tailieunhanh - Báo cáo "On the stability of the distribution function of the composed random variables by their index random variable "

Let us consider the composed random variable η = k=1 ξk , where ξ1 , ξ2 , . are independent identically distributed random variables and ν is a positive value random, independent of all ξk . In [1] and [2], we gave some the stabilities of the distribution function of η in the following sense: the small changes in the distribution function of ξ k only lead to the small changes in the distribution function of η. In the paper, we investigate the distribution function of η when we have the small changes of the distribution of ν. . | VNU Journal of Science Mathematics - Physics 23 2007 70-74 On the stability of the distribution function of the composed random Variables by their index random variable Nguyen Huu Bao Faculty of Infomation Technology Water Resources University 175 Tay Son Dong Da Hanoi Vietnam Received 15 November 2006 received in revised form 2 August 2007 Abstract. Let us consider the composed random variable n Sk i k where 1 2 are independent identically distributed random variables and V is a positive value random independent of all k. In 1 and 2 we gave some the stabilities of the distribution function of n in the following sense the small changes in the distribution function of k only lead to the small changes in the distribution function of n. In the paper we investigate the distribution function of n when we have the small changes of the distribution of V . 1. Introduction Let us consider the random variable V n E k k 1 1 where 1 2 are independent identically distributed random variables with the distribution function F x V is a positive value independent of all k and V has the distribution function A x . In 1 and 2 n is called to be the composed and V is called to be its index . If R x is the distribution function of n with the characteristic function f x respecrively then see 1 or 2 x a F t 2 where a z is the generating function of V and p t is the characteristic function of k. In 1 and 2 we gave some the stabilities of R x in the following sence the small changes in the distribution function F x only lead to the small changes in the distribution function x . In this paper we shall investigate the stability of n s distribution function when we have the small change of the distribution of the index V. Tel. 84-4-5634255. E-mail nhuubao@ 70 Nguyen Huu Bao VNU Journal of Science Mathematics - Physics 23 2007 70-75 71 2. Stability theorem Let us consider the now V1 ni E k 3 k 1 where V1 has the distribution function A1 x with the generating .