Asymptotic Expansions for Distributions of Compound Sums of Random Variables with Rapidly Varying Subexponential Distribution

Abstract

We derive an asymptotic expansion for the distribution of a compound sum of independent random variables, all having the same rapidly varying subexponential distribution. The examples of a Poisson and geometric number of summands serve as an illustration of the main result. Complete calculations are done for a Weibull distribution, with which we derive, as examples and without any difficulties, seven-term expansions.

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

“…Since the step from fixed n to a random N is straight-forward in this case (cf. Barbe et al [11]), we will only focus on the proof for fixed n.…”

“…Before we outline the ideas of the proofs given in [7,11] one should mention that all proofs for higher order expansion (except the proof in [20] which uses Banach algebra techniques) use some decomposition of P(S n > s) and then asymptotically evaluate involved convolution integrals. For a distribution function F (s) and an 0 < η < 1, define the operators…”

“…For the choice g i (s) = F (i) (s) it becomes clear that one has to evaluate T F,1/2 F (i) (s) and T F,1/2 K(s), so that we end up in the same situation as in [7,11].…”

“…Further work on approximations of compound distributions in the heavy-tailed case can be found in Mikosch & Nagaev [22], Willekens [31] and Baltrūnas [5]. Higher-order asymptotic expansions were finally given in Geluk et al [18], Borovkov and Borovkov [14], Barbe and McCormick [7] and Barbe et al [11].…”

“…The paper is organized as follows: in Section 2 we give an overview of available results on higher-order approximations with a particular emphasis on the used methodology, including a discussion of the main steps of the proof of Barbe & McCormick [7] and Barbe et al [11]. In Section 3 we present two alternative approaches to derive asymptotic expansions and extend the expansion given in [7] by adding an additional term.…”