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Lithuanian Mathematical Journal

, Volume 58, Issue 4, pp 421–440 | Cite as

Expectation of the truncated randomly weighted sums with dominatedly varying summands

  • Eglė Jaunė
  • Olena Ragulina
  • Jonas Šiaulys
Article
  • 29 Downloads

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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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of MathematicsVilnius UniversityVilniusLithuania
  2. 2.Department of Probability Theory, Statistics and Actuarial MathematicsTaras Shevchenko National University of KyivKyivUkraine

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