I am wondering what's the distribution of $$\frac{n_1+n_2+n_3}{\sqrt{N_1N_2N_3}} \text{ with } n_i=0,\ldots,N_{i}-1$$ A quick check using matlab gives that the histogram of this sequence follows a Gaussian distribution. I am wondering how I can prove this analytically.
Thanks for your response in advance!
If $A,B,C$ are independent random variables, uniformly distributed over $(0,a),(0,b),(0,c)$, the characteristic function of $A+B+C$ is given by $$ \frac{i(1-e^{iat})}{at}\cdot \frac{i(1-e^{ibt})}{bt}\cdot \frac{i(1-e^{ict})}{ct} $$ and the PDF of $A+B+C$ can be recovered by (Fourier) inversion, i.e. by computing $$ \frac{1}{2\pi}\int_{-\infty}^{+\infty}\frac{i(1-e^{iat})}{at}\cdot \frac{i(1-e^{ibt})}{bt}\cdot \frac{i(1-e^{ict})}{ct} e^{-itx}\,dt, $$ usually through the residue theorem. We may also notice that the PDF of $A+B$ is the convolution of the PDFs of $A$ and $B$, hence it is a continuous and piecewise-linear function. Similarly, the PDF of $A+B+C$ is a $C^1$, piecewise-quadratic function.
If $A,B,C\sim U\left(-1,1\right)$ the PDF of $A+B+C$ is explicitly given by $$\frac{(3-x)|3-x|-3(1-x)|1-x|-3(1+x)|1+x|+(3+x)|3+x|}{32}$$ and the following diagram provides a comparison between the PDF of $A+B+C$ (blue) and the PDF of the standard normal distribution (purple). Indeed they are pretty close, as the Berry-Esseen theorem ensures.
Actually a better Gaussian approximation of the PDF of $A+B+C$ is $\sqrt{\frac{7}{16\pi}}\exp\left(-\frac{7x^2}{16}\right)$.
We may deal with discrete distributions along the same lines, I just picked continuous distributions since the Fourier inversion theorem has a more compact form.