So I know the characteristic function of my random variable is given by:
$$ \phi_{Z}(t) = \frac{ e^{ \frac{ j(N-1)A^{2}t - (N+1) A^{2} \sigma^{2} t^{2} }{1 + \sigma^{4} t^{2} } } }{ (1 + \sigma^{4} t^{2} )^{N}} $$
Since I am having trouble finding the pdf by taking its Fourier transform, I figured I could just find the moments and use some sort of approximation method.
From the characteristic function, the first four raw moments are:
$$ \mu^{'}_{1} = E\left[Z\right] = A^{2} (N-1) $$ $$ \mu^{'}_{2} = E\left[Z^{2}\right] = A^{4} (N-1)^2+2 A^{2} {\sigma^{2}} (N+1)+2 {\sigma^{4}} N $$ $$\mu^{'}_{3} = E\left[Z^{3}\right] = A^{2} (N-1) \left(A^{4} (N-1)^2+6 A^{2} {\sigma^{2}} (N+1)+6 {\sigma^{4}} (N+1)\right)$$ $$ \mu^{'}_{4} = E\left[Z^{4}\right] = A^{8} (N-1)^4+12 A^{6} {\sigma^{2}} (N+1) (N-1)^2+12 A^{4} {\sigma^{4}} \left(N^3+N^2-N+3\right)+24 A^{2} {\sigma^{6}} (N+1)^2+12 {\sigma^{8}} N (N+1)$$
From everything I have seen, the Gram-Charlier expansion requires that the mean is equal to 0 and the variance equal to 1. However, this is clearly not the case for this random variable.
Questions:
- How can I fit the Gram-Charlier series to this random variable, given these raw moments?
- If I can't, what other expansion should I use?
- Could you solve the pdf via the Fourier transform?
Thanks!