The expected ratio of the sum of squares to the square of the sum. (English) Zbl 0545.60034

For positive i.i.d. random variables \(X_ 1,X_ 2,..\). define \(S_ n=X_ 1+...+X_ n\), \(T_ n=X^ 2_ 1+...+X^ 2_ n\), and \(R_ n=S_ n^{-2}T_ n\). It is shown that \(ER_ n\to 0\) if and only if the function \(x\to EX_ 11_{(X<x)}\) is slowly varying. In order to obtain reasonable rates for the convergence of \(ER_ n\), moment conditions must be imposed. So, e.g., \(ER_ n=O(1/n)\) if and only if \(EX^ 2_ 1<\infty\), and \(ER_ n=o(1/\log n)\) if \(EX_ 1\log X_ 1<\infty\). If the function \(x\to P(X>x)\) is regularly varying with exponent -1 in a strict sense (formula (26)), then \(ER_ n=O(1/\log n)\).
Reviewer: Ch.Hipp


60F15 Strong limit theorems
60E15 Inequalities; stochastic orderings
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