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Probabilities of very large deviations

Published online by Cambridge University Press:  09 April 2009

Stephen A. Book
Stephen A. Book
Affiliation:
Department of Mathematics, California State College, Dominguez Hills, California 90747.
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If {Xn: 1 ≦ n < ∞} are independent, identically distributed random variables having E(X1) = 0 and Var(X1) = 1, the most elementary form of the central limit theorem implies that P(n-½Sn≧ zn) → 0 as n → ∞, where Sn = Σnk=1 X,k, for all sequences {zn:1 ≧ n gt; ∞} for which zn → ∞. The probability P(n-½ Sn ≧ zn) is called a “large deviation probability”, and the rate at which it converges to 0 has been the subject of much study. The objective of the present article is to complement earlier results by describing its asymptotic behavior when n-½zn → ∞ as n → ∞, in the case of absolutely continuous random variables having moment-generating functions.

Keywords

large deviationsasymptotic expansionsmoment-generating functionslaws of large numbers.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 25 , Issue 3 , May 1978 , pp. 332 - 347
Copyright
Copyright © Australian Mathematical Society 1978

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