The extremal t process was proposed in the literature for modeling spatial
extremes within a copula framework based on the extreme value limit of
elliptical t distributions (Davison, Padoan and Ribatet (2012)). A major
drawback of this max-stable model was the lack of a spectral representation
such that for instance direct simulation was infeasible. The main contribution
of this note is to propose such a spectral construction for the extremal t
process. Interestingly, the extremal Gaussian process introduced by Schlather
(2002) appears as a special case. We further highlight the role of the extremal
t process as the maximum attractor for processes with finite-dimensional
elliptical distributions. All results naturally also hold within the
multivariate domain