Asymptotic expansions for large deviation probabilities in the strong law of large numbers
Let \(X_ 1,X_ 2,..\). be a sequence of independent random variables with common distribution function F having zero mean, and let \((S_ n)\) be the random walk of partial sums. The weak and strong laws of large numbers, respectively, imply that for any \(\alpha\) \(\in {\mathbb{R}}\) and \(\epsilon >0\) the probabilities \(P\{S_ m>\alpha +\epsilon m\}\) and \(p_ m:=P\{S_ n>\alpha +\epsilon n\) for some \(n\geq m\}\) tend to 0 as m tends to \(\infty.\) Building upon work of \textit{R. R. Bahadur} and \textit{R. Ranga Rao} [Ann. Math. Stat. 31, 1015-1027 (1960; Zbl 0101.126)], \textit{D. Siegmund} [Z. Wahrscheinlichkeitstheor. verw. Geb. 31, 107-113 (1975; Zbl 0326.60028)], and the author and \textit{M. J. Wichura} [see the review above (Zbl 0643.60024)], we produce complete asymptotic expansions for the probabilities \(P\{S_ m>\alpha +\epsilon m\}\) and \(p_ m\).
scientific article; zbMATH DE number 4048762