Random Sums of Random Variables and Vectors
Let fX;Xi; i = 1; 2; :::g denote independent positive random variables having a common distribution function F(x) and, independent of X, let N denote an integer valued random variable. Using S(0) = 0 and S(n) = S(n ?? 1) + Xn, the random sum S(N) has distribution function G(x) = 1Xi=0 P(N = i)P(S(i) _ x) and tail distribution G(x) = 1 ?? G(x). In which case, we say that the distribution function G is subordinated to F with subordinator N. Under suitable conditions, it can be proved that G(x) s E(N)F(x) as x ! 1. In this paper, we extend some of the existing results. In the place of i.i.d. random variables, we use variables that are independent or variables that are asymptotically in- dependent. We also consider multivariate subordinated distribution functions.
|Date of creation:||15 May 2009|
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