3 An example based on the $\chi^2$ distribution

 A good example for illustrating the fast divergence of the Gram-Charlier series is given by its application to the $\chi^2_\nu$ distribution with $\nu$ degrees of freedom, since the moments of this distribution are known analytically and its PDF tends to the Gaussian one for large $\nu$.If X₁, X₂, ..., $X_\nu$, are independent, normally distributed random variables with zero expectation and unit dispersion, then the variable $\chi^2=\chi^2_\nu \equiv \sum_{i=1}^\nu X_i^2$has the PDF  

$$\rho(\chi^2)={(\chi^2)^{\nu/2-1}\exp(-\chi^2/2) \over 2^{\nu/2}\Gamma(\nu/2)}, \qquad \chi^2\gt \; .$$ (14)

The expectation value of $\chi^2$ is $\nu$ and its dispersion is $2\nu$.If $x=(\chi^2-\nu)/\sqrt{2\nu}$, then x has zero expectation and unit dispersion and its PDF p(x) asymptotically tends to the Gaussian distribution Z(x). Transforming $\rho(\chi^2)$ from (14) to p(x), we obtain for $x\gt-\sqrt{\nu/2}$:  

$$p(x)=\sqrt{2\nu}{(\sqrt{2\nu}x+\nu)^{\nu/2-1}\exp{-(\sqrt{2\nu}x+\nu)/2} \over 2^{\nu/2}\Gamma(\nu/2)} \; .$$ (15)

The $\chi^2$ distribution was employed by Matsubara & Yokoyama (1996) for a representation of the cosmological density field (see also Luo 1995 for an application of the $\chi^2$ distribution in the context of cosmic microwave background temperature anisotropies). It can also serve as a crude model for approximating the profiles of spectral lines in moving media. If the lines are nearly Gaussian in the matter at rest, then the motion with a high velocity gradient, like that in supernova envelopes, leads to distortions which can be approximated by the $\chi^2_\nu$ PDF (with $\nu=2$ for highly non-Gaussian profiles, see e.g. Blinnikov 1996, and references therein).

The comparison of the Gram-Charlier approximations of p(x) with the exact results is presented in Figs. 1 and 2 for an increasing number of terms in the expansion. It is clear that the series quickly becomes inaccurate with a larger number of terms included.

  

Figure 1: The normalized $\chi^2$ PDF (15) for $\nu=5$ (dashed line), and its Gram-Charlier approximations with 2 and 6 terms in the expansion (solid line)

  

Figure 2: The same as in Fig. 1 but for 12 and 36 terms in the Gram-Charlier expansion