When you sum N IID samples from a univariate distribution, the sum approaches a normally-distributed value.
Likewise, when you sum variables together, the moment-generating functions of their distribution can be multiplied to produce the resulting moment-generating function of the sum.
It would seem that you could exponentiate a variable's moment-generating function by a large N, and the resulting function would be close to that of a normal distribution's.
However, from a mathematical perspective, this makes no sense, as you are simply
For simplicity's sake, let's use variables from continuous distributions that have an MGF (to avoid problems with the domain).
One example I am having trouble with is the Chi-squared distribution, with an MGF of
The resulting MGF of the sum of many of these variables would be
If we assume
N*k/2 is an integer,
C, then the MGF is
(1-2t)^C = 1-2t * C + 4t^2 * C*(C-1)/2 +...
C ≈ C-1 ≈ C-2 ≈ ... for large C, this can simplify to
sum((-2t*C)^n/n!) from n=0 to C ≈ e^(-2t*C)
Of course, the later terms deviate more strongly from the normal distribution's, but those derivatives don't affect the function's value very much.
Unfortunately, this MGF is degenerate, because it simply describes that of a variable that is always
C. There is no
t^2 term in the exponent, which is required for a normal MGF. Did I make a mistake in my calculation, or is there some fundamental assumption that I overlooked/violated?
Edit: I replaced "added" with "multiplied" in one instance because I was referring to multiplying the MGFs, not adding them.
Edit 2: I realized I forgot the minus sign in the exponent of the Chi-squared MGF. That resulted in a computational error.