Asymptotic expansions of convolutions of regularly varying distributions

Abstract

In this paper we derive precise tail-area approximations for the sum of an arbitrary finite number of independent heavy-tailed random variables. In order to achieve second-order asymptotics, a mild regularity condition is imposed on the class of distribution functions with regularly varying tails.Higher-order asymptotics are also obtained when considering a semiparametric subclass of distribution functions with regularly varying tails. These semiparametric subclasses are shown to be closed under convolutions and a convolution algebra is constructed to evaluate the parameters of a convolution from the parameters of the constituent distributions in the convolution. A Maple code is presented which does this task.2000 Mathematics subject classification: primary 62E17; secondary 62E20, 60G70.

Higher-order expansions for compound distributions and ruin probabilities with subexponential claims

“…See [16], [17] and [9] for more details aboutâ 2 (s). If f (s) is not continuous, but f is of bounded total variation, it follows from Grübel [20] that if there exists a monotonically decreasing function τ (s) with…”

“…Following another approach, Barbe & McCormick [7,10] (see also [8,9]) recently used the concept of smoothly varying functions to derive higher-order expansions: …”

Tail asymptotic expansions for L-statistics

“…The last step is due to 6) which follows from Potter bounds for j > α and the dominated convergence theorem. Now, we consider the case of α = l + 1.…”

“…Clearly, when c 1 = · · · = c n = 1 we have S n (c) = n i=1 X i =: S n ; the second-order tail behavior of S n has been investigated under some smoothness conditions by Degen et al [10], Mao et al [23], Omey and Willekens [24]. Further results on the higher-order tail asymptotics can be found in Albrecher et al [1], Barbe and McCormick [6], Geluk et al [14]. Results for the second-order tail asymptotics of S n under some second-order regular variation conditions are derived in Geluk et al [13], Kortschak [17], Mao and Hu [22] even for dependent cases.…”

Second-order properties of tail probabilities of sums and randomly weighted sums

“…For general n, presented a second-order result for the sum S n of n independent and identically distributed (iid) random variables with a common distribution in the Hall-Weissman class which is a special case of 2RV. A remarkable result was given by Barbe and McCormick (2005) which investigated the second-order expansion of P(S n > t) under the assumption that all margins are identical and possess a property which is called the asymptotic smoothness property besides the first-order regular variation property. By exploiting the main result in Barbe and McCormick (2005), under the additional 2RV assumption, Degen et al (2010) and Mao et al (2012) derived second-order expansions of the risk concentrations based on risk measures, Value-at-Risk (VaR) and conditional tail expectation (CTE), respectively.…”