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A Frequency-function form of the central limit theorem

Published online by Cambridge University Press:  24 October 2008

Walter L. Smith
Walter L. Smith
Affiliation:
Statistical LaboratoryCambridge
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Extract

The central limit theorem in the calculus of probability has been extensively studied in recent years. In its simplest form the theorem states that if X1, X2,… is a sequence of independent, identically distributed random variables of mean zero, then under general conditions the distribution function of Zm = (X1 + … + Xn)/√ n converges as n → ∞ to the normal or Gaussian distribution function. This form of the theorem in terms of distribution functions is the one required in statistical work, since it enables statements to be made about the limiting behaviour of prob {a ≤ Zn ≤ b}.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 49 , Issue 3 , July 1953 , pp. 462 - 472
Copyright
Copyright © Cambridge Philosophical Society 1953

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References

REFERENCES

(1)Cramér, H.Skand. AktuarTidskr. (1928), p. 13.Google Scholar
(2)Cramér, H.Random variables and probability distributions (Camb. Tracts Math. no. 36, 1937).Google Scholar
(3)Hammersley, J. M.Proc. Camb. phil. Soc. 48 (1952), 592–9.Google Scholar
(4)Khintchine, A. I.Mathematical foundations of statistical mechanics (New York, 1949), pp. 166–74.Google Scholar
(5)McShane, E. J.Integration (Princeton, 1947), p. 202.Google Scholar
(6)Titchmarsh, E. C.Introduction to the theory of Fourier integrals (Oxford, 1948), p. 172.Google Scholar