  The Marshall-Olkin (MO) distribution has been considered a key model in
reliability theory and in risk analysis, where it is used to model the
lifetimes of dependent components or entities of a system and dependency is
induced by "shocks" that hit one or more components at a time. Of particular
interest is the L\'evy-frailty subfamily of the Marshall-Olkin (LFMO)
distribution, since it has few parameters and because the nontrivial dependency
structure is driven by an underlying L\'evy subordinator process. The main
contribution of our work is that we derive the precise asymptotic behavior of
the upper order statistics of the LFMO distribution. More specifically, we
consider a sequence of $n$ univariate random variables jointly distributed as a
multivariate LFMO distribution and analyze the order statistics of the sequence
as $n$ grows. Our main result states that if the underlying L\'evy subordinator
is in the normal domain of attraction of a stable distribution with index of
stability $\alpha$ then, after certain logarithmic centering and scaling, the
upper order statistics converge in distribution to a stable distribution if
$\alpha>1$ or a simple transformation of it if $\alpha\leq1$. Our result is
especially useful in network reliability and systemic risk, when modeling the
lifetimes of components in a system using the LFMO distribution, as it allows
to understand the behavior of systems that rely on its last working components.
Our result can also give easily computable confidence intervals for these
components, provided that a proper convergence analysis is carried out first.
