## Abstract

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 ^{-1}Σ ^{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 language | English |
---|---|

Pages (from-to) | 403-419 |

Number of pages | 17 |

Journal | Theory of Probability and its Applications |

Volume | 56 |

Issue number | 3 |

DOIs | |

State | Published - 2012 |

## Keywords

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