A class of formal series expansions in derivatives of a distribution which may (but need not) be the normal distribution function
(1)
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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
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(Wallace 1958). Integrating by parts gives as the characteristic function of , so the formal identity corresponds pairwise to the identity
(3)
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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).