Reversing the BerryEsseen inequality
Authors:
Peter Hall and A. D. Barbour
Journal:
Proc. Amer. Math. Soc. 90 (1984), 107110
MSC:
Primary 60F05; Secondary 60E15, 60G50
MathSciNet review:
722426
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Abstract: We derive a lower bound to the rate of convergence in the central limit theorem. Our result is expressed in terms similar to those of the BerryEsséen inequality, with the distance between two distributions on one side of the inequality and an easily calculated function of the summands on the other, related by a universal constant. The proof is based on Stein's method.
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 A. C. Berry, The accuracy of the Gaussian approximation to the sum of independent variates, Trans. Amer. Math. Soc. 49 (1941), 122136. MR 0003498 (2:228i)
 [2]
 A. Bikyalis, Estimates of the remainder term in the central limit theorem, Litovsk. Mat. Sb. 6 (1966), 323346. (Russian) MR 0210173 (35:1067)
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 C.G. Esséen, Fourier analysis of distribution functions. A mathematical study of the LaplaceGaussian law, Acta Math. 77 (1945), 1125. MR 0014626 (7:312a)
 [4]
 W. Feller, On the BerryEsséen theorem, Z. Wahrsch. Verw. Gebiete 10 (1968), 261268. MR 0239639 (39:996)
 [5]
 P. Hall, Characterizing the rate of convergence in the central limit theorem, Ann. Probab. 8 (1980), 10371048. MR 602378 (82k:60045a)
 [6]
 , Rates of convergence in the central limit theorem, Pitman, London, 1982.
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 L. V. Osipov and V. V. Petrov, On an estimate of the remainder term in the central limit theorem, Theor. Probab. Appl. 12 (1967), 281286. MR 0216552 (35:7383)
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 L. V. Rozovskii, A lower bound to the remainder term in the central limit theorem, Mat. Zametki 24 (1978), 403410. (Russian) MR 511643 (80c:60040)
 [9]
 , On the precision of an estimate of the remainder term in the central limit theorem, Theor. Probab. Appl. 23 (1978), 712730.
 [10]
 C. Stein, A bound for the error in the normal approximation to the distribution of a sum of dependent variables, Proc. Sixth Berkeley Sympos. Math. Statist. Prob., Vol. 2, Univ. of California Press, Berkeley, Calif., 1970.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002993919840722426X
PII:
S 00029939(1984)0722426X
Keywords:
BerryEsséen inequality,
central limit theorem,
lower bound,
rate of convergence,
sums of independent variables
Article copyright:
© Copyright 1984
American Mathematical Society
