The tail behaviour of a random sum of subexponential random variables and vectors. (English) Zbl 1150.60029
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.
Reviewer: R. E. Maiboroda (Kyïv)
MSC:
| 60G70 | Extreme value theory; extremal stochastic processes |
| 60E07 | Infinitely divisible distributions; stable distributions |
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