Tran Loc Hung


The  main  goal of this  note  is to establish the Le Cam  type  bounds in general  Poisson  approximation for distributions of sums  (and  random sums) of independent, non-negative integer-valued random variables with respect to a probability distance based  on a linear  operator originated by R´enyi.

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R. Arratia, L. Goldstein and L. Gordon. Poisson approximation and the Chen-Stein method, Statist. Sci., 5 (1990), No. 4, 403–434.

A. D. Barbour, L. Holst and S. Janson. ”Poisson approximation”, Oxford Univ. Press, New York, 1992.

Louis H. Y. Chen, On the convergence of Poisson binomial to Poisson distributions,

Ann. Probability, 2(1) (1974), 178–180.

P. Deheuvels, A. Karr, D. Pfeifer and R. Serfling, Poisson approximations in selected metrics by coupling and semi-group methods with applications, J. Statist. Plann. Infer- ence 20 (1988), No. 1, 1–22.

Boris V. Gnedenko and Victor Yu. Korolev, ”Random summation. Limit theorems and applications”, CRC Press, Boca Raton, FL, 1996.

Tran Loc Hung and Vu Thi Thao, Bounds for the approximation of Poisson- binomial distribution by Poisson distribution, J. Ineq. Appl. (2013), 2013:30, 10 pp, http://www.journalofinequalitiesandapplications.com/content/2013/1/30.

Tran Loc Hung and Le Truong Giang, On bounds in Poisson approximation for integer-valued independent random variables, J. Ineq. Appl. (2014), 2014:291, 11 pp, http://www.journalofinequalitiesandapplications.com/content/2014/1/291.

Tran Loc Hung and Le Truong Giang, On the bounds in Poisson approximation for independent geometric distributed random variables, Bulletin of the Irannian Mathe- matical Society, 42(5)(2016), 1087–1096.

Johannes Kerstan, Verallgemeinerung eines Satzes von Prochorow und Le Cam, (Ger- man) Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 173–179.

V. M. Kruglov and V. Yu. Korolev, ”Limit theorems for random sums” (Russian), Moskov. Gos. Univ., Moscow, 1990.

L. Le Cam, An approximation theorem for the Poisson binomial distribution, Pacific

J. Math. 10 (1960), 1181–1197.

K. Neammanee, A nonuniform bound for the approximation of Poisson binomial by

Poisson distribution, Int. J. Math. Sci. (2003), No. 48, 3041–3046.

A. Renyi, ”Probability theory”, Translated by L. Vekerdi, North-Holland Series in Ap- plied Mathematics and Mechanics, Vol. 10. North-Holland Publishing Co., Amsterdam- London, American Elsevier Publishing Co., Inc., New York, 1970.

R. J. Serfling, A general Poisson approximation theorem, Ann. Probability 3 (1975), No. 4, 726–731.

J. Michael Steele, Le Cam’s inequality and Poisson approximations, Amer. Math.

Monthly, 101(1) (1994), 48–54.

K. Teerapabolarn K and P. Wongkasem, Poisson approximation for independent geo- metric random variables, Int. Math. Forum, 2 (2007), No. 65-68, 3211–3218.

K. Teerapabolarn, A note on Poisson approximation for independent geometric random variables, Int. Math. Forum 4 (2009), no. 9-12, 531–535.

K. Teerapabolarn, A pointwise approximation of generalized binomial by Poisson dis- tribution, Appl. Math. Sci. (Ruse), 6, (2012), No. 21-24, 1095–1104.

N. S. Upadhye and P. Vellaisamy, Improved bounds for approximations to compound distributions, Statist. Probab. Lett. 83 (2013), No. 2, 467–473.

N. S. Upadhye and P. Vellaisamy, Compound Poisson approximation to convolutions of compound negative binomial variables, Methodol. Comput. Appl. Probab. 16(4) (2014),


Nikos Yannaros, Poisson approximation for random sums of Bernoulli random vari- ables, Statistics and Probability Letters 11 (1991), 161–165.

V. Zacharovas and H. Hwang, A Charlier-Parseval approach to Poisson approximation and its applications, Lith. Math. J., 50)1) (2010), 88–119.


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