Geodesic
Expectation of the Ratio of the Sum of Squares to the Square of the Sum: Exact and Asymptotic Results
Teoriâ veroâtnostej i ee primeneniâ, Tome 46 (2001) no. 2, pp. 297-310

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Let $X_i$, $i=1,\dots,n$, be a sequence of positive independent identically distributed random variables. Define $$ R_n:=\mathbf{E}\frac{X_1^2+X_2^2+\dots+X_n^2}{(X_1+X_2+\dots+X_n)^2}. $$ Let $\varphi(s)=\mathbf{E}e^{-sX}$. We give an explicit representation of $R_n $ in terms of $\varphi$, and with the help of the Karamata theory of functions of regular variation, we study the asymptotic behavior of $R_n$ for large $n$.
Keywords: Karamata theory, functions of regular variation, domain of attraction of a stable law, Doeblin's universal law. 
@article{TVP_2001_46_2_a4,
     author = {A. Fuchs and A. Joffe and J. L. Teugels},
     title = {Expectation of the {Ratio} of the {Sum} of {Squares} to the {Square} of the {Sum:} {Exact} and {Asymptotic} {Results}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {297--310},
     publisher = {mathdoc},
     volume = {46},
     number = {2},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2001_46_2_a4/}
}
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A. Fuchs; A. Joffe; J. L. Teugels. Expectation of the Ratio of the Sum of Squares to the Square of the Sum: Exact and Asymptotic Results. Teoriâ veroâtnostej i ee primeneniâ, Tome 46 (2001) no. 2, pp. 297-310. http://geodesic.mathdoc.fr/item/TVP_2001_46_2_a4/