  Let $\bm X=(X_1,...,X_d)$ be a random vector, whose components are not
necessarily independent nor are they required to have identical distribution
functions $F_1,...,F_d$. Denote by $N_s$ the number of exceedances among
$X_1,...,X_d$ above a high threshold $s$. The fragility index, defined by
$FI=\lim_{s\nearrow}E(N_s\mid N_s>0)$ if this limit exists, measures the
asymptotic stability of the stochastic system $\bm X$ as the threshold
increases. The system is called stable if $FI=1$ and fragile otherwise. In this
paper we show that the asymptotic conditional distribution of exceedance counts
(ACDEC) $p_k=\lim_{s\nearrow}P(N_s=k\mid N_s>0)$, $1\le k\le d$, exists, if the
copula of $\bm X$ is in the domain of attraction of a multivariate extreme
value distribution, and if
$\lim_{s\nearrow}(1-F_i(s))/(1-F_\kappa(s))=\gamma_i\in[0,\infty)$ exists for
$1\le i\le d$ and some $\kappa\in{1,...,d}$. This enables the computation of
the FI corresponding to $\bm X$ and of the extended FI as well as of the
asymptotic distribution of the exceedance cluster length also in that case,
where the components of $\bm X$ are not identically distributed.
