The Subexponentiality For The Extremums Of Two Subexponential Distributions Under Some Classes Of Dependence Structure

Subexponential distribution, as one of the most important class of heavy-tailed distribution, plays a very important role in actuarial science and mathe-matical finance. For example, Weibull distribution is often applied to reliability analysis and the theoretic base of life test and the Lognormal distribution pro-vides important basis for stock investors to analysis, judge market and make a prediction. As its special properties, subexponential distribution has specif-ic applications in the queuing theory,renewal theory and infinitely divisible distribution theory, one can see Hutchinson, Joe, Nelsen, Huang, Cuadras,Durante, Cuadras, Amblard.The subexponentiality of subexponential distributions under some oper-ations has a long history. Embrechts and Goldie proved the convolution closure of two independent subexponential distribution; Cline gave some sufficient conditions of the product of two independent subexponential distri-butions concerning the closure of subexponential distribution subclass; Sub-sequently, Su and Chen, Tang, Chen and so on extend the results of Cline; By using some theoretical results of the subexponential distribution in Yakymiv, Geluk proved that two independent subexponential dis-tribution are still subexponential under the operation of minimum, and gave a sufficient and necessary condition of two independent subexponential distri-butions concerning the closure of subexponential distribution class under the operation of maximum. Subexponential distribution has an important proper-ty named maximum-equivalence. Therefore, recently, the asymptotic behavior of partial sums of subexponential tail probabilities had been widely applied in the theory of the risk, M/G/1 queuing system.Obviously, the research concerning subexponential distribution under the operation of minimum and maximum is helpful to describe asymptotic behavior of tail probabilities of partial sums.Yakymiv, Geluk proved the subexponentiality of two independent subexponential random variables under the operation of minimum and gave a sufficient and necessary condition concerning the closure of subexponential distribution under the operation of maximum. However, note that all the re-sults mentioned above are based on the independent assumption between the subexponential random variables, which does not meet the objective reality ob-viously. Therefore, the subexponentiality of two subexponential distributions under the operation of minimum and maximum together with some depen-dence structure has become an important research topic.Based on some existing literature, assuming that two subexponential ran-dom variables follow bivariate FGM, Sarmanov and a wide class of dependence structure, respectively, this paper proves that the minimum of two dependent random variables is still subexponential, gives a sufficient and necessary condi-tion for the subexponentiality of the maximum of two random variables under dependence structure FGM, Sarmanov and a wide class of dependence struc-ture. The obtained results extend the ones of Yakymiv and Geluk.