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Fourier Analysis of Periodic Weakly Stationary Processes: A Note on Slutsky’s Observation

Mini Course

Part of the Advances in Mathematical Economics book series (MATHECON,volume 20)


The periodic behavior of a specific weakly stationary stochastic process (w.s.p.) is examined from a viewpoint of classical Fourier analysis.(1) A w.s.p. has a spectral measure which is absolutely continuous with respect to the Lebesgue measure if and only if it is a moving average of a white noise. (2) A periodic or almost periodic w.s.p. must have a “discrete” spectral measure. Combining these two, we can conclude that any moving average of a white noise can neither be periodic nor almost periodic.However any w.s.p. can be approximated by a sequence of almost periodic w.s.p.’s in some specific sense.


  • Weakly stationary process
  • Periodicity
  • Almost periodicity
  • Spectral measure

The earlier draft of this paper was read at the 20th Conference of the International Federation of Operational Research Societies, which was held in Barcelona, (July 13–18, 2014). It is a pleasure for me to express my cordial gratitude to the late Professor Tatsuo Kawata for incessant encouragement, which led me to the field of Fourier analysis. I am much indebted to Professor Shigeo Kusuoka for his kind suggestions concerning probability theory, which I am not familiar with very well. Helpful comments by Dr. Yuhki Hosoya and Mr.Chaowen Yu are also gratefully acknowledged.

JEL Classification: CO2, E32

Mathematics Subject Classification (2010): 42A38, 42A82, 60G10

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Fig. 1


  1. 1.

    Frisch [11] also deserves a special attention.

  2. 2.

    Kawata [17, 18], Maruyama [23] and Wold [31] are classical works on Fourier analysis of stationary stochastic processes, which provided me with all the basic mathematical background. Among more recent literatures, I wish to mention Brémaud [5]. Granger and Newbold [12] Chap. 2, Hamilton [13] Chap. 3 and Sargent [26] Chap. XI are textbooks written from the standpoint of economics.

  3. 3.

    cf. Kawata [19] p. 46.

  4. 4.

    Schwartz [27] Chap. VII, §9, and Lax [20] Chap. 14 discuss the Bochner-Herglotz theorem in the spirit of the theory of distributions. Rudin [25] Chap. 1 provides a proof based upon the theory of Banach algebras. Naimark [24] Chap. 6 gives an abstract version, following D.A. Raikov.

    “Let G be a locally compact commutative topological group with unit e. Then any continuous positive semi-definite function \(\varphi: G \rightarrow \mathbb{C}\) is uniquely representable in the form

    $$\displaystyle{\varphi (g) =\int _{\,\widehat{G}}\chi (g)d\mu (\chi ),}$$

    where μ is a measure on the character group \(\widehat{G}\) of G, which satisfies \(\mu (\widehat{G}) =\varphi (e).\)

    The converse also holds true.

  5. 5.

    According to Itô [15] p. 255, Kolmogorov’s important article was published in C.R. Acad. Sci. URSS, 26 (1940), 115–118. However I have never read it yet, very regrettably. That is why I dropped it from the reference list.

  6. 6.

    In case X(t, ω) is real-valued, is it possible to give an expression of it in terms of an orthogonal measure without complex function like \(e^{i\lambda t}\)? This probelm was advocated by Slutsky [28, 29] and completed by Doob [10] and Maruyama [23]. See also Itô [15] pp. 263–266.

  7. 7.

    See Katznelson [16] p. 194. Loomis [21] is also beneficial.

  8. 8.

    This problem was studied by Doob [9] Chap. X, §8, Chap. XI, §8 and Kawata [19] pp. 69–73. I try to clarify the subtle details embedded in their works.

  9. 9.

    See Doob [9] Chap. II, §2.

  10. 10.

    The convergence of the series in (3) is in \(\mathfrak{L}^{2}(\mathbb{T}, \mathbb{C})\). However the series is, actually, convergent a.e. and is equal to \(\alpha (\lambda )\) according to the Carleson theorem. cf. Carleson [6].

  11. 11.

    In case \(\mathbb{T}_{0} =\emptyset\) (and so \(\alpha (\lambda )\) never vanishes), the discussion becomes much easier, since it is enough to define γ(S, ω) simply by

    $$\displaystyle{\gamma (S,\omega ) =\int _{S} \frac{1} {\alpha (\lambda )}\xi (d\lambda,\omega )}$$

    for any \(S \in \mathcal{B}(\mathbb{T})\). Clearly ν γ (S) = m(S).

  12. 12.
    $$\displaystyle\begin{array}{rcl} & & \mathbb{E}_{(\omega,\omega ')}\vert \gamma '(S,\omega,\omega ')\vert ^{2} {}\\ & & \qquad = \mathbb{E}_{\omega }\Big\vert \int _{S}\alpha _{1}(\lambda )\xi (d\lambda,\omega )\Big\vert ^{2} {}\\ & & \qquad \quad + \mathbb{E}_{\omega '}\Big\vert \int _{S}\alpha _{2}(\lambda )\eta (d\lambda,\omega ')\Big\vert ^{2} {}\\ & & \qquad \quad + 2\mathcal{R}e\mathbb{E}_{(\omega,\omega ')}\int _{S}\alpha _{1}(\lambda )\xi (d\lambda,\omega )1(\omega ')\int _{S}\overline{\alpha _{2}(\lambda )\eta (d\lambda,\omega ')1(\omega )} {}\\ & & \qquad =\int _{S}\vert \alpha _{1}(\lambda )\vert ^{2}\nu _{\xi }(d\lambda ) +\int _{ S}\vert \alpha _{2}(\lambda )\vert ^{2}\nu _{\eta }(d\lambda ) {}\\ & & \qquad \quad + 2\mathcal{R}e\mathbb{E}_{(\omega,\omega ')}\int _{S\cap \mathbb{T}_{+}}\alpha _{1}(\lambda )\xi (d\lambda,\omega )\int _{S\cap \mathbb{T}_{0}}\overline{\alpha _{2}(\lambda )\eta (d\lambda,\omega ')} {}\\ & & \qquad = m(S \cap \mathbb{T}_{+}) + m(S \cap \mathbb{T}_{0}) + 0. {}\\ \end{array}$$
  13. 13.
    $$\displaystyle\begin{array}{rcl} & & \int _{\mathbb{T}}e^{-in\lambda }\alpha (\lambda )\gamma (d\lambda,\omega ) {}\\ & & \quad =\int _{\mathbb{T}_{+}}e^{-in\lambda }\alpha (\lambda ) \cdot \frac{1} {\alpha (\lambda )}\xi (d\lambda,\omega ) + \mathbb{E}_{\omega '}\int _{\mathbb{T}_{0}}e^{-in\lambda }\alpha (\lambda ) \cdot \alpha _{ 2}(\lambda )\eta (d\lambda,\omega ') {}\\ & & \quad =\int _{\mathbb{T}_{+}}e^{-in\lambda }\xi (d\lambda,\omega ) =\int _{\mathbb{T}}e^{-in\lambda }\xi (d\lambda,\omega ). {}\\ \end{array}$$

    The final equality is justified by

    $$\displaystyle\begin{array}{rcl} & & \mathbb{E}_{\omega }\Big\vert \int _{\mathbb{T}_{0}}e^{-in\lambda }\xi (d\lambda,\omega )\Big\vert ^{2} =\int _{\mathbb{T}_{ 0}}\nu _{\xi }(d\lambda )\quad \text{(by D-K formula)} {}\\ & & \qquad =\int _{\mathbb{T}_{0}}p(\lambda )dm = 0\quad (\,p(\lambda ) = 0\;\text{on}\;\mathbb{T}_{0}). {}\\ \end{array}$$
  14. 14.

    Since Z n (ω) is a white noise, the covariance is given by

    $$\displaystyle{\mathbb{E}Z_{n+u}(\omega )\overline{Z_{n}(\omega )} = \left \{\begin{array}{@{}l@{\quad }l@{}} 1\quad \text{if}\quad u = 0,\quad \\ 0\quad \text{if} \quad u\neq 0.\quad \end{array} \right.}$$
  15. 15.

    The convergence of \((1/\sqrt{2\pi })\sum _{k=-p}^{q}c_{ k}e^{-ik\lambda }\) to \(C(\lambda )\) also holds true “almost everywhere” thanks to the Carleson theorem. Hence c k is the Fourier coefficient of \(C(\lambda )\) corresponding to \((1/\sqrt{2\pi })e^{-ik\lambda }\).

  16. 16.

    The orthogonality, for instance, can be verified as follows. If S and \(S' \in \mathcal{B}(\mathbb{T})\) are disjoint,

    $$\displaystyle\begin{array}{rcl} \mathbb{E}\theta (S,\omega )\overline{\theta (S',\omega )}& =& \mathbb{E}\int _{\mathbb{T}}C(\lambda )\chi _{S}(\lambda )\xi (d\lambda,\omega )\int _{\mathbb{T}}\overline{C(\lambda )\chi _{S'}(\lambda )\xi (d\lambda,\omega )} {}\\ & =& \int _{\mathbb{T}}\vert C(\lambda )\vert ^{2}\chi _{ S}(\lambda )\chi _{S'}(\lambda )d\nu _{\xi } = 0\quad (\text{by D-K formula}). {}\\ \end{array}$$
  17. 17.

    We can also establish

    $$\displaystyle{ \frac{1} {\sqrt{2\pi }}\int _{\mathbb{R}}f(\lambda )g(\lambda )e^{-iz\lambda }dm(\lambda ) = \frac{1} {\sqrt{2\pi }}(\mathfrak{F}_{2}f {\ast} \mathfrak{F}_{2}g)(z).}$$
  18. 18.

    The third line of (21)

    $$\displaystyle\begin{array}{rcl} & =& \frac{1} {\sqrt{2\pi }}\Big\{\int _{\mathbb{R}_{+}}\alpha (\lambda -(t + u))\alpha (\lambda -t)dm(\lambda ) +\int _{\mathbb{R}_{0}}\mathop{ \underbrace{\alpha (\lambda -(t + u))\alpha (\lambda -t)}}\limits _{(\dag )}d\nu _{\gamma }\Big\} {}\\ & =& \frac{1} {\sqrt{2\pi }}\Big\{\int _{\mathbb{R}}(\dag )dm(\lambda ) -\int _{\mathbb{R}_{0}}(\dag )dm(\lambda ) +\int _{\mathbb{R}_{0}}(\dag )d\nu _{\gamma }\Big\} {}\\ {}\\ {}\\ & =& \frac{1} {\sqrt{2\pi }}\Big\{\int _{\mathbb{R}}(\dag )dm(\lambda ) {}\\ & & \qquad \quad -\int _{\begin{array}{l}\lambda \in \mathbb{R}_{0} \\ \lambda -t\in \mathbb{R}_{+}\end{array}}(\dag )dm(\lambda ) -\int _{\begin{array}{l}\lambda \in \mathbb{R}_{0} \\ \lambda -t\in \mathbb{R}_{0}\end{array}}(\dag )dm(\lambda ) +\int _{\begin{array}{l}\lambda \in \mathbb{R}_{0} \\ \lambda -t\in \mathbb{R}_{+}\end{array}}(\dag )d\nu _{\gamma }(\lambda ) +\int _{\begin{array}{l}\lambda \in \mathbb{R}_{0} \\ \lambda -t\in \mathbb{R}_{0}\end{array}}(\dag )d\nu _{\gamma }(\lambda )\Big\} {}\\ & =& \frac{1} {\sqrt{2\pi }}\Big\{\int _{\mathbb{R}}(\dag )dm(\lambda ) {}\\ & & \qquad \quad -\int _{(\mathbb{R}_{0}-t)\cap \mathbb{R}_{+}}\alpha (\lambda ' - u)\alpha (\lambda ')dm(\lambda ') +\int _{(\mathbb{R}_{0}-t)\cap \mathbb{R}_{+}}\alpha (\lambda ' - u)\alpha (\lambda ')d\nu _{\gamma }(\lambda ')\Big\}. {}\\ \end{array}$$

    The last two terms cancel out. So we obtain (21).

  19. 19.

    However we have to be more careful about a couple of subtle reasonings. (a) \((\varOmega,\mathcal{E},P)\) can not be fixed a priori. It must be chosen suitably. (b) A justification must be given for the construction of independent random variables.

  20. 20.

    Let Φ 1, Φ 2, ⋯ be a sequence of Borel probability measures on \(\mathbb{R}\). Then there exists a sequence of independent random variables defined on some probability space \((\varOmega,\mathcal{E},P)\), the distributions of which are Φ 1, Φ 2, ⋯ (cf. Itô [15] p. 68).

  21. 21.

    Let X 1, X 2, ⋯ be a sequence of independent real-valued random variables, and \(g_{1},g_{2},\cdots: \mathbb{R} \rightarrow \mathbb{C}\) Borel measurable functions. Then \(Y _{1} = g_{1}(X_{1}),Y _{2} = g_{2}(X_{2}),\cdots \) are also independent random variables (cf. Itô [15] p. 66). So Y (ω) and e iZ(ω)t in the text are independent.

  22. 22.

    Katznelson [16] p. 197.

  23. 23.

    A function of the form

    $$\displaystyle{\,f(x) =\sum _{ j=1}^{n}a_{ j}e^{-i\xi _{j}u}\quad (\xi _{ j} \in \mathbb{R})}$$

    is called a trigonometric polynomial. Any trigonometric polynomial is almost periodic.


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We reviewed, in the first few sections, some basic concepts and results to be required in our main concerns (Theorems 1, 2, and 3). I expect this exposition to be a small help for the readers who are not familiar with them. There can be no communications without common language. Of course, these materials are, more or less, known to every mathematician working in this discipline. That is why we just stated necessary facts and dropped proofs in the text.

However it does not seem to be a waste of time to give here complete proofs of selected three propositions (Propositions 5’6’ and 7’) as exceptions, taking account of their indispensable roles in our problem.

Proof of Proposition 5’.

The proof is basically due to Kawata [19] pp. 56–57.Footnote 19 If we define \(\theta: \mathcal{B}(\mathbb{R}) \rightarrow \mathbb{R}\) by

$$\displaystyle{ \theta (E) = \frac{\nu (E)} {\nu (\mathbb{R})},\quad E \in \mathcal{B}(\mathbb{R}), }$$

then \(\theta\) is a Radon probability measure. There exist independent real random variables Z(ω) and Y (ω) defined on some probability space \((\varOmega,\mathcal{E},P)\) such that the distribution of Z(ω) is equal to \(\theta\), \(\mathbb{E}Y (\omega ) = 0\), and finally \(\mathbb{E}Y (\omega )^{2} = (1/\sqrt{2\pi })\nu (\mathbb{R})\). Footnote 20 Define a stochastic process X(t, ω) by

$$\displaystyle{ X(t,\omega ) = Y (\omega )e^{iZ(\omega )t}. }$$

Then \(\mathbb{E}X(t,\omega )\) and ρ(t + u, t) are calculated as follows.Footnote 21

$$\displaystyle\begin{array}{rcl} \mathbb{E}X(t,\omega )& =& \mathbb{E}Y (\omega )\mathbb{E}e^{iZ(\omega )t} = 0\quad \text{(by independence)}, {}\\ \rho (t + u,t)& =& \mathbb{E}X(t + u,\omega )\overline{X(t,\omega )} = \mathbb{E}[\,y(\omega )^{2}e^{iZ(\omega )u}] {}\\ & =& \frac{1} {\sqrt{2\pi }}\nu (\mathbb{R})\int _{\mathbb{R}}e^{i\lambda u}d\theta (\lambda ) = \frac{1} {\sqrt{2\pi }}\int _{\mathbb{R}}e^{i\lambda u}d\nu (\lambda ). {}\\ \end{array}$$

Thus the covariance ρ(t + u, t) of X(t, ω) depends only upon u, and it is expressed as the Fourier transform of ν. Since the Fourier transform of ν is uniformly continuous, X(t, ω) is strongly continuous on \(\mathbb{R}\) by Proposition 1. The measurability of X(t, ω) is obvious. Q.E.D.

Proof of Proposition 6’.

(i)\(\Rightarrow \) (ii): Assume that ρ(u) is τ-periodic. We then prove that

$$\displaystyle{ \mathbb{E}\vert X(t+\tau,\omega ) - X(t,\omega )\vert ^{2} = 0 }$$

which is equivalent to (ii). By direct computation, we have

$$\displaystyle\begin{array}{rcl} & & \mathbb{E}\vert X(t+\tau,\omega ) - X(t,\omega )\vert ^{2} {}\\ & & \ = \mathbb{E}\vert X(t+\tau,\omega )\vert ^{2} + \mathbb{E}\vert X(t,\omega )\vert - 2\mathcal{R}e\,\mathbb{E}X(t+\tau,\omega )\overline{X(t,\omega )} {}\\ & & \ = 2\rho (0) - 2\mathcal{R}e\rho (\tau ) {}\\ & & \ = 2(\rho (0) -\mathcal{R}e\rho (\tau )) {}\\ & & \ = 0\quad (\text{by (i)}). {}\\ \end{array}$$

(ii)\(\Rightarrow \) (i): Assume (ii). then we have

$$\displaystyle\begin{array}{rcl} \vert \rho (u+\tau ) -\rho (u)\vert ^{2}& =& \vert \mathbb{E}[X(u+\tau,\omega )\overline{X(0,\omega )} - X(u,\omega )\overline{X(0,\omega )}]\vert ^{2} {}\\ & =& 0.\quad (\text{by (ii)}) {}\\ \end{array}$$

(i)\(\Rightarrow \) (iii): If ρ(u) is τ-periodic, then we have

$$\displaystyle{ 0 = 2\rho (0) -\rho (\tau ) -\rho (-\tau ) = \frac{2} {\sqrt{2\pi }}\int _{\mathbb{R}}(1 -\cos t\tau )d\nu (t). }$$

Taking account of \(1 -\cos t\tau \geqq 0\), we must have ν(E) = 0 for any \(E \in \mathcal{B}(\mathbb{R})\) such that

$$\displaystyle{ E \cap \{ t \in \mathbb{R}\vert 1 -\cos t\tau = 0\} =\emptyset, }$$

which is equivalent to

$$\displaystyle{ E \cap \{ 2k\pi /\tau \vert k \in \mathbb{Z}\} =\emptyset. }$$

(iii)\(\Rightarrow \) (i): By definition of the spectral measure,

$$\displaystyle{ \rho (u) = \frac{1} {\sqrt{2\pi }}\int _{\mathbb{R}}e^{-iut}d\nu (t). }$$

So putting \(a_{k} =\nu (\{2\pi k/\tau \})\;(k \in \mathbb{Z})\), we obtain, by (iii), that

$$\displaystyle{ \rho (u) = \frac{1} {\sqrt{2\pi }}\sum a_{k}e^{-iu\cdot 2\pi k/\tau }. }$$

This is clearly τ-periodic. Q.E.D.

The following well-known lemmaFootnote 22 will be used in the course of the proof of Proposition 7’.


Let μ be a Radon measure on \(\mathbb{R}\) . If the Fourier transform \(\hat{\mu }\) is almost periodic, then μ is discrete.

Proof of Proposition 7’.

(i)\(\Rightarrow \)(ii): Assume that ρ(μ) is almost periodic. Then we have

$$\displaystyle\begin{array}{rcl} \sup _{t\in \mathbb{R}}\mathbb{E}\vert X(t+\tau,\omega ) - X(t,\omega )\vert ^{2}& =& 2[\rho (0) -\mathcal{R}e\rho (\tau )] \\ & =& 2\mathcal{R}e[\rho (0) -\rho (\tau )] \leqq 2\vert \rho (0) -\rho (\tau )\vert \\ &\leqq & 2\sup _{u\in \mathbb{R}}\vert \rho (u+\tau ) -\rho (u)\vert. {}\end{array}$$

Since ρ(⋅ ) is almost periodic, there exists a number \(\varLambda (\varepsilon /2,\rho ) > 0\), for each \(\varepsilon > 0\), such that any interval the length of which is \(\varLambda (\varepsilon /2,\rho )\) contains an \(\varepsilon /2\)- almost period, τ. Since

$$\displaystyle\begin{array}{rcl} \{\tau \in \mathbb{R}\vert & \sup _{u\in \mathbb{R}}& \vert \rho (u+\tau ) -\rho (u)\vert < \frac{\varepsilon } {2}\} {}\\ & \subset &\{\tau \in \mathbb{R}\vert \sup _{t\in \mathbb{R}}\mathbb{E}\vert X(t+\tau,\omega ) - X(t,\omega )\vert ^{2} <\varepsilon \} {}\\ \end{array}$$

by (1), we see that (ii) is satisfied by setting \(\varGamma (\varepsilon,X) =\varLambda (\varepsilon /2,\rho )\).

(ii)\(\Rightarrow \)(i): Assume (ii). By a simple calculation, we obtain the inequality

$$\displaystyle\begin{array}{rcl} \vert \rho (u+\tau ) -\rho (u)\vert ^{2}& =& \vert \mathbb{E}[X(u+\tau,\omega )\overline{X(0,\omega )} - X(u,\omega )\overline{X(0,\omega )}]\vert ^{2} {}\\ & \leqq & \mathbb{E}\vert X(u+\tau,\omega ) - X(u,\omega )\vert ^{2}\mathbb{E}\vert X(0,\omega )\vert ^{2} {}\\ & & \qquad \qquad \text{(by Schwarz inequality)}. {}\\ \end{array}$$

It follows that

$$\displaystyle{ \vert \rho (u+\tau ) -\rho (u)\vert \leqq [\mathbb{E}\vert X(t+\tau,\omega ) - X(u,\omega )\vert ^{2}]^{1/2}\rho (0)^{1/2}. }$$

Hence we have

$$\displaystyle\begin{array}{rcl} & & \Big\{\tau \in \mathbb{R}\Big\vert \sup _{u\in \mathbb{R}}\mathbb{E}\vert X(u+\tau,\omega ) - X(u,\omega )\vert ^{2} < \frac{\varepsilon ^{2}} {\rho (0)}\Big\} {}\\ & & \phantom{\Big\{\tau \in \mathbb{R}\Big\vert \sup _{u\in \mathbb{R}}\mathbb{E}\vert X(u+\tau,\omega )-}\subset \{\tau \in \mathbb{R}\vert \sup _{u\in \mathbb{R}}\vert \rho (u+\tau ) -\rho (u)\vert <\varepsilon \}. {}\\ \end{array}$$

Thus (i) holds true by setting \(\varLambda (\varepsilon,\rho ) =\varGamma (\varepsilon ^{2}/\rho (0),X)\).

(i)\(\Rightarrow \) (iii): This is a direct consequence of the lemma. The covariance function ρ is represented by the Fourier transform of some positive Radon measure ν on \(\mathbb{R}\) (Proposition 4’). Since \(\rho (u) =\hat{\nu } (u)\) is almost periodic, ν must be discrete by the lemma.

(iii)\(\Rightarrow \) (i): Assume that the spectral measure ν of X is discrete, say

$$\displaystyle{ \nu =\sum _{ n=1}^{\infty }a_{ n}\delta _{\xi _{n}}\quad (\delta _{\xi _{n}}: \text{Dirac measure}). }$$

Then the covariance can be expressed as

$$\displaystyle{ \rho (u) = \frac{1} {\sqrt{2\pi }}\int _{\mathbb{R}}e^{-i\lambda u}d\nu (\lambda ) = \frac{1} {\sqrt{2\pi }}\sum _{n=1}^{\infty }a_{ n}e^{-i\xi _{n}u}. }$$

Since \(a_{n} \geqq 0\), the series (2) is absolutely and so uniformly convergent. Thus ρ(u) is the uniform limit of trigonometric polynomials,Footnote 23 and hence almost periodic.


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Maruyama, T. (2016). Fourier Analysis of Periodic Weakly Stationary Processes: A Note on Slutsky’s Observation. In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics Volume 20. Advances in Mathematical Economics, vol 20. Springer, Singapore.

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