Rates in Approximations to Ruin Probabilities for Heavy-Tailed Distributions

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\Sigma _{n = 1}^\infty p_n F^{n*} (x)$$ \end{document} is approximated by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$(1 - F(x))\Sigma _{n = 1}^\infty np_n $$ \end{document} 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).