The subexponentiality of products revisited. (English) Zbl 1142.60012

A CDF \(F\) is called subexponential if \(\overline{F*F}(x)\sim 2\bar F(x)\) as \(x\to+\infty\) (\(\bar F(x)=1-F(x)\), \(*\) means convolution). \(F\) belongs to the class \(A\) iff \(F\) is subexponential and \( \lim\sup_{x\to+\infty}\bar F(vx)/\bar F(x)<1 \) for some \(v>1\). The main result of the paper is that if \(X\in R\) and \(Y\in R_{+}\) are independent r.v.s with CDFs \(F\) and \(G\) respectively, then the CDF \(H\) of \(X\cdot Y\) belongs to \(A\) if \(F\in A\) and \(\bar G(ux)=o(\bar H(x))\) for all \(u>0\).


60E05 Probability distributions: general theory
60G70 Extreme value theory; extremal stochastic processes
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