## 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:

  Barbe, Ph., McCormick, W.P.: Asymptotic expansions for infinite weighted convolutions of rapidly varying subexponential distributions. Probab. Theory Relat. Fields 141(1–2), 155–180 (2008) · Zbl 1142.60034  Chistyakov, V.P.: A theorem on sums of independent positive random variables and its applications to branching random processes. Theory Probab. Appl. 9, 640–648 (1964) · Zbl 0203.19401  Chover, J., Ney, P., Wainger, S.: Functions of probability measures. J. Anal. Math. 26, 255–302 (1973a) · Zbl 0276.60018  Chover, J., Ney, P., Wainger, S.: Degeneracy properties of subcritical branching processes. Ann. Probab. 1, 663–673 (1973b) · Zbl 0387.60097  Cline, D.B.H.: Convolution tails, product tails and domains of attraction. Probab. Theory Relat. Fields 72(4), 529–557 (1986) · Zbl 0577.60019  Cline, D.B.H., Samorodnitsky, G.: Subexponentiality of the product of independent random variables. Stoch. Process. Their Appl. 49(1), 75–98 (1994) · Zbl 0799.60015  Embrechts, P., Goldie, C.M.: On closure and factorization properties of subexponential and related distributions. J. Aust. Math. Soc. Ser. A 29(2), 243–256 (1980) · Zbl 0425.60011  Emmer, S., Klüppelberg, C., Korn, R.: Optimal portfolios with bounded capital at risk. Math. Financ. 11(4), 365–384 (2001) · Zbl 1038.91044  Foss, S., Korshunov, D.: Lower limits and equivalences for convolution tails. Ann. Probab. 35(1), 366–383 (2007) · Zbl 1129.60014  Hashorva, E.: Extremes of asymptotically spherical and elliptical random vectors. Insur. Math. Econ. 36(3), 285–302 (2005) · Zbl 1110.62023  Kalashnikov, V., Norberg, R.: Power tailed ruin probabilities in the presence of risky investments. Stoch. Process. Their Appl. 98(2), 211–228 (2002) · Zbl 1058.60095  Klüppelberg, C., Kostadinova, R.: Integrated insurance risk models with exponential Lévy investment. Insur. Math. Econ. 42(2), 560–577 (2008) · Zbl 1152.60325  Norberg, R.: Ruin problems with assets and liabilities of diffusion type. Stoch. Process. Their Appl. 81(2), 255–269 (1999) · Zbl 0962.60075  Pakes, A.G.: Convolution equivalence and infinite divisibility. J. Appl. Probab. 41(2), 407–424 (2004) · Zbl 1051.60019  Shimura, T., Watanabe, T.: Infinite divisibility and generalized subexponentiality. Bernoulli 11(3), 445–469 (2005) · Zbl 1081.60016  Su, C., Chen, Y.: On the behavior of the product of independent random variables. Sci. China Ser. A 49(3), 342–359 (2006) · Zbl 1106.60018  Tang, Q.: On convolution equivalence with applications. Bernoulli 12(3), 535–549 (2006a) · Zbl 1114.60015  Tang, Q.: The subexponentiality of products revisited. Extremes 9(3–4), 231–241 (2006b) · Zbl 1142.60012  Tang, Q., Tsitsiashvili, G.: Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stoch. Process. Their Appl. 108(2), 299–325 (2003) · Zbl 1075.91563  Tang, Q., Tsitsiashvili, G.: Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments. Adv. Appl. Probab. 36(4), 1278–1299 (2004) · Zbl 1095.91040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.