Second order behavior of the tails of compound sums of regularly varying random variables

  • Gerd Christoph
  • Nadja Malevich


We consider a compound sum $S_{\nu}=X_{1}+X_{2}+\ldots+X_{\nu},$ where $X, X_{1}, X_{2},\ldots$ are independent and identical distributed random variables with common distribution function $F$ and $\nu \in \{1,2,3,\ldots\} $ is an integer-valued counting random variable, independent of $X_{1}, X_{2},\ldots$ It is known, that for subexponential distributions under some conditions $$\Delta(x):=\frac{P(X_{1}+X_{2}+\ldots+X_{\nu}>x)}{P(X>x)}-E\nu \to 0 \quad \mbox{as} \quad x \to \infty\,.$$ We are interested in the rate of convergence of $\Delta(x)$ as $x\rightarrow\infty$. In case when the mean $\mu:= E\,X$ is finite some authors have obtained results related to the asymptotic behaviour of $\Delta(x)$, see Baltr\^{u}nas and Omey (2002), Christoph (2004, 2005), Mikosch and Nagaev (2001). The case of infinite mean has been studied less and therefore we consider it here. Improving the method of Christoph and Wolf (1993) we obtain the first and higher order results for $\Delta(x)$ for some classes of distributions, including Pareto distributions.