I'm currently working through a script on time series analysis and happend across this theorem of kolmogorov:

For every positive kernel \(\displaystyle \phi: T \times T \to \mathbb{C} \) there exists a stochastic process \(\displaystyle (X_t)_{t \in T}\) where \(\displaystyle X_t \in L^2(P), E(X_t) = 0\) and \(\displaystyle \phi(s,t) = Cov(X_s,X_t)\).

Sadly, neither did the script provide a proof, nor did it refer to literature where I might find it.

So, I would be happy, if someone could point me to some literature or, if you happen to know the proof, outline it.

Thanks.