Geodesic
On the Difference Between the Distribution Function of the sum and the Maximum of Real Random Variables
Publications de l'Institut Mathématique, _N_S_71 (2002) no. 85, p. 63 .

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Let $X$ denote a nonnegative random variable with distribution function (d.f.) $F(x)$. If $F(x)$ is a subexponential d.f. it is well known that the tails of the d.f. of the partial sums and the partial maxima are asymptotically the same. In this paper among others we analyse subexponential d.f. on the real line. It is easy to prove that again partial sums and partial maxima have asymptotically the same d.f.. In this paper we analyse the difference between these two distribution functions. In the main part of the paper we consider independent real random variables $X$ and $Y$ with d.f. $F(x)$ and $G(x)$. Under various conditions we obtain a variety of $O$-, $o$- and exact (asymptotic) estimates for $D(x)=F(x)G(x)-F\star G(x)$ and $R(x)=P(X+Y>x)-P(X>x)-P(Y>x)$. Our results generalize the results of Omey (1994) and Omey and Willekens (1986) where the case $X\geq 0$, $Y\geq 0$ was treated.
DOI : 10.2298/PIM0271063O
Classification : 60E99 60G50 26A12
Keywords: regular variation, subexponential distributions, O-regular variation 
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     journal = {Publications de l'Institut Math\'ematique},
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Edward Omey. On the Difference Between the Distribution Function of the sum and the Maximum of Real Random Variables. Publications de l'Institut Mathématique, _N_S_71 (2002) no. 85, p. 63 . doi : 10.2298/PIM0271063O. http://geodesic.mathdoc.fr/articles/10.2298/PIM0271063O/

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