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  "body": "\n## Statement\n\nLet $(X_t)_{t \\in \\R}$ be a complex-valued weakly [[stationary process|DT0QXJE705]]: $E[\\abs{X_t}^2] \u003c \\infty$, $E[X_t] = m$ for all $t$, and the autocovariance $r(\\tau) = E[(X_{t+\\tau} - m)\\overline{(X_t - m)}]$ does not depend on $t$.\nAssume that $r$ is [[continuous|0X3WGX95FE]] at $0$.\nThen there is a unique finite Borel measure $F$ on $\\R$, the *spectral measure* of $X$, such that\n$$r(\\tau) = \\int_{\\R} e^{i \\tau \\omega} \\, F(d\\omega) \\quad \\text{for all } \\tau \\in \\R.$$\nIf moreover $r$ is integrable on $\\R$, then $F$ has the continuous density\n$$S(\\omega) = \\frac{1}{2\\pi} \\int_{\\R} r(\\tau) \\, e^{-i \\tau \\omega} \\, d\\tau,$$\nthe *power spectral density* of $X$, and $S(\\omega) \\ge 0$ for all $\\omega$.\n\n## Remarks\n\nThe function $r$ is positive definite: for all $\\tau_1, \\dots, \\tau_k \\in \\R$ and $c_1, \\dots, c_k \\in \\C$, $\\sum_{j, l} c_j \\overline{c_l} \\, r(\\tau_j - \\tau_l) = E\\bigl[\\abs{\\sum_j c_j (X_{\\tau_j} - m)}^2\\bigr] \\ge 0$.\nA positive definite function that is continuous at $0$ is continuous everywhere, and Bochner's theorem then gives the measure $F$.\nThe density formula is Fourier inversion, see the Fourier transform.\n\nFor 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.\n\nExample: if $r(\\tau) = e^{-\\abs{\\tau}}$, as for a suitably normalised stationary [[Ornstein–Uhlenbeck process|HZM2ASCDY4]], then $\\int_{\\R} e^{-\\abs{\\tau}} e^{-i\\tau\\omega} \\, d\\tau = \\dfrac{2}{1 + \\omega^2}$, so $S(\\omega) = \\dfrac{1}{\\pi(1 + \\omega^2)}$.\n\n## Explanations\n\n### L1 human\n\nA stationary random signal can be analysed in two ways: by how its values at two times are correlated, or by how much of its power lies at each frequency.\nThe Wiener–Khinchin theorem says that these two descriptions carry the same information: the power spectrum is the Fourier transform of the autocovariance.\nThis is why engineers can measure the spectrum of noise from its correlations.\n\n### L2 human\n\nThe theorem is the basis of spectral analysis of time series.\nIt is used to estimate the spectrum from data and to classify noise; for example, white noise has a flat spectrum.\nIt is also used to design filters, since a linear time-invariant filter multiplies the spectral density by the squared modulus of its transfer function.\nThe spectral measure also leads to the spectral representation of the process itself as a stochastic integral against a process with orthogonal increments.\n",
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