10:1181–1197, 1960. · Zbl 0118.33601
[55] H. Makabe, On the approximations to some limiting distributions with some applications, Kōdai Math. Semin. Rep., 14:123–133, 1962. · Zbl 0241.60024
[56] T. Matsunawa, Uniform {\(\phi\)}-equivalence of probability distributions based on information and related measures of discrepancy, Ann. Inst. Stat. Math., 34:1–17, 1982. · Zbl 0486.62007
[57] K. Neammanee, A nonuniform bound for the approximation of Poisson binomial by Poisson distribution, Int. J. Math. Math. Sci., 48:3041–3046, 2003. · Zbl 1033.60027
[58] K. Neammanee, Pointwise approximation of Poisson distribution, Stoch. Model. Appl., 6:20–26, 2003. · Zbl 1033.60027
[59] D. Pfeifer, A semigroup setting for distance measures in connexion with Poisson approximation, Semigroup Forum, 31:201–205, 1985. · Zbl 0551.60039
[60] J. Pitman, Probabilistic bounds on the coefficients of polynomials with only real zeros, J. Comb. Theory Ser. A, 77:279–303, 1997. · Zbl 0866.60016
[61] S.D. Poisson, Recherches sur la probabilité des jugements en matière criminelle et en matière civile: Précedés des règles générales du calcul des probabilités, Bachelier, Paris, 1837.
[62] H. Pollaczek-Geiringer, Die Charlier’sche Entwicklung willkürlicher Verteilungen, Skand. Aktuarietidskr., 11:98–111, 1928. · JFM 54.0560.02
[63] H.V. Poor, The maximum difference between the binomial and Poisson distributions, Stat. Probab. Lett., 11:103–106, 1991. · Zbl 0712.62012
[64] È.L. Presman, Approximation of binomial distributions by infinitely divisible ones, Theory Probab. Appl., 28:393–403, 1983. · Zbl 0533.60018
[65] È.L. Presman, The variation distance between the distribution of a sum of independent Bernoulli variables and the Poisson law, Teor. Veroyatn. Primen., 30(2):391–396, 1985 (in Russian). · Zbl 0568.60026
[66] Y.V. Prokhorov, Asymptotic behavior of the binomial distribution, Usp. Mat. Nauk, 8:135–142, 1953. Also in Sel. Transl. Math. Stat. Probab., 1:87–95.
[67] A. Röllin, Translated Poisson approximation using exchangeable pair couplings, Ann. Appl. Probab., 17:1596–1614, 2007. · Zbl 1143.60020
[68] M. Romanowska, A note on the upper bound for the distance in total variation between the binomial and the Poisson distribution, Stat. Neerl., 31:127–130, 1977. · Zbl 0369.60025
[69] B. Roos, A semigroup approach to Poisson approximation with respect to the point metric, Stat. Probab. Lett., 24:305–314, 1995. · Zbl 0832.62014
[70] B. Roos, Asymptotics and sharp bounds in the Poisson approximation to the Poisson-binomial distribution, Bernoulli, 5:1021–1034, 1999. · Zbl 0951.60040
[71] B. Roos, Sharp constants in the Poisson approximation, Stat. Probab. Lett., 52:155–168, 2001. · Zbl 0991.60014
[72] B. Roos, Poisson approximation via the convolution with Kornya–Presman signed measures, Theory Probab. Appl., 48:555–560, 2004. · Zbl 1064.60033
[73] B. Roos, On variational bounds in the compound Poisson approximation of the individual risk model, Insur. Math. Econ., 40:403–414, 2007. · Zbl 1183.91075
[74] E. Schmidt, Uber die Charlier–Jordansche Entwicklung einer willkurlichen Funktion nach der Poissonschen Funktion und ihren Ableitungen, Z. Angew. Math., 13:139–142, 1933. · JFM 59.0310.02
[75] E. Seneta, Modern probabilistic concepts in the work of E. Abbe and A. de Moivre, Math. Sci., 8:75–80, 1983. · Zbl 0528.60003
[76] R.J. Serfling, Probability inequalities for the sum in sampling without replacement, Ann. Stat., 2:39–48, 1974. · Zbl 0288.62005
[77] R.J. Serfling, Some elementary results on Poisson approximation in a sequence of Bernoulli trials, SIAM Rev., 20:567–579, 1978. · Zbl 0383.60027
[78] S.Y. Shorgin, Approximation of a generalized binomial distribution, Theory Probab. Appl., 22:846–850, 1977. · Zbl 0392.60021
[79] R. Siegmund-Schultze, Hilda Geiringer-von Mises, Charlier series, ideology, and the human side of the emancipation of applied mathematics at the University of Berlin during the 1920s, Hist. Math., 20:364–381, 1993. · Zbl 0805.01017
[80] J.M. Steele, Le Cam’s inequality and Poisson approximations, Am. Math. Mon., 101:48–54, 1994. · Zbl 0802.60019
[81] C. Stein, Approximate Computation of Expectations, IMS, Hayward, CA, 1986. · Zbl 0721.60016
[82] G. Szegö, Orthogonal Polynomials, Amer. Math. Soc., New York, 1939.
[83] K. Teerapabolarn and K. Neammanee, Poisson approximation for sums of dependent Bernoulli random variables, Acta Math. Acad. Paedagog. Nyházi. (N.S.), 22:87–99, 2006. · Zbl 1120.60303
[84] J.V. Uspensky, On Ch. Jordan’s series for probability, Ann. Math., 32:306–312, 1931. · Zbl 0002.20004
[85] W. Vervaat, Upper bounds for the distance in total variation between the binomial or negative binomial and the Poisson distribution, Stat. Neerl., 23:79–86, 1969. · Zbl 0162.22203
[86] M. Weba, Bounds for the total variation distance between the binomial and the Poisson distribution in the case of medium-sized success probabilities, J. Appl. Probab., 36:97–104, 1999. · Zbl 0936.62018
[87] H.-J. Witte, A unification of some approaches to Poisson approximation, J. Appl. Probab., 27:611–621, 1990. · Zbl 0721.60022
[88] A. Xia, On using the first difference in the Stein–Chen method, Ann. Appl. Probab., 7(4):899–916, 1997. · Zbl 0903.60012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.