Let a distribution to be approximated be the distribution 
of standardized sums
 |
(1)
|
In the Charlier series, take the component random variables identically distributed with mean 
, variance 
, and higher cumulants 
for 
. Also, take the developing function 
as the standard normal
distribution function 
, so we have
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
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)
|
(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 
.
