## 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 $$u>0$$.

### MSC:

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

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