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Zero bias transformation and asymptotic expansions. (English. French summary) Zbl 1238.60050

Summary: Let \(W\) be a sum of independent random variables. We apply the zero bias transformation to deduce recursive asymptotic expansions for \(\mathbb{E}[h(W)]\) in terms of normal expectations, or of Poisson expectations for integer-valued random variables. We also discuss the estimates of remaining errors.

MSC:

60G50 Sums of independent random variables; random walks
60F05 Central limit and other weak theorems
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[1] R. Arratia, L. Goldstein and L. Gordon. Two moments suffice for Poisson approximations: The Chen-Stein method. Ann. Probab. 17 (1989) 9-25. · Zbl 0675.60017
[2] A. D. Barbour. Asymptotic expansions based on smooth functions in the central limit theorem. Probab. Theory Related Fields 72 (1986) 289-303. · Zbl 0572.60029
[3] A. D. Barbour. Asymptotic expansions in the Poisson limit theorem. Ann. Probab. 15 (1987) 748-766. · Zbl 0622.60049
[4] A. D. Barbour and V. Čekanavičius. Total variation asymptotics for sums of independent integer random variables. Ann. Probab. 30 (2002) 509-545. · Zbl 1018.60049
[5] A. D. Barbour, L. H. Y. Chen and K. P. Choi. Poisson approximation for unbounded functions. I. Independent summands. Statist. Sinica 5 (1995) 749-766. · Zbl 0826.60024
[6] A. D. Barbour, L. Holst and S. Janson. Poisson Approximation . Oxford Univ. Press, Oxford, 1992. · Zbl 0746.60002
[7] L. H. Y. Chen. Poisson approximation for dependent trials. Ann. Probab. 3 (1975) 534-545. · Zbl 0335.60016
[8] L. H. Y. Chen and Q.-M. Shao. A non-uniform Berry-Esseen bound via Stein’s method. Probab. Theory Related Fields 120 (2001) 236-254. · Zbl 0996.60029
[9] L. H. Y. Chen and Q.-M. Shao. Stein’s method for normal approximation. In An Introduction to Stein’s Method 1-59. Lecture Notes Series, IMS, National University of Singapore 4 . Singapore Univ. Press, Singapore, 2005.
[10] L. H. Y. Chen and Q.-M. Shao. Normal approximation for nonlinear statistics using a concentration inequality approach. Bernoulli 13 (2007) 581-599. · Zbl 1146.62310
[11] N. El Karoui and Y. Jiao. Stein’s method and zero bias transformation for CDOs tranches pricing. Finance Stoch. 13 (2009) 151-180. · Zbl 1199.91063
[12] T. Erhardsson. Stein’s method for Poisson and compound Poisson approximation. In An Introduction to Stein’s Method 61-113. Lecture Notes Series, IMS, National University of Singapore 4 . Singapore Univ. Press, Singapore, 2005.
[13] L. Goldstein. L 1 bounds in normal approximation. Ann. Probab. 35 (2007) 1888-1930. · Zbl 1144.60018
[14] L. Goldstein. Bounds on the constant in the mean central limit theorem. Ann. Probab. 38 (2010) 1672-1689. · Zbl 1195.60034
[15] L. Goldstein and G. Reinert. Stein’s method and the zero bias transformation with application to simple random sampling. Ann. Appl. Probab. 7 (1997) 935-952. · Zbl 0903.60019
[16] L. Goldstein and G. Reinert. Distributional transformations, orthogonal polynomials, and stein characterizations. J. Theoret. Probab. 18 (2005) 237-260. · Zbl 1072.62002
[17] F. Götze and C. Hipp. Asymptotic expansions in the central limit theorem under moment conditions. Z. Wahrsch. Verw. Gebiete 42 (1978) 67-87. · Zbl 0369.60027
[18] C. Hipp. Edgeworth expansions for integrals of smooth functions. Ann. Probab. 5 (1977) 1004-1011. · Zbl 0375.60032
[19] Y. Jiao. Risque de crédit: modélisation et simulation numérique. PhD thesis, Ecole Polytechnique, 2006. Available at . · Zbl 1188.91008
[20] A. Kolmogolov and S. Fomine. Éléments de la théorie des fonctions et de l’analyse fonctionnelle . Éditions Mir., Moscow, 1974.
[21] V. V. Petrov. Sums of Independent Random Variables . Springer, New York, 1975. · Zbl 0322.60043
[22] V. Rotar. Stein’s method, Edgeworth’s expansions and a formula of Barbour. In Stein’s Method and Applications 59-84. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 5 . Singapore Univ. Press, Singapore, 2005.
[23] P. Smith. A recursive formulation of the old problem of obtaining moments from cumulants and vice versa. Amer. Statist. 49 (1995) 217-218. · Zbl 04527603
[24] C. Stein. A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proc. Sixth Berkeley Symp. Math. Statist. Probab. 583-602. California Univ. Press, Berkeley, 1972. · Zbl 0278.60026
[25] C. Stein. Approximate Computation of Expectations . IMS, Hayward, CA, 1986. · Zbl 0721.60016
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