EXPANSIONS FOR NEARLY GAUSSIAN DISTRIBUTIONS

Various types of expansions in series of Chebyshev-Hermite polynomials currently used in astrophysics for weakly non-normal distributions ar e compared, namely the Gram-Charlier, Gauss-Hermite and Edgeworth expa nsions. It is shown that the Gram-Charlier series is most suspect beca use 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 attentio n to the form of this expansion found by Petrov for arbitrary order of the asymptotic parameter and present a simple algorithm realizing Pet rov's prescription for the Edgeworth expansion. The results are illust rated by examples similar to the problems arising when fitting spectra l line profiles of galaxies, supernovae, or other stars, and for the c ase of approximating the probability distribution of peculiar velociti es in the cosmic string model of structure formation.