Wiener–Khinchin theorem

Remarks

The function $r$ is positive definite: for all $\tau_1, \dots, \tau_k \in \R$ and $c_1, \dots, c_k \in \C$, $\sum_{j, l} c_j \overline{c_l} \, r(\tau_j - \tau_l) = E\bigl[\abs{\sum_j c_j (X_{\tau_j} - m)}^2\bigr] \ge 0$. A positive definite function that is continuous at $0$ is continuous everywhere, and Bochner's theorem then gives the measure $F$. The density formula is Fourier inversion, see the Fourier transform.

For a weakly stationary sequence $(X_n)_{n \in \Z}$, the analogue is Herglotz's theorem, with a finite measure on $[-\pi, \pi)$ and no continuity assumption.

Example: if $r(\tau) = e^{-\abs{\tau}}$, as for a suitably normalised stationary Ornstein–Uhlenbeck process, then $\int_{\R} e^{-\abs{\tau}} e^{-i\tau\omega} \, d\tau = \dfrac{2}{1 + \omega^2}$, so $S(\omega) = \dfrac{1}{\pi(1 + \omega^2)}$.

Going deeper

For a student

The theorem is the basis of spectral analysis of time series. It is used to estimate the spectrum from data and to classify noise; for example, white noise has a flat spectrum. It is also used to design filters, since a linear time-invariant filter multiplies the spectral density by the squared modulus of its transfer function. The spectral measure also leads to the spectral representation of the process itself as a stochastic integral against a process with orthogonal increments.