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The convergence of a sum of independent random variables
Published online by Cambridge University Press: 24 October 2008
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Let Xν, ν= l, 2, …, n be n independent random variables in k-dimensional (real) Euclidean space Rk, which have, for each ν, finite fourth moments β4ii = l,…, k. In the case when the Xν are identically distributed, have zero means, and unit covariance matrices, Esseen(1) has discussed the rate of convergence of the distribution of the sums
If 
denotes the projection of 
on the ith coordinate axis, Esseen proves that if
and ψ(a) denotes the corresponding normal (radial) distribution function of the same first and second moments as μn(a), then
where 
and C is a constant depending only on k. (C, without a subscript, will denote everywhere a constant depending only on k.)
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 59 , Issue 2 , April 1963 , pp. 411 - 416
- Copyright
- Copyright © Cambridge Philosophical Society 1963
References
REFERENCES
(1)Esseen, C. G., Fourier analysis of distribution functions. Acta Math. 77 (1945), 1–125.CrossRefGoogle Scholar
(2)Gnedenko, B. V., and Kolmogorov, A. N., Limit distributions for sums of independent random variables (Addison Wesley; Reading, Mass., 1954).Google Scholar
(3)Bergström, H., On the central limit theorem in the case of not equally distributed random variables. Skand. Aktuarietidskr. 32 (1949), 37–62.Google Scholar
(4)Kolmogorov, A. N., Deux théorèmes asymptotiques uniformes pour des sommes des variables aléatoires. Teor. Veroyatnost. i Primenen, 1 (1956), 426–436. (Russian, French summary.)Google Scholar