Probabilities of very large deviations

Journal of the Australian Mathematical Society

Stephen A. Book¹ 

关键词: large deviations;    asymptotic expansions;    moment-generating functions;    laws of large numbers.;   

DOI  :  10.1017/S144678870002108X

学科分类：数学（综合）

来源: Cambridge University Press

【 摘 要 】

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.

【 授权许可】

Unknown