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Subexponentiality of the product of independent random variables. (English) Zbl 0799.60015
A distribution function (d.f.) \(F\) on \([0,\infty)\) is called subexponential if \(F(t)<1\;\forall t\) and \((1-F *F(t))/(1-F(t))\to 2\) as \(t\to\infty\), where \(*\) denotes convolution. The class of subexponential d.f.’s, which is denoted by \({\mathcal S}\), has been widely studied, see e.g. C. M. Goldie and S. Resnick [Adv. Appl. Probab. 20, No. 4, 706-718 (1988; Zbl 0659.60028)]. The present authors deal principally with the following question. If \(X\) has d.f. \(F\in{\mathcal S}\) and \(Y\) is independent of \(X\), what are sufficient conditions on the d.f. of \(Y\) for the d.f. of the product \(XY\) (rather than the sum \(X+Y\)) to be in \({\mathcal S}\)? The relationship between \(\overline F(t)=P(X>t)\) and \(P(XY>t)\) is also studied for special cases where \(\overline F\) satisfies one of the extensions of regular variation.
Reviewer: M.Quine

MSC:
60E05 Probability distributions: general theory
60F99 Limit theorems in probability theory
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