Tails of the Moments for Sums with Dominatedly Varying Random Summands

Abstract

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.

Product Convolution of Generalized Subexponential Distributions

“…Then, the random sum S θξ n would correspond to the present value of the total loss of a portfolio at the present moment in the former case, and the total weighted portfolio loss in the later case. For details, see [34,[44][45][46][47][48][49][50].…”

“…where the symbol a denotes the integer part of the real number a. It follows from this (for details see [34]) that…”

Randomly Stopped Sums, Minima and Maxima for Heavy-tailed and Light-Tailed Distributions

“…for some c 1 > 0, all k ≥ 1 and all x ∈ R. Therefore, by the alternative expectation formula (see, for instance, [39]), we derive from (10) that…”

“…Similarly, we are interested when F X (ν) , F X (ν) , F S (ν) and F S (ν) are heavy-tailed or light tailed. For various distribution classes, similar questions were studied in [1][2][3][4][5][6][7][8][9][10], [11][12][13][14][15][16][17][18][19][20], [21][22][23][24][25][26][27][28][29][30]. We mention also the paper [31], where two independent heavy-tailed r.v.s, such that their minimum is not heavy tailed, were constructed.…”

Markov and Semi-Markov Chains, Processes, Systems, and Emerging Related Fields

“…We provide a brief description of the articles that could be categorized in this section: (iii1) Tails of the Moments for Sums with Dominatedly Varying Random Summands, by Dirma, M., Paukstys, S., and Siaulys, J. [24] This paper investigates the asymptotic behaviour of tails of the moments for randomly weighted sums with possibly dependent dominatedly varying summands. The findings improve and generalise other related results of the relevant literature.…”