Variations on the berry-esseen theorem

B. Klartag*, S. Sodin

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


Suppose that X 1,...,X n are independent, identically distributed random variables of mean zero and variance one. Assume that E|X 1| 4 ≦ δ 4. We observe that there exist many choices of coefficients θ 1,..., θ n ∈ R with Σ j θ 2 j = 1 for which sup, where C > 0 is a universal constant. This inequality should be compared with the classical Berry-Esseen theorem, according to which the left-hand side may decay with n at the slower rate of O(1/ √ n) for the unit vector θ = (1, ..., 1)/ √ n. An explicit, universal example for coefficients θ = (θ1, ..., θn) for which this inequality holds is θ = (1, √ 2,-1,- √ 2, 1, √ 2,-1,- √ 2, ...)(3n/2) -1/2, when n is divisible by four. Parts of the argument are applicable also in the more general case, in which X 1, ..., X n are independent random variables of mean zero and variance one yet are not necessarily identically distributed. In this general setting, the bound above holds with δ 4 = n -1Σ n j=1 E|X j | 4 for most selections of a unit vector θ = (θ 1, ..., θ n) ∈ R n. Here "most" refers to the uniform probability measure on the unit sphere.

Original languageEnglish
Pages (from-to)403-419
Number of pages17
JournalTheory of Probability and its Applications
Issue number3
StatePublished - 2012


  • Berry-esseen theorem
  • Central limit theorem
  • Gaussian distribution


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