Assume that I have some probability density function, $f\left(x\right)$. If I want to approximate it using a Fourier series I can use the identity:
$c_{k}=\frac{1}{2\pi}\int_{-\pi}^{\pi} f\left(x\right)e^{-ikx}\text{d}x$
If I don't know what my true pdf is, but I have $N$ samples, $\left\{x_1,x_2,\dots,x_N\right\}$ from my distribution, I can get an approximation for my Fourier coefficients by the following:
$c_k\approx\hat{c_k}=\frac{1}{2\pi N} \sum_{n=1}^{N} e^{-ikx_n}$
My question is, how can I prove whether or not $\lim_{N\rightarrow\infty} \hat{c_k} = c_k$?