  Moving average processes driven by exponential-tailed L\'evy noise are
important extensions of their Gaussian counterparts in order to capture
deviations from Gaussianity, more flexible dependence structures, and sample
paths with jumps. Popular examples include non-Gaussian Ornstein--Uhlenbeck
processes and type G Mat\'ern stochastic partial differential equation random
fields. This paper is concerned with the open problem of determining their
extremal dependence structure. We leverage the fact that such processes admit
approximations on grids or triangulations that are used in practice for
efficient simulations and inference. These approximations can be expressed as
special cases of a class of linear transformations of independent,
exponential-tailed random variables, that bridge asymptotic dependence and
independence in a novel, tractable way. This result is of independent interest
since models that can capture both extremal dependence regimes are scarce and
the construction of such flexible models is an active area of research. This
new fundamental result allows us to show that the integral approximation of
general moving average processes with exponential-tailed L\'evy noise is
asymptotically independent when the mesh is fine enough. Under mild assumptions
on the kernel function we also derive the limiting residual tail dependence
function. For the popular exponential-tailed Ornstein--Uhlenbeck process we
prove that it is asymptotically independent, but with a different residual tail
dependence function than its Gaussian counterpart. Our results are illustrated
through simulation studies.
