## 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:

 6e+06 Probability distributions: general theory 6.2e+21 Asymptotic distribution theory in statistics

### Citations:

Zbl 1095.91040; Zbl 1106.60018; Zbl 1114.60015; Zbl 1142.60012
Full Text:

### References:

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