Spectral density and Wiener–Khinchin

For a centered real weakly stationary process, take gamma(h)=Cov(X at t+h,X at t). If the covariance sequence is absolutely summable, its Fourier series defines a continuous nonnegative even density on minus pi to pi. We put one over two pi in the forward transform and no prefactor in the inverse integral. Therefore the integral of the density equals gamma(0), the variance. Frequencies are radians per observation, not cycles per unit of physical time.

Many time-series references instead use cycles nu between minus one half and one half, with the exponent minus two pi i nu h. Their cycles density S(nu) equals two pi times our angular density evaluated at two pi nu. In particular, weak white noise of variance sigma-squared has cycles density sigma-squared and angular density sigma-squared divided by two pi. These are the same variance allocation after the change of variable in the integral.

Without the summability assumption, the positive-semidefinite covariance sequence still has a finite nonnegative spectral measure by Herglotz's theorem. The inverse uses integration with respect to that measure. A random constant of variance v has covariance v at every lag and a spectral atom of mass v at zero; it has no ordinary density representing that mass. Random sinusoidal components similarly give spectral lines at paired nonzero frequencies under suitable stationary amplitude laws.

The measure is not a probability distribution unless normalized by the total variance. Its density, when present, can vanish at some frequencies without invalidating the process. Nor does having an absolutely continuous spectral measure by itself establish pure nondeterminism: the innovation factorization has additional regularity requirements. Wold's linear-past condition and a graph that looks smooth in frequency should therefore not be treated as interchangeable definitions.

A stable positive AR(1) coefficient produces a density concentrated toward zero frequency, reflecting slowly decaying positive covariances. A negative coefficient shifts emphasis toward pi, reflecting alternating lag correlations. These are properties of a stationary process and do not by themselves imply a deterministic trend. A near-unit root can create a very sharp stationary low-frequency peak, whereas a genuine random walk has no stationary covariance density of the form used here.

Differencing multiplies a stationary input spectrum by the squared modulus of one minus exp of minus i omega, suppressing zero frequency. It can be an appropriate transformation for integrated data, but applying it to already stationary white noise creates a negatively correlated MA process. Spectral filtering changes the information available for a task; it is not automatically a cure for every visual trend or a proof of a causal relation between two series with similar peaks.