Let a distribution to be approximated be the distribution 
of standardized sums
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(1)
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In the Charlier series, take the component random variables independent
and identically distributed with mean 
, variance 
, and higher cumulants 
for 
. Also, take the developing function 
as the standard normal
distribution function 
, so we have
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(2)
|
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(3)
|
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(4)
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Then the Edgeworth series is obtained by collecting terms to obtain the asymptotic expansion of the characteristic function of the form
![f_n(t)=[1+sum_(r=1)^infty(P_r(it))/(n^(r/2))]e^(-t^2/2),](/images/equations/EdgeworthSeries/NumberedEquation2.svg) |
(5)
|
where 
is a polynomial of degree 
with coefficients
depending on the cumulants of orders 3 to 
. If the powers of 
are interpreted as derivatives,
then the distribution function expansion
is given by
 |
(6)
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(Wallace 1958). The first few terms of this expansion are then given by
![f(t)=Psi(t)-(lambda_3Psi^((3))(t))/(6sqrt(n))
+1/n[(lambda_4Psi^((4))(t))/(24)+(lambda_3^2Psi^((6))(t))/(72)]+....](/images/equations/EdgeworthSeries/NumberedEquation4.svg) |
(7)
|
Cramér (1928) proved that this series is uniformly valid in 
.
