Tails of the Moments for Sums with Dominatedly Varying Random Summands

The asymptotic behaviour of the tail expectation ?E(Snξ)α?{Snξ>x} is investigated, where exponent α is a nonnegative real number and Snξ=ξ1+…+ξn is a sum of dominatedly varying and not necessarily identically distributed random summands, following a specific dependence structure. It turns out that the tail expectation of such a sum can be asymptotically bounded from above and below by the sums of expectations ?Eξiα?{ξi>x} with correcting constants. The obtained results are extended to the case of randomly weighted sums, where collections of random weights and primary random variables are independent. For illustration of the results obtained, some particular examples are given, where dependence between random variables is modelled in copulas framework.

On the Kesten-type inequality for randomly weighted sums with applications to an operational risk model

This paper considers the randomly weighted sums generated by some dependent subexponential primary random variables and some arbitrarily dependent random weights. To study the randomly weighted sums with infinitely many terms, we establish a Kesten-type upper bound for their tail probabilities in presence of subexponential primary random variables and under a certain dependence among them. Our result extends the study of Chen [5] to the dependent case. As applications, we derive some asymptotic formulas for the tail probability and the Value-at-Risk of total aggregate loss in a multivariate operational risk cell model.

On The Distribution of Partial Sums of Randomly Weighted Powers of Uniform Spacings

Objectives: To study the asymptotic theory of the randomly wieghted partial sum process of powers of k-spacings from the uniform distribution. Methods: Earlier results on the distribution of the uniform incremental randomly weighted sums. Methods: Based on theorems on weak and strong approximations of partial sum processes. Results and conculsions: Our main contribution is to prove the weak convergence of weighted sum of powers of uniform spacings.