On the behavior of the product of independent random variables  

作　　者：SU Chun CHEN Yu 

机构地区：[1]Department of Statistics and Finance, University of Science and Technology of China, Hefei 230026, China Department of Statistics and Finance, University of Science and Technology of China, Hefei 230026, China

出　　处：《Science China Mathematics》2006年第3期342-359,共18页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant No.10371117);the Doctoral Program Foundation of Ministry of Education and the Special Foundation of USTC.

摘　　要：For two independent non-negative random variables X and Y, we treat X as the initial variable of major importance and Y as a modifier (such as the interest rate of a portfolio).Stability in the tail behaviors of the product compared with that of the original variable X is of practical interests. In this paper, we study the tail behaviors of the product XY when the distribution of X belongs to the classes L and S, respectively. Under appropriate conditions, we show that the distribution of the product XY is in the same class as X when X belongs to class L or S, in other words, classes L and S are stable under some mild conditions on the distribution of Y. We also show that if the distribution of X is in class L(γ) (γ＞ 0) and continuous, then the product XY is in L if and only if Y is unbounded.For two independent non-negative random variables X and Y, we treat X as the initial variable of major importance and Y as a modifier (such as the interest rate of a portfolio). Stability in the tail behaviors of the product compared with that of the original variable X is of practical interests. In this paper, we study the tail behaviors of the product XY when the distribution of X belongs to the classes L and S, respectively. Under appropriate conditions, we show that the distribution of the product XY is in the same class as X when X belongs to class L or S, in other words, classes L and S are stable under some mild conditions on the distribution of Y. We also show that if the distribution of X is in class L(γ) (γ> 0) and continuous, then the product XY is in L if and only if Y is unbounded.

关 键 词：INDEPENDENT product  stability  TAILS of a distribution  heavy-tailed. 

分 类 号：N[自然科学总论]