Charlier Series
A class of formal series expansions in derivatives of a distribution 
which may (but need not) be the normal distribution function
 |
(1)
|
and moments or other measured parameters. Edgeworth series are known as the Charlier series or Gram-Charlier series. Let 
be the characteristic function of the function 
, and 
its cumulants. Similarly, let 
be the distribution to be approximated, 
its characteristic function, and 
its cumulants. By definition, these quantities are connected by the formal series
![f(t)=exp[sum_(r=1)^infty(kappa_r-gamma_r)((it)^r)/(r!)]psi(t)](https://mathworld.wolfram.com/images/equations/CharlierSeries/NumberedEquation2.gif) |
(2)
|
(Wallace 1958). Integrating by parts gives 
as the characteristic function of 
, so the formal identity corresponds pairwise to the identity
![F(x)=exp[sum_(r=1)^infty(kappa_r-gamma_r)((-D)^r)/(r!)]Psi(x),](https://mathworld.wolfram.com/images/equations/CharlierSeries/NumberedEquation3.gif) |
(3)
|
where 
is the differential operator. The most important case 
was considered by Chebyshev (1890), Charlier (1905-06), and Edgeworth (1905).
Expanding and collecting terms according to the order of the derivatives gives the so-called Gram-Charlier A-Series, which is identical to the formal expansion of 
in Hermite polynomials. The A-series converges for functions 
whose tails approach zero faster than 
(Cramér 1925, Wallace 1958, Szegö 1975).
REFERENCES:
Charlier, C. V. L. "Über das Fehlergesetz." Ark. Math. Astr. och Phys. 2, No. 8, 1-9, 1905-06.
Chebyshev, P. L. "Sur deux théorèmes relatifs aux probabilités." Acta Math. 14, 305-315, 1890.
Cramér, H. "On Some Classes of Series Used in Mathematical Statistics." Proceedings of the Sixth Scandinavian Congress of Mathematicians, Copenhagen. pp. 399-425, 1925.
Edgeworth, F. Y. "The Law of Error." Cambridge Philos. Soc. 20, 36-66 and 113-141, 1905.
Gram, J. P. "Über die Entwicklung reeler Funktionen in Reihen mittelst der Methode der kleinsten Quadrate." J. reine angew. Math. 94, 41-73, 1883.
Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975.
Wallace, D. L. "Asymptotic Approximations to Distributions." Ann. Math. Stat. 29, 635-654, 1958.
