Expectation of the truncated randomly weighted sums with dominatedly varying summands |
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| Authors: | Egl? Jaun? Olena Ragulina Jonas ?iaulys |
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| Institution: | 1.Institute of Mathematics,Vilnius University,Vilnius,Lithuania;2.Department of Probability Theory, Statistics and Actuarial Mathematics,Taras Shevchenko National University of Kyiv,Kyiv,Ukraine |
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| Abstract: | We consider the asymptotic behavior of the values P(S > x), E(S 1{S>x}), and E(S | S > x). Here S = θ1X1 + θ2X2 + · · · + θnXn is a randomly weighted sum of the basic random variables X1,X2, . . . , Xn, which follow some special dependence structure, and {θ1, θ2, . . . , θn} is a collection of nonnegative and arbitrarily dependent random weights; the collections {X1,X2, . . .,Xn} and {θ1, θ2, . . . , θn} are supposed to be independent. We derive asymptotic formulas in the case where the number of summands n is fixed and the distributions of the basic random variables are dominatedly varying.We apply them to some values related to the risk measures of certain weighted sums. |
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