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Rates in Approximations to Ruin Probabilities for Heavy-Tailed Distributions

Authors:	Thomas Mikosch Alexander Nagaev

Affiliation:	(1) Laboratory of Actuarial Mathematics, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark;(2) Faculty of Mathematics and Informatics, Copernicus University, UL. Chopina 12/18, 87-100 Torun, Poland

Abstract:	A well known result by Embrechts and Veraverbeke [3] says that, for subexponential distribution functions F(x), the tail of the compound sum distribution function $Sigma _{n = 1}^infty p_n F^{n*} (x)$ is approximated by $(1 - F(x))Sigma _{n = 1}^infty np_n$ as x . We show that the rate of convergence in this result can be arbitrarily slow. On the other hand, if F satisfies some smoothness condition (for example if F is an integrated tail distribution function) then the rate cannot be worse than O(x^-1).

Keywords:	heavy tails total claim amount Pollaczek-Khintchine formula Cramé r– Lundberg model ruin probability convergence rates
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