Non-Gaussian elliptically contoured autoregressive moving average (EC-
ARMA) processes are defined as ARMA models with non-Gaussian noise. Th
e driving noise series is uncorrelated but generally dependent, and fo
llows elliptically contoured distributions. Moments, characteristic fu
nctions, parametric representations, and other probabilistic character
istics are given for elliptically contoured vectors and EC-ARMA proces
ses. These characteristics are used to outline methods for generating
samples of EC-ARMA processes. The analysis includes classical and dege
nerated elliptically contoured driving noises. The Cauchy noise is an
example of a degenerated noise, in the sense that it does not have mom
ents. It is shown that EC-ARMA models with finite first two moments be
come Gaussian as time increases. However, degenerate EC-ARMA models re
main non-Gaussian at any time. Several numerical examples are used to
demonstrate features of EC-ARMA processes and methods to generate real
izations of these processes.