Spectral Lines and Periodic Components

From a Spectral Atom to a Covariance Oscillation

A spectral measure describes how the variance of a weakly stationary sequence is distributed across frequencies. When the measure has an atom, it places positive variance at one exact frequency rather than spreading that variance continuously over a range. The resulting covariance contribution does not fade as the lag grows: it oscillates with a fixed amplitude. This is why an atom is also called a spectral line.

Let \((X_n)_{n\in\mathbb Z}\) be a real-valued weakly stationary sequence on a probability space \((\Omega,\mathcal F,\mathbb P)\). Write its mean as \(\mu\) and its autocovariance as \(\gamma(h)=\operatorname{Cov}(X_{n+h},X_n)\), which does not depend on \(n\). Use the discrete-time frequency circle, represented by \([-\pi,\pi)\), with frequencies understood modulo \(2\pi\). In the Fourier convention used here, its finite positive spectral measure \(F\) satisfies

A point \(\lambda\) is an atom if \(F(\{\lambda\})>0\); its mass is the variance assigned to that line. The integral formula alone gives its covariance contribution: a mass \(a\) at \(\lambda\) contributes \(a e^{ih\lambda}\) at lag \(h\). For a real-valued sequence, the full covariance is real and even. Accordingly, its spectral measure is symmetric on the frequency circle: a line at \(\omega\), for \(0<\omega<\pi\), is accompanied by one at \(-\omega\) with the same mass. If each has mass \(a\), their combined contribution is \(2a\cos(h\omega)\). The frequencies \(0\) and \(\pi\) are their own negatives modulo \(2\pi\), so they need not occur in pairs.

The symmetry statement follows from the standard uniqueness of the spectral measure: the reflected measure has Fourier coefficients \(\gamma(-h)\), which equal \(\gamma(h)\) for a real stationary sequence, so it is the same measure. The uniqueness of the spectral measure is a standard result from the spectral representation of stationary sequences.

This conclusion uses the covariance formula for the spectral measure; it does not require asserting that a random-measure representation exists on the original probability space. In particular, for a sequence indexed only by \(n\geq0\), one must not assume without further conditions that such a representation is available on that same space. The covariance conclusions below are stated directly in terms of \(F\).

The Atomic Part and Its Covariance

Proof. Since \(F\) is finite, for each positive integer \(m\) there can be only finitely many atoms with mass at least \(1/m\): otherwise, arbitrarily large finite collections of such distinct atoms would have total mass at least their size divided by \(m\), contradicting \(F([-\pi,\pi))<\infty\). Every positive mass is at least \(1/m\) for some \(m\), so \(D\) is a countable union of finite sets and is at most countable. For any finite subset \(D_0\) of \(D\), \(\sum_{\lambda\in D_0}a_\lambda\leq F([-\pi,\pi))\). Taking increasing finite subsets shows \(\sum_{\lambda\in D}a_\lambda\leq F([-\pi,\pi))\). Thus the series defining the atomic measure has finite total mass. Since \(|e^{ih\lambda}|=1\), its covariance series is absolutely convergent, with \(\sum_{\lambda\in D}|a_\lambda e^{ih\lambda}|=\sum_{\lambda\in D}a_\lambda<\infty\). Integrating against the sum of point masses gives the displayed series. For a real sequence the measure is symmetric, so the terms at \(\omega\) and \(-\omega\) have equal mass and sum to \(a e^{ih\omega}+a e^{-ih\omega}=2a\cos(h\omega)\). At \(0\), the exponential is \(1\); at \(\pi\), it is \(e^{ih\pi}=(-1)^h\). \(\square\)

The theorem concerns the covariance contributed by the atomic part of the measure. It does not say that the remainder of the covariance vanishes, nor that the sequence itself is just a sum of sinusoidal sample paths. A measure may have both atomic and non-atomic parts. The atomic contribution can be calculated separately because integration is additive over measures.

Proof. For a single atom, \(|a e^{ih\lambda}|=a>0\) for every integer \(h\). For a real pair with \(0<\omega<\pi\), put \(c_h=2a\cos(h\omega)\). For positive integer \(N\), the average of its squared values is $$ \frac1N\sum_{h=0}^{N-1}c_h^2 =\frac{4a^2}{N}\sum_{h=0}^{N-1}\cos^2(h\omega) =2a^2+\frac{2a^2}{N}\sum_{h=0}^{N-1}\cos(2h\omega). $$ Because \(e^{i2\omega}\ne1\), the finite geometric-sum formula gives \(\left|\sum_{h=0}^{N-1}e^{i2h\omega}\right|=\left|(1-e^{i2N\omega})/(1-e^{i2\omega})\right|\leq2/|1-e^{i2\omega}|\). The absolute value of the cosine sum is no larger than this bound, so its contribution divided by \(N\) tends to zero. The average of \(c_h^2\) therefore tends to \(2a^2>0\). If \(c_h\) converged to zero, then its squared values would converge to zero, and their averages would also converge to zero; this contradicts the positive limit. The endpoint lines give the constant sequence \(a\) or the alternating sequence \(a(-1)^h\), neither of which tends to zero. \(\square\)

A line contribution is periodic whenever its frequency is a rational multiple of \(2\pi\): if \(q\omega\) is an integer multiple of \(2\pi\), then \(e^{i(h+q)\omega}=e^{ih\omega}\), so the contribution repeats after \(q\) lags. An irrational frequency ratio need not give an exactly repeating sequence, but the contribution still does not decay. “Persistent” therefore refers to non-decay, not necessarily to exact periodicity.

Worked Covariance Calculations

Interpreting the Lines Carefully

Two distinctions prevent common errors. First, the mass of a single line is not always the amplitude of a real cosine. At an interior frequency, the matching line at the negative frequency must also be included. Second, an atomic covariance contribution is only one part of the total covariance if the spectral measure also has non-atomic mass. The atomic and non-atomic parts add; neither should be mistaken for the whole sequence without checking the measure.

Common mistake. In the first example, one might keep only the \(+\pi/3\) line and claim that the real covariance is \(\tfrac14\cos(h\pi/3)\). That calculation both discards the negative-frequency line and gives the wrong variance: at \(h=0\) it returns \(\tfrac14\), although the total mass, and therefore the variance, is \(\tfrac14+\tfrac14=\tfrac12\). The correction is to add both terms:

Finally, a spectral line identifies a persistent covariance pattern. It does not, by itself, prove that the observed sequence has a particular sinusoidal sample path or that a sinusoidal component is independent of the rest. Such stronger decomposition claims require additional representation and probability-space conditions. For covariance calculations, the safe procedure is direct: identify each atomic mass, multiply it by \(e^{ih\lambda}\), sum the contributions, and for a real sequence combine conjugate lines into cosines.