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Teoriya Veroyatnostei i ee Primeneniya, 2001, Volume 46, Issue 2, Pages 297–310
DOI: https://doi.org/10.4213/tvp3919 (Mi tvp3919) 		 
This article is cited in 15 scientific papers (total in 15 papers)

Expectation of the Ratio of the Sum of Squares to the Square of the Sum: Exact and Asymptotic Results

A. Fuchsa, A. Joffeb, J. L. Teugelsc

a Université de Strasbourg
b Université de Montréal, Département de Mathématiques et de Statistique
c Katholieke Universiteit Leuven
Full-text PDF (1328 kB) Citations (15)
DOI: https://doi.org/10.4213/tvp3919
Abstract: 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.
Received: 19.05.2000
English version:
Theory of Probability and its Applications, 2002, Volume 46, Issue 2, Pages 243–255
DOI: https://doi.org/10.1137/S0040585X97978919
Bibliographic databases:
Language: Russian
Citation: 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”, Teor. Veroyatnost. i Primenen., 46:2 (2001), 297–310; Theory Probab. Appl., 46:2 (2002), 243–255
Citation in format AMSBIB
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\jour Theory Probab. Appl.
\yr 2002
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\issue 2
\pages 243--255
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  • https://doi.org/10.4213/tvp3919
  • https://www.mathnet.ru/eng/tvp/v46/i2/p297
    • This publication is cited in the following 15 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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