Tail behaviour of random sums under consistent variation with applications to the compound renewal risk model
Let \(\xi_1\),\dots,\(\xi_n\),\dots be i.i.d. with cdf \(F_\xi\) and \(\eta\) be a nonnegative integer-valued r.v. independent of \(\xi_i\) with cdf \(F_\eta\), \(\bar F=1-F\). The authors show that as \(x\to\infty\) \[ {\mathbf P}(\xi_1+\dots+\xi_\eta>x)\sim{\mathbf P}(\eta>x\;| \;{\mathbf E}\xi) \] if \(\bar F_\xi(x)=o(\bar F_\eta(x))\) and \(F_\eta\) is a function with consistent variation, i.e., \[ \lim_{y\uparrow 1}\lim\sup_{x\to\infty}{\bar F_\eta(xy)\over \bar F(x)}=1. \] Some refinements of this statement and an application to the ruin probability evaluation in the compound renewal risk model are considered.
sum of random number of random variables
scientific article; zbMATH DE number 5520601