Some time ago, I derived the function of random variable for the physical process: mixing of diffusing particles (see the link for more details) to take into account the finite number N of such particles. $F_N (x)=Φ(x)+\frac{3}{50\sqrt{2π}} e^{-x^2/2} (x^3-3x) \frac {6N^3+21N^2+31N+31} {N(2N+5)^2 (N-1)}+O(\frac{1}{N^2} )$
The function was derived as asymptotic expansion in the Central Limit Theorem. Due to the asymptotic expansion, there is an error like $O(\frac{1}{N^2})$. I guess that for practical usage, the error $\epsilon \le 1$% of such approximation should be fine.
My question: what assumptions are required to get probability distribution function and what would be the final pdf?
The difference with normal distribution depends on the second summand: $\frac {1}{\sqrt{2\pi}}e^{-x^2/2}+\frac{3}{50\sqrt{2π}}\frac {6N^3+21N^2+31N+31} {N(2N+5)^2 (N-1)}e^{-x^2/2} (-x^4+6x^2-3)$
I think that the assumptions have to be in line with the case $N \to \infty$ when we have normal distribution.
PS. pre-kidney user advised to post a separate question about this topic.