  A bivariate random vector can exhibit either asymptotic independence or
dependence between the largest values of its components. When used as a
statistical model for risk assessment in fields such as finance, insurance or
meteorology, it is crucial to understand which of the two asymptotic regimes
occurs. Motivated by their ubiquity and flexibility, we consider the extremal
dependence properties of vectors with a random scale construction
$(X_1,X_2)=R(W_1,W_2)$, with non-degenerate $R>0$ independent of $(W_1,W_2)$.
Focusing on the presence and strength of asymptotic tail dependence, as
expressed through commonly-used summary parameters, broad factors that affect
the results are: the heaviness of the tails of $R$ and $(W_1,W_2)$, the shape
of the support of $(W_1,W_2)$, and dependence between $(W_1,W_2)$. When $R$ is
distinctly lighter tailed than $(W_1,W_2)$, the extremal dependence of
$(X_1,X_2)$ is typically the same as that of $(W_1,W_2)$, whereas similar or
heavier tails for $R$ compared to $(W_1,W_2)$ typically result in increased
extremal dependence. Similar tail heavinesses represent the most interesting
and technical cases, and we find both asymptotic independence and dependence of
$(X_1,X_2)$ possible in such cases when $(W_1,W_2)$ exhibit asymptotic
independence. The bivariate case often directly extends to higher-dimensional
vectors and spatial processes, where the dependence is mainly analyzed in terms
of summaries of bivariate sub-vectors. The results unify and extend many
existing examples, and we use them to propose new models that encompass both
dependence classes.
