  Various types of expansions in series of Chebyshev-Hermite polynomials
currently used in astrophysics for weakly non-normal distributions are
compared, namely the Gram-Charlier, Gauss-Hermite and Edgeworth expansions. It
is shown that the Gram-Charlier series is most suspect because of its poor
convergence properties. The Gauss-Hermite expansion is better but it has no
intrinsic measure of accuracy. The best results are achieved with the
asymptotic Edgeworth expansion. We draw attention to the form of this expansion
found by Petrov for arbitrary order of the asymptotic parameter and present a
simple algorithm realizing Petrov's prescription for the Edgeworth expansion.
The results are illustrated by examples similar to the problems arising when
fitting spectral line profiles of galaxies, supernovae, or other stars, and for
the case of approximating the probability distribution of peculiar velocities
in the cosmic string model of structure formation.
