the spectral representation of a stationary Gaussian process
A stationary process looks statistically the same no matter when you start watching it — its covariance depends only on the time lag, not on absolute time. For stationary Gaussian processes there is a beautiful Fourier-analytic answer to 'what do they look like': every such process is a superposition of random sinusoids whose amplitudes across frequencies are governed by a spectral measure. This is the frequency-domain picture of Gaussian noise, and it makes the covariance and the process two sides of a Fourier transform.
Let (X_t)_(t in R) be a centered, stationary Gaussian process: K(s, t) = r(t - s) depends only on the lag h = t - s. By Bochner's theorem, the covariance function r is positive-definite, hence the Fourier transform of a finite non-negative measure mu on the frequency axis: r(h) = integral exp(i lambda h) mu(d lambda). This mu is the spectral measure; when it has a density f, called the spectral density, r(h) = integral exp(i lambda h) f(lambda) d lambda and the total variance is r(0) = mu(R). Cramer's spectral representation lifts this to the process itself: X_t = integral exp(i lambda t) dZ(lambda), where Z is a complex Gaussian random measure (orthogonal increments) with E[ |dZ(lambda)|^2 ] = mu(d lambda). So the process is literally an integral of random-amplitude complex exponentials, with power at frequency lambda set by the spectrum. Concentrated spectrum means slowly varying, highly correlated paths; spread-out (flat) spectrum means rough, weakly correlated paths — white noise is the formal limit of a flat spectral density.
Why it matters: the spectral representation diagonalizes a stationary Gaussian process — frequencies become independent coordinates — turning convolution-type operations into multiplication and making prediction, filtering, and sample-path regularity computable from the tail of the spectral measure (heavier high-frequency tail = rougher paths). It is the foundation of time-series analysis, signal processing, and the Karhunen-Loeve idea. Two honest points: 'stationary' here is strict stationarity, which for a Gaussian process is EQUIVALENT to weak (covariance) stationarity precisely because the whole law is determined by the covariance; and Bochner requires r to be continuous (or the measure may have to include atoms / a discrete spectral line for periodic components).
The stationary Ornstein-Uhlenbeck process has covariance r(h) = exp(-|h|). Its spectral density is the Cauchy/Lorentzian f(lambda) = (1/pi) / (1 + lambda^2): the Fourier transform of exp(-|h|). The slow polynomial decay of f in lambda reflects OU's rough (non-differentiable) but stationary paths; a faster-decaying spectrum (e.g. Gaussian f) would give smoother paths.
A stationary Gaussian process is a Fourier integral of random sinusoids; its spectral density's tail dictates path roughness.
For a Gaussian process strict and weak (covariance) stationarity coincide, because the covariance determines the whole law — a luxury that fails for non-Gaussian processes.