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An Asymptotic Formula for a Class of Distribution Functions
Published online by Cambridge University Press: 20 January 2009
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If x1, x2, …., xk, …. are independent random variables each of which is subjected to a distribution law σ = σ(x) independent of k and having a finite positive dispersion, then x1 + x2 + …. + xn is known to obey the Gauss law as n→ + ∞, no matter how σ (x) be chosen. There arises, however, the question whether it is nevertheless possible to determine the elementary law σ (x) from the asymptotic behaviour of the distribution law of x1 + x2 + …. + xn for very large but finite values of n. It will be shown that the answer is affirmative under very general conditions.
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- Research Article
- Information
- Proceedings of the Edinburgh Mathematical Society , Volume 4 , Issue 3 , August 1935 , pp. 138 - 143
- Copyright
- Copyright © Edinburgh Mathematical Society 1935
References
page 138 note 1 CfLévy, P., Calcul des Probabilités, Paris, 1925, pp. 233–235.Google Scholar
page 139 note 1 Lévy, P., op. dt., p. 167.Google Scholar
page 139 note 2 Lévy, P., op. dt., p. 171.Google Scholar
page 138 note 3 There exists a derivative σ′(x) up to a set of measure zero even if σ (x) is not absolutely continuous, but
holds if and only if σ(x) is absolutely continuous. It is meaningless to regard σ(x) as a density of probability if (i) is not valid.
page 140 note 1 CfMenchoff, D., “Sur l'unicité du développemenf trigonométrique,” Comptes Rendus, 163 (1916), pp. 433–436.Google Scholar
page 140 note 2 Cf., e.g., Lévy, P., op. cit., pp. 184–185.Google Scholar
page 140 note 3 Ibid, pp. 233–235.
page 140 note 4 Cf., e.g., Whittaker, E.T. and Robinson, G., The Calculus of Observations, London, 1924, p. 172Google Scholar; cf. also Zernike, F., Zernike, Handbuch der Physik, 3 (1928) 450–51, where further references are also given.Google Scholar
page 141 note 1 It is not true, however, that L(t; σ) is necessarily regular-analytic along the iaxis.
page 141 note 2 In virtue of the Schwarz inequality it is sufficient to consider even values of m.
page 141 note 3 It may be mentioned that (8) is actually false in the second case. In fact, L(t; σ) is then a periodic function so that L(t; σ) = 1 holds for some t ╪ 0 since it holds for t = 0.
page 142 note 1 CfTchebychef, P.L., Oeuvres, vol. 1, St. Pétersburg, 1899, pp. 251–270, where analyticity of the functions is not required.Google Scholar
page 142 note 2 CfWintner, A., “On the asymptotic formulae of Riemann and of Laplace,” Proceedings of the National Academy of Sciences, 20 (1934), pp. 57–62.Google Scholar
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