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From light tails to heavy tails through multiplier. (English) Zbl 1199.60040

Let \(X\) and \(Y\) be two independent nonnegative random variables with distributions \(F\) and \(G\), respectively, and let \(H\) be the distribution of their product \(Z=XY\). The author studies how the tail behaviour of the product \(Z\) is effected by the tail behaviour of \(X\) given that \(F\) belongs to the class \(\mathcal L(\gamma)\) or \(\mathcal S(\gamma)\) for some \(\gamma\geq0\). Recall that a distribution \(F\) on \([0,\infty)\) is said to belong to the class \(\mathcal L(\gamma)\) for some \(\gamma\geq0\) if the relation \(\lim_{x\to\infty}\bar{F}(x-u)/\bar{F}(x)=e^{\gamma u}\) holds for all \(u\). A distribution \(F\) on \([0,\infty)\) is said to belong to the class \(\mathcal S(\gamma)\) if \(F\in\mathcal L(\gamma)\) and \(\lim_{x\to\infty}\overline{F^{\ast2}}(x)/\bar{F}(x)=2c\) exists and is finite, where \(F^{\ast2}\) denotes the convolution of \(F\) with itself, the constant \(c\) is equal to \(\int_{0-}^{\infty}e^{\gamma x}F(dx)\). If \(\gamma=0\), then the limiting relations describe the well-known long-tailed distribution class \(\mathcal L(0)\) and subexponential distribution class \(\mathcal S(0)\), respectively. Consider \(F\in\mathcal L(\gamma)\) for some \(\gamma\geq0\). Define the (upper) endpoint of \(Y\) as \(\hat{y}=\sup\{y:P(Y\leq y)<1\}\). Lemma A.4 of Q. Tang and G. Tsitsiashvili [Adv. Appl. Probab. 36, No. 4, 1278-1299 (2004; Zbl 1095.91040)] shows that if \(F\in\mathcal L(\gamma)\) for some \(\gamma\geq0\) and \(0<\hat{y}<\infty\), then \(H\in\mathcal L(\gamma/\hat{y})\). This leads to conjecture that if \(F\in\mathcal L(\gamma)\) for \(\gamma\geq0\) and \(\hat{y}=\infty\), then \(H\in\mathcal L(0)\). This is not true, in general. The author gives a counterexample. The corresponding theorem is proved.
Theorem 1. Let \(F\in\mathcal L(\gamma)\) for \(\gamma\geq0\) and \(\hat{y}=\infty\). Then \(H\in\mathcal L(0)\) if and only if either: (A) \(D[F]=\emptyset\), or (B) \(D[F]\not=\emptyset\) and the relation \(G\left(x/d,(x+1)/d\right]=o(1)\bar{H}(x)\) holds true for all \(d\in D[F]\), where \(D[F]\) is the set of all possible discontinuities of \(F\).
Note, that in their Theorems 2.1 and 2.2, C. Su and Y. Chen [Sci. China, Ser. A 49, No. 3, 342–359 (2006; Zbl 1106.60018)] have obtained that if \(F\in\mathcal L(\gamma)\) for \(\gamma\geq0\), \(\hat{y}=\infty\), and \(D[F]=\emptyset\), then \(H\in\mathcal L(0)\).
Next consider \(F\in\mathcal S(\gamma)\) for some \(\gamma\geq0\). Theorem 1.1 of Q. Tang [Bernoulli 12, No. 3, 535–549 (2006; Zbl 1114.60015)] shows that if \(F\in\mathcal S(\gamma)\) for \(\gamma\geq0\) and \(0<\hat{y}<\infty\), then \(H\in\mathcal S(\gamma/\hat{y})\). The aim is an extension of this result to the case \(\hat{y}=\infty\). Based on the proposed counterexample, we see that, in general, the conditions \(F\in\mathcal S(\gamma)\) for \(\gamma\geq0\) and \(\hat{y}=\infty\) can not guarantee \(H\in\mathcal S(0)\). Theorem 2.1 of Q. Tang [Extremes 9, No. 3-4, 231–241 (2006; Zbl 1142.60012)] shows that if \(F\in\mathcal L(0)\), \(\lim\sup\bar{F}(vx)/\bar{F}(x)<1\) for some \(v>1\), and relation \(\lim_{x\to\infty}\bar{G}(cx)/\bar{H}(x)=0\) holds for all \(c>0\), then \(H\in\mathcal S(0)\). In this paper the author obtains the following result.
Theorem 2. Let \(F\in\mathcal S(\gamma)\) for \(\gamma>0\) and \(\hat{y}=\infty\). If the indicated limiting relation holds true for all \(c>0\), then \(H\in\mathcal S(0)\).
For more relative results see Q. Tang [“Subexponential tails in the world of dependence”, Proceedings of the 5th Conference in Actuarial Science and Finance on Samos, 98–139 (2008) http://www.actuar.aegean.gr/samos2008].

MSC:

60E05 Probability distributions: general theory
62E20 Asymptotic distribution theory in statistics
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