1 Introduction

 The normal, or Gaussian, distribution plays a prominent role in statistical problems in various fields of astrophysics and general physics. This is quite natural, since the sums of random variables tend to a normal distribution when the quite general conditions of the central limit theorem are satisfied. In many applications, to extract useful information on the underlying physical processes, it is more interesting to measure the deviations of a probability density function (hereafter PDF) from the normal distribution than to prove that it is close to the Gaussian one. This has been done for example in the work on peculiar velocities and cosmic microwave background anisotropies in various cosmological models (Scherrer & Bertschinger 1991; Kofman et al. 1994; Moessner et al. 1994; Juszkiewicz et al. 1995; Bernardeau & Kofman 1995; Amendola 1994; Colombi 1994; Moessner 1995; Ferreira et al. 1997; Gaztañaga et al. 1997), in analyzing the velocity distributions and fine structure in elliptical galaxies (Rix & White 1992; van der Marel & Franx 1993; Gerhard 1993; Heyl et al. 1994), and in studies of large Reynolds number turbulence (Tabeling et al. 1996). In this paper we present a unified approach to the formalism used in those applications, illustrating it by examples of similar problems arising in cosmology and in the theory of supernova line spectra.

The first investigation of slightly non-Gaussian distributions was undertaken by Chebyshev in the middle of the 19th century. He studied in detail a family of orthogonal polynomials which form a natural basis for the expansions of these distributions. A few years later the same polynomials were also investigated by Hermite and they are now called Chebyshev-Hermite or simply Hermite polynomials (their definition was first given by Laplace, see e.g. Encyclopaedia of Mathematics 1988).

There are several forms of expansions using Hermite polynomials, namely the Gram-Charlier, Gauss-Hermite and Edgeworth expansions. We introduce the notation in Sect. 2 and use the simple example of the $\chi^2$ distribution for various degrees of freedom to illustrate the properties of Gram-Charlier expansions in Sect. 3. In subsequent sections the $\chi^2$ distribution is used for testing the other two expansions. We show in Sect. 4 that the Gram-Charlier series is just a Fourier expansion which diverges in many situations of practical interest, whereas the Gauss-Hermite series has much better convergence properties. However, we point out that another series, the so called Edgeworth expansion, is more useful in many applications even if it is divergent, since, first, it is directly connected to the moments and cumulants of a PDF (the property which is lost in the Gauss-Hermite series) and, second, it is a true asymptotic expansion, so that the error of the approximation is controlled.

Some applications (e.g. Bernardeau 1992, 1994; Moessner 1995) involve cumulants of higher order. These require the use of a correspondingly higher order Edgeworth series, but only the first few terms of the series are given in standard references (Cramér 1957; Abramowitz & Stegun 1972; Juszkiewicz et al. 1995; Bernardeau & Kofman 1995). In Sect. 5 we popularize a derivation of the Edgeworth expansion due to Petrov (1962, 1972, 1987) for an arbitrary order of the asymptotic parameter, and we present a slightly simpler and more straightforward way of obtaining his result. The formula found by Petrov (see Sect. 5 below) requires a summation over indices with non-trivial combinatorics which hindered its direct application. We find a simple algorithm realizing Petrov's prescription for any order of the Edgeworth expansion; this algorithm is easily coded, e.g. with standard Fortran, eliminating the need for symbolic packages.

We find that the Edgeworth-Petrov expansion is indeed very efficient and reliable. In Sect. 6 we apply the formalism to the problem of peculiar velocities resulting from cosmic strings studied by Moessner (1995) and we show how this technique allows one to reliably extract deviations from Gaussianity, even when they are tiny.