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On accuracy of approximation in limit theorems for large deviations. (English) Zbl 0582.60037

From the author’s abstract: Let \(X_ 1,X_ 2,..\). be a sequence of independent random variables which have a common distribution function and are such that \(EX_ 1=0\) and \(EX^ 2_ 1=1\). In this article, necessary and sufficient conditions are investigated under which various asymptotic representations of the ratio of \(P\{X_ 1+...+X_ n\geq x\}\) to 1-\(\Phi\) (x/\(\sqrt{n})\) can be obtained as \(n\to \infty\) when x/\(\sqrt{n}\uparrow \infty\), x remaining in an interval of the type ”r\(\Lambda\) (\(\sqrt{n})\) to \(\Lambda\) (\(\sqrt{n})''\), for example. Here \(\Phi\) is the standard normal distribution function, while \(0<r<1\) and the function \(\Lambda\) (z) is such that, for z sufficiently large, \(\Lambda\) (z)/z increases monotonically to \(\infty\), and for some \(\epsilon_ 0\), \(0<\epsilon_ 0<1\), \(\Lambda (z)/z^{1+\epsilon_ 0}\) decreases monotonically.
Reviewer: J.Steinebach

MSC:

60F10 Large deviations
60F05 Central limit and other weak theorems
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