# zbMATH — the first resource for mathematics

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
##### Keywords:
subexponential; independent; regular variation
Full Text:
##### References:
 [1] Athreya, K.B.; Ney, P.E., Branching processes, (1972), Springer New York · Zbl 0259.60002 [2] Balkema, A.A.; de Haan, L., On R. von Mises’ condition for the domain of attraction of expe^−x, Ann. math. statist., 43, 1352-1354, (1972) · Zbl 0239.60018 [3] Berman, S.M., Sojourns and extremes of a diffusion process on a fixed interval, Adv. appl. probab., 14, 811-832, (1982) · Zbl 0494.60076 [4] Berman, S.M., Extreme sojourns of diffusion processes, Ann. probab., 16, 361-374, (1988) · Zbl 0637.60089 [5] Berman, S.M., The tail of the convolution of densities and its application to a model of HIV-latency time, Ann. appl. probab., 2, 481-502, (1992) · Zbl 0752.62014 [6] Bingham, N.H.; Goldie, C.M.; Teugels, J.L., Regular variation. encyclopedia of mathematics and its applications, (1989), Cambridge Univ. Press Cambridge, UK, paperback with additions · Zbl 0667.26003 [7] Borovkov, A.A., Stochastic processes in queueing theory, (1976), Springer New York · Zbl 0319.60057 [8] Breiman, L., On some limit theorems similar to the arc-sine law, Theory probab. appl., 10, 323-331, (1965) · Zbl 0147.37004 [9] 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 [10] Chover, J.; Ney, P.; Wainger, S., Functions of probability measures, J. anal. math., 26, 255-302, (1973) · Zbl 0276.60018 [11] Chover, J.; Ney, P.; Wainger, S., Degeneracy properties of subcritical branching processes, Ann. probab., 1, 663-673, (1973) · Zbl 0387.60097 [12] Cline, D.B.H., Convolution tails, product tails and domains of attraction, Probab. theory rel. fields, 72, 529-557, (1986) · Zbl 0577.60019 [13] Cline, D.B.H., Convolutions of distributions with exponential and subexponential tails, J. austral. math. soc. ser. A, 43, 347-365, (1987) · Zbl 0633.60021 [14] Cline, D.B.H., Consistency for least squares regression estimators with infinite variance, J. statist. plann. inference, 23, 163-179, (1989) · Zbl 0679.62054 [15] Cline, D.B.H., Intermediate regular- and II-variation, (1991), to appear in: Proc. London Math. Soc. [16] Cline, D.B.H.; Hsing, T., Large deviation probabilities for sums of random variables with heavy or subexponential tails, Tech. rept., (1990), Texas A&M Univ College Station, TX [17] Cline, D.B.H.; Resnick, S.I., Multivariate subexponential distributions, Stochastic process. appl., 42, 49-72, (1992) · Zbl 0751.62025 [18] Davis, R.A.; Resnick, S.I., Limit theory for moving averages of random variables with regularly varying tail probabilities, Ann. probab., 13, 179-195, (1985) · Zbl 0562.60026 [19] Davis, R.A.; Resnick, S.I., More limit theory for the sample correlation function of moving averages, Stochastic process. appl., 20, 257-279, (1985) · Zbl 0572.62075 [20] Davis, R.A.; Resnick, S.I., Limit theory for the sample covariance and correlation function of moving averages, Ann. statist., 14, 533-558, (1986) · Zbl 0605.62092 [21] Embrechts, P.; Goldie, C.M., On closure and factorization properties of subexponential distributions, J. austral. math. soc. ser. A, 29, 243-256, (1980) · Zbl 0425.60011 [22] Embrechts, P.; Goldie, C.M., On convolution tails, Stochastic process. appl., 13, 263-278, (1982) · Zbl 0487.60016 [23] Embrechts, P.; Omey, E., A property of long tailed distributions, J. appl. probab., 21, 80-87, (1984) · Zbl 0534.60015 [24] Goldie, C.M.; Resnick, S.I., Distributions that are both exponential and in the domain of attraction of an extreme-value distribution, Adv. appl. probab., 20, 706-718, (1988) · Zbl 0659.60028 [25] Klüppelberg, C., Subexponential distributions and integrated tails, J. appl. probab., 25, 132-141, (1988) · Zbl 0651.60020 [26] Leslie, J.R., On the non-closure under convolution of the subexponential family, J. appl. probab., 26, 58-66, (1989) · Zbl 0672.60027 [27] Pakes, A.G., On the tails of waiting-time distributions, J. appl. probab., 12, 555-564, (1975) · Zbl 0314.60072 [28] Pinelis, I.F., Asymptotic equivalence of the probabilities of large deviations for sums and maxima of independent random variables, (), 144-173, [In Russian.] [29] Pitman, E.J.G., Subexponential distribution functions, J. austral. math. soc. ser. A, 29, 337-343, (1980) · Zbl 0425.60012 [30] Rosinski, J.; Samorodnitsky, G., Distributions of subadditive functionals of sample paths of infinitely divisible processes, Ann. probab., 21, 994-1014, (1993) · Zbl 0776.60049 [31] Teugels, J.L., The class of subexponential distributions, Ann. probab., 1, 1001-1011, (1975) · Zbl 0374.60022
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.