The Annals of Probability

Exact convergence rate and leading term in central limit theorem for student’s t statistic

Peter Hall and Qiying Wang

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The leading term in the normal approximation to the distribution of Student’s t statistic is derived in a general setting, with the sole assumption being that the sampled distribution is in the domain of attraction of a normal law. The form of the leading term is shown to have its origin in the way in which extreme data influence properties of the Studentized sum. The leading-term approximation is used to give the exact rate of convergence in the central limit theorem up to order n1/2, where n denotes sample size. It is proved that the exact rate uniformly on the whole real line is identical to the exact rate on sets of just three points. Moreover, the exact rate is identical to that for the non-Studentized sum when the latter is normalized for scale using a truncated form of variance, but when the corresponding truncated centering constant is omitted. Examples of characterizations of convergence rates are also given. It is shown that, in some instances, their validity uniformly on the whole real line is equivalent to their validity on just two symmetric points.

Article information

Ann. Probab., Volume 32, Number 2 (2004), 1419-1437.

First available in Project Euclid: 18 May 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 62E20: Asymptotic distribution theory

Berry–Esseen theorem characterization of rate of convergence domain of attraction Edgeworth expansion random norm rate of convergence self-normalized sum Studentize


Hall, Peter; Wang, Qiying. Exact convergence rate and leading term in central limit theorem for student’s t statistic. Ann. Probab. 32 (2004), no. 2, 1419--1437. doi:10.1214/009117904000000252.

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