The tail behaviour of a random sum of subexponential random variables and vectors (Q928490)

Let \(G\) be the CDF of \(Z=\sum_{i=0}^\nu X_i\), where \(X_i\) are i.i.d. r.v.s with CDF \(F\) and \(\nu\) is independent of \(X_i\). The authors demonstrate that \[ {1-G(x)\over 1-F(x)}\to E\nu \] if \(1-F(x)\) is an \(O\)-regularly varying subexponential function with the lower Matusewska index \(\beta<-1\) and \(E\nu^{| \beta| +1+\varepsilon}<\infty \) (for some \(\varepsilon>o\)). If \(\beta>-1\) then \(E\nu<\infty\) is sufficient. Special attention is paid to the nonnegative stable \(F\). A generalization to the bivariate case is considered.