Gram-Charlier series

2010 Mathematics Subject Classification: Primary: 60E99 [MSN][ZBL]

A series defined by the expression (1)

or (2)

where is the normalized value of a random variable.

The series (1) is known as the Gram–Charlier series of type ; here  is the -th derivative of , which can be represented as where are the Chebyshev–Hermite polynomials. The derivatives and the polynomials are orthogonal, owing to which the coefficients can be defined by the basic moments of the given distribution series. If one restricts to the first few terms of the series (1), one obtains  The series (2) is known as a Gram–Charlier series of type ; here while are polynomials analogous to the polynomials .

If one restricts to the first terms of the series (2), one obtains  Here are the central moments of the distribution, while .

Gram–Charlier series were obtained by J.P. Gram [G] and C.V.L. Charlier [Ch] in their study of functions of the form These are convenient for the interpolation between the values of the general term of the binomial distribution, where is the characteristic function of the binomial distribution. The expansion of in powers of yields a Gram–Charlier series of type for , whereas the expansion of in powers of yields a Gram–Charlier series of type .