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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

MSC:
 60F15 Strong limit theorems 60E15 Inequalities; stochastic orderings
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