Rates in approximations to ruin probabilities for heavy-tailed distributions
Consider \(S_\nu=\sum_{i=1}^\nu X_i\), where \(X_i\) are i.i.d. positive random variables independent of r.v. \(\nu\). Let \(H\) be a d.f. of \(S_\nu\) and let \(F\) be d.f. of \(X_i\). The authors analyze the rate of convergence \[ \Delta(x)={1-H(x)\over 1-F(x)}- E\nu\to 0,\;x\to\infty. \] It is shown that, in general, this rate can be arbitrarily slow but if \(1-F(x)\) is of dominated variation, \(E\nu<\infty\), the moment generating function of \(\nu\) exists in a neighborhood of the origin and \(\lim\inf_{x\to\infty} x^\alpha(1-F(x))>0\) for some \(\alpha>0\), then \(\Delta(x)=O(x^{-1})\) as \(x\to\infty\).
scientific article; zbMATH DE number 1833859