The Annals of Probability

L1 bounds in normal approximation

Larry Goldstein

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The zero bias distribution W* of W, defined though the characterizing equation EWf(W)=σ2Ef'(W*) for all smooth functions f, exists for all W with mean zero and finite variance σ2. For W and W* defined on the same probability space, the L1 distance between F, the distribution function of W with EW=0 and Var(W)=1, and the cumulative standard normal Φ has the simple upper bound


This inequality is used to provide explicit L1 bounds with moderate-sized constants for independent sums, projections of cone measure on the sphere S(np), simple random sampling and combinatorial central limit theorems.

Article information

Ann. Probab., Volume 35, Number 5 (2007), 1888-1930.

First available in Project Euclid: 5 September 2007

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Mathematical Reviews number (MathSciNet)

Primary: 60F05: Central limit and other weak theorems 60F25: $L^p$-limit theorems 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60C05: Combinatorial probability

Stein’s method Berry–Esseen cone measure sampling combinatorial CLT


Goldstein, Larry. L 1 bounds in normal approximation. Ann. Probab. 35 (2007), no. 5, 1888--1930. doi:10.1214/009117906000001123.

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  • Anttila, M., Ball, K. and Perissinaki, I. (2003). The central limit problem for convex bodies. Trans. Amer. Math. Soc. 355 4723--4735.
  • von Bahr, B. (1976). Remainder term estimate in a combinatorial limit theorem. Z. Wahrsch. Verw. Gebiete 35 131--139.
  • Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Univ. Press.
  • Bickel, P. and Doksum, K. (1977). Mathematical Statistics: Basic Ideas and Selected Topics. Holden-Day, San Francisco.
  • Bobkov, S. (2003). On concentration of distributions of random weighted sums. Ann. Probab. 31 195--215.
  • Bolthausen, E. (1984). An estimate of the reminder in a combinatorial central limit theorem. Z. Wahrsch. Verw. Gebiete 66 379--386.
  • Diaconis, P. and Freedman, D. (1987). A dozen de Finetti-style results in search of a theory. Ann. Inst. H. Poincaré Probab. Statist. 23 397--423.
  • Goldstein, L. (2004). Normal approximation for hierarchical sequences Ann. Appl. Probab. 14 1950--1969.
  • Goldstein, L. (2005). Berry--Esseen bounds for combinatorial central limit theorems and pattern occurrences, using zero and size biasing. J. Appl. Probab. 42 661--683.
  • Goldstein, L. and Reinert, G. (1997). Stein's method and the zero bias transformation with application to simple random sampling Ann. Appl. Probab. 7 935--952.
  • Goldstein, L. and Rinott, Y. (1996). On multivariate normal approximations by Stein's method and size bias couplings. J. Appl. Probab. 33 1--17.
  • Hoeffding, W. (1951). A combinatorial central limit theorem. Ann. Math. Statist. 22 558--566.
  • Hoeffding, W. (1955). The extrema of the expected value of a function of independent random variables. Ann. Math. Statist. 26 268--275.
  • Ho, S. T. and Chen, L. H. Y. (1978). An $L_p$ bound for the remainder in a combinatorial central limit theorem. Ann. Probab. 6 231--249.
  • Lefvre, C. and Utev, S. (2003). Exact norms of a Stein-type operator and associated stochastic orderings. Probab. Theory Related Fields 127 353--366.
  • Meckes, M. and Meckes, E. (2007). The central limit problem for random vectors with symmetries. J. Theoret. Probab. To appear. Available at arXiv:math.PR/0505618.
  • Motoo, M. (1957). On the Hoeffding's combinatorial central limit theorem. Ann. Inst. Statist. Math. Tokyo 8 145--154.
  • Naor, A. and Romik, D. (2003). Projecting the surface measure of the sphere of $\ell_p^n$. Ann. Inst. H. Poincaré Probab. Statist. 39 241--261.
  • Rachev, S. T. (1984). The Monge--Kantorovich transference problem and its stochastic applications. Theory Probab. Appl. 29 647--676.
  • Schechtman, G. and Zinn, J. (1990). On the volume of the intersection of two $\ell_p^n$ balls. Proc. Amer. Math. Soc. 110 217--224.
  • Stein, C. (1972). 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. 2 583--602. Univ. California Press, Berkeley.
  • Stein, C. (1981). Estimation of the mean of a multivariate normal distribution. Ann. Statist. 9 1135--1151.
  • Stein, C. (1986). Approximate Computation of Expectations. IMS, Hayward, CA.
  • Sudakov, V. (1978). Typical distributions of linear functionals in finite-dimensional spaces of higher dimension. Soviet Math. Dokl. 19 1578--1582. [Translated from Dokl. Akad. Nauk SSSR 243 (1978) 1402--1405.]
  • Wald, A. and Wolfowitz, J. (1944). Statistical tests based on permutations of the observations. Ann. Math. Statist. 15 358--372.