7 Conclusions

 We have shown that the Gram-Charlier series has a limited domain of applicability for nearly normal distributions because of its rather poor convergence properties. The Gauss-Hermite expansion can give good results in problems like fitting profiles of spectral lines of galaxies, supernovae, or ordinary stars. In advanced calculations of stellar atmospheres (e.g. Hauschildt et al. 1997; Hubeny & Lanz 1995) the profiles of thousands or even millions of lines must be integrated for up to hundreds of Doppler widths, and the Gauss-Hermite expansion can perhaps be useful for saving information of the line profiles in an economical way. But since it has no intrinsic measure of accuracy, the number of terms needed in the expansion must be examined carefully for each individual problem.

For situations where the estimate of a deviation of a PDF from a Gaussian one is needed, the asymptotic Edgeworth expansion is indispensable, and for high order moments the form of this expansion found by Petrov is necessary. We found a workable algorithm for Petrov's formula of the Edgeworth expansion and applied it to several examples. The source codes used in this work are available on request from the authors.

Acknowledgements

We are grateful to Ya.M. Kazhdan for valuable advice on references, to N.N. Pavlyuk for assistance, and to the referee for useful comments. The work of SB is supported in part by INTAS grant ``Thermonuclear Supernovae" and by a grant from the Research Center for the Early Universe, University of Tokyo, and he cordially thanks K. Nomoto, as well as W.Hillebrandt, MPA, Garching, for their hospitality.

This research has made use of NASA's Astrophysics Data System Abstract Service.