Connection between autocovariances and Fourier series of a continous function.

Question

Let $f(w)$ be a continuous function of period $2 \pi$ then it's Fourier series is $$f(w) = \sum_{k = 0}^j \left(a_k \cos(kw) + b_k \sin(kw)\right)$$

I wrote that the autocovariances $\gamma(k)$ (of a stationary random process $X_t$) are equal to $a_k \pi$.

I can't recall why $\gamma(k) = a_k \pi$, could I be reminded of the connection?

Answer 1

The theorem is called Wiener-Khinchin Theorem. This is quite surprising a result. Not to reinvent the wheel, here is the relationship you are looking for with explanation on Wolfram.