Lithuanian Mathematical Journal

, Volume 58, Issue 4, pp 421–440

# Expectation of the truncated randomly weighted sums with dominatedly varying summands

• Eglė Jaunė
• Olena Ragulina
• Jonas Šiaulys
Article

## 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 1, θ2, . . . , θn} is a collection of nonnegative and arbitrarily dependent random weights; the collections {X1,X2, . . .,Xn} and 1, 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.

## Keywords

truncated distribution asymptotic bound tail expectation random weight dominatedly varying distribution quasiasymptotic independence value at risk conditional tail expectation

## MSC

91B30 62E20 60E15 91G10

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