= Wiener-Khinchin theorem
{c}
{title2=$\Gamma(\tau)=\int S(\Omega)e^{-i\Omega\tau}\,d\Omega$}
{wiki=Wiener–Khinchin_theorem}
For a complex <weakly stationary process> with finite second moment and continuous uncentered <autocorrelation> $\Gamma(\tau)=\mathbb E[\overline{X_t}X_{t+\tau}]$, there is a unique finite positive <spectral measure of a stationary random field>. With the optical sign convention, write it as
$$
\Gamma(\tau)=\int_{\mathbb R}e^{-i\omega\tau}\,dF(\omega),\qquad F(\mathbb R)=\mathbb E|X_t|^2.
$$
The sign reverses the frequency coordinate relative to the alternative positive-exponential convention. To see the positivity behind the theorem, expand $\mathbb E|\sum_j c_jX_{t_j}|^2\geq0$. This makes the correlation a <positive-definite function>, so <Bochner's theorem> supplies the measure.<245C, Notes 2: The Fourier transform by Terence Tao>[Bochner theorem immediately before Exercise58; applied here to the positive-definite correlation function]{ref}
An ordinary <power spectral density> $S$ exists when $dF=S(\omega)d\omega$. If $\Gamma$ is integrable, $S(\omega)=(2\pi)^{-1}\int\Gamma(\tau)e^{i\omega\tau}d\tau$. Stationarity alone does not guarantee such a density. For $X_t=Z$ at every time and $\mathbb E|Z|^2=1$, the correlation is one and $F=\delta_0$.
The normalized <Lorentzian line shape> $S(\omega)=\kappa/[\pi((\omega-\omega_0)^2+\kappa^2)]$ gives $\Gamma(\tau)=e^{-\kappa|\tau|}e^{-i\omega_0\tau}$. Its width therefore describes decay of temporal coherence. A persistent spectral line instead contributes an atom, which cannot be treated as an ordinary integrable density.
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