  We develop an asymptotic theory for extremes in decomposable graphical models
by presenting results applicable to a range of extremal dependence types.
Specifically, we investigate the weak limit of the distribution of suitably
normalised random vectors, conditioning on an extreme component, where the
conditional independence relationships of the random vector are described by a
chordal graph. Under mild assumptions, the random vector corresponding to the
distribution in the weak limit, termed the tail graphical model, inherits the
graphical structure of the original chordal graph. Our theory is applicable to
a wide range of decomposable graphical models including asymptotically
dependent and asymptotically independent graphical models. Additionally, we
analyze combinations of copula classes with differing extremal dependence in
cases where a normalization in terms of the conditioning variable is not
guaranteed by our assumptions. We show that, in a block graph, the distribution
of the random vector normalized in terms of the random variables associated
with the separators converges weakly to a distribution we term tail noise. In
particular, we investigate the limit of the normalized random vectors where the
clique distributions belong to two widely used copula classes, the Gaussian
copula and the max-stable copula.
