Reversing the Berry-Esseen inequality

Authors:
Peter Hall and A. D. Barbour

Journal:
Proc. Amer. Math. Soc. **90** (1984), 107-110

MSC:
Primary 60F05; Secondary 60E15, 60G50

DOI:
https://doi.org/10.1090/S0002-9939-1984-0722426-X

MathSciNet review:
722426

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Abstract | References | Similar Articles | Additional Information

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 Berry-Esséen 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.

**[1]**A. C. Berry,*The accuracy of the Gaussian approximation to the sum of independent variates*, Trans. Amer. Math. Soc.**49**(1941), 122-136. MR**0003498 (2:228i)****[2]**A. Bikyalis,*Estimates of the remainder term in the central limit theorem*, Litovsk. Mat. Sb.**6**(1966), 323-346. (Russian) MR**0210173 (35:1067)****[3]**C.-G. Esséen,*Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law*, Acta Math.**77**(1945), 1-125. MR**0014626 (7:312a)****[4]**W. Feller,*On the Berry-Esséen theorem*, Z. Wahrsch. Verw. Gebiete**10**(1968), 261-268. MR**0239639 (39:996)****[5]**P. Hall,*Characterizing the rate of convergence in the central limit theorem*, Ann. Probab.**8**(1980), 1037-1048. MR**602378 (82k:60045a)****[6]**-,*Rates of convergence in the central limit theorem*, Pitman, London, 1982.**[7]**L. V. Osipov and V. V. Petrov,*On an estimate of the remainder term in the central limit theorem*, Theor. Probab. Appl.**12**(1967), 281-286. MR**0216552 (35:7383)****[8]**L. V. Rozovskii,*A lower bound to the remainder term in the central limit theorem*, Mat. Zametki**24**(1978), 403-410. (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), 712-730.**[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:
https://doi.org/10.1090/S0002-9939-1984-0722426-X

Keywords:
Berry-Esséen inequality,
central limit theorem,
lower bound,
rate of convergence,
sums of independent variables

Article copyright:
© Copyright 1984
American Mathematical Society