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KMID : 0380319740120000009
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Journal of Korean Research Institute for Better Living
1974 Volume.12 No. 0 p.9 ~ p.16
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On Convergence Properties of a Gram-Charlier A Series Expansion
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Abstract
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We may consider a type of random variable X with pdf g(x) so that
g(x) = αf(x) + βf(α^2x)
where f(x) = 1/σ√2π exp(-x^2/2), α= 1/1+σ^2 and β=σ^4/1+σ^2.
Then g(x) represents a pdf of a standardized variable.
We obtained the expansion of g(x) in gram-charlier A series as follows:
g(x) = 1/(1+σ^2)√2π^∞∑_n=0H_2n(x)/2^n(N!){(σ^2-1)^n +σ_4(1/σ^2-1)^n}exp(-x^2/2)
where H_2n(x) is the Hermite polynomial of degree 2n.
A condition for the convergence of the series (2), obtained by analytical methods, is as follows:
1/2≤σ^2≤2
And the condition directly deduced from the convergence theorem (suggested by Harald Crame´r) is as follows:
1/2＜σ^2＜2
Since interval (3) contains interval (4) as proper subset, (3) is a better condition than (4).
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