The Gram-Charlier A series is a formal expansion of a probability density function 
around the standard normal
distribution density 
in rescaled Hermite
polynomials. Writing 
, where 
has the normalization used for the Hermite
polynomial, the expansion has the form
 |
where
 |
For a standardized random variable, 
and 
, while the higher coefficients
can be expressed in terms of its moments or cumulants.
A truncated Gram-Charlier A series need not be nonnegative and therefore need not itself be a probability density function; convergence also requires additional hypotheses. The Edgeworth series uses related terms but reorders them according to their asymptotic dependence on sample size, so the two series are not synonyms.
