On the difference between the distribution function of the sum and the maximum of real random variables

Abstract

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 analyze 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*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>=0, Y>=0 was treated. .