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).
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Mikosch, T., Nagaev, A. Rates in Approximations to Ruin Probabilities for Heavy-Tailed Distributions. Extremes 4, 67–78 (2001). https://doi.org/10.1023/A:1012237524316
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DOI: https://doi.org/10.1023/A:1012237524316