Abstract
During the decade around 1930, the world economy was thrown into a serious depression that nobody had previously experienced.
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- 1.
R. Frisch [6] also deserves special attention.
- 2.
- 3.
Kawata [12, 13], Maruyama [20] 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 literature, I wish to mention Brémaud [1]. Granger and Newbold [7] Chap. 2, Hamilton [8] Chap. 3 and Sargent [26] Chap. XI are textbooks written from the standpoint of economics.
- 4.
- 5.
That is, both the real and imaginary parts are rationals.
- 6.
We should use distinct notations for one-variable function ρ(u) and two variable function ρ(s, t). However, we use the same notation because there seems no possibility of confusion. ρ(u) is also called the autocovariance and ρ(u)∕ρ(0) is called the autocorrelation of the process.
- 7.
In this evaluation, we are examining the situation where the upper limits (q, q′) of the sums tend to ∞, and the lower limits (p, p′) tend to −∞. So, without loss of generality, we may assume \({p', p<0,\; 0\leqq q, q'}\). There are various cases other than p′< p and q < q′. However, we can treat them in the same manner.
- 8.
It is known that the right-hand side converges strongly. Hence we can specify the upper and lower limits of \(\sum \) as p and −p. l.i.m. denotes the limit in \(\mathfrak {L}^2\).
- 9.
Crum [3].
- 10.
It is well-known that
$$\displaystyle \begin{aligned} |\alpha -\beta|{}^2 \leqq 2(|\alpha |{}^2+|\beta|{}^2) \end{aligned}$$for any \(\alpha , \beta \in \mathbb {C}\), in general. Specifying α and β as
$$\displaystyle \begin{aligned} \alpha =e^{-t^2}X(t,\omega),\quad \beta=e^{-(t+u)^2}X(t+u,\omega), \end{aligned}$$we obtain the desired result.
- 11.
The graph of the function t↦e −2tu is depicted as follows according to the sign of u.
-
(a)
In the case of u > 0, e −2tu monotonically converges to 1 at each t as u → 0. To say more in detail, it is monotonically increasing in the region t > 0, and decreasing in the region t < 0. Since we may assume \(0<u\leqq 1\), without loss of generality, \(e^{-2t^2-2tu}\leqq e^{-2t^2-2t}\) in the region t < 0. Hence applying the dominated convergence theorem and the monotone convergence theorem to the first and the second terms, respectively, of the right-hand side of the integral
$$\displaystyle \begin{aligned} \int_{\mathbb{R}} e^{-2t^2-2tu}dt = \int_{-\infty}^0 e^{-2t^2} e^{-2tu}dt + \int_0^\infty e^{-2t^2} e^{-2tu}dt, \end{aligned}$$we obtain
$$\displaystyle \begin{aligned} \int_{\mathbb{R}} e^{-2t^2-2tu}dt\rightarrow \int_{\mathbb{R}} e^{-2t^2}dt \end{aligned}$$(†)as u → 0 (u > 0).
-
(b)
A similar argument applies to the case of u < 0, u → 0.
In general, assume that a numerical sequence {u n} (u n may either be positive or negative) converges to 0. Suppose
$$\displaystyle \begin{aligned} \int_{\mathbb{R}}e^{-2t^2}e^{-2tu_n}dt \nrightarrow \int_{\mathbb{R}}e^{-2t^2}dt. \end{aligned} $$Then for sufficiently small ε > 0, there exists a subsequence \(\{u_{n'}\}\) of {u n} such that
$$\displaystyle \begin{aligned} \bigg|\int_{\mathbb{R}}e^{-2t^2-2tu_{n'}}dt-\int_{\mathbb{R}}e^{-2t^2}dt \bigg|\geqq \varepsilon \quad \text{for all}\; n'. \end{aligned} $$If we choose a further subsequence \(\{u_{n''}\}\) of \(\{u_{n'}\}\) of the type (a) or (b), then we must have
$$\displaystyle \begin{aligned} \bigg|\int_{\mathbb{R}}e^{-2t^2-2tu_{n''}}dt-\int_{\mathbb{R}}e^{-2t^2}dt\bigg|\geqq \varepsilon \quad \text{for all}\; n''. \end{aligned} $$A contradiction occurs.
-
(a)
- 12.
Kawata [16], pp. 150–151.
- 13.
X(t, ω) is strongly continuous by Theorem 8.5.
- 14.
Let Φ 1, Φ 2, ⋯ be a sequence of Borel probability measures. Then there exist a certain probability space \((\varOmega , \mathbb {E}, P)\) and mutually independent random variables, the distributions of which are Φ 1, Φ 2, ⋯( Itô [10], p. 68).
- 15.
Let X 1, X 2, ⋯ be independent real random variables and \(g_1, g_2, \cdots : \mathbb {R}\rightarrow \mathbb {C}\) Borel measurable. Then Y 1 = g 1(X 1), Y 2 = g 2(X 2), ⋯ are independent random variables ( Itô [10], p. 66). Based upon this result, Y (ω) and e −iZ(ω)t in the text are independent.
- 16.
If we denote by \(\hat {\nu }\) the Fourier transform of ν, we have
$$\displaystyle \begin{aligned} |\hat{\nu}(t+h)-\hat{\nu}(t)| &=\frac{1}{\sqrt{2\pi}}\bigg|\int_{\mathbb{R}}(e^{-i(t+h)x}-e^{-itx})d\nu(x)\bigg| \\ &\leqq \frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}|e^{-ihx}-1|d\nu(x) \\ &\rightarrow 0\quad \text{as}\quad h\rightarrow 0, \end{aligned} $$by the dominated convergence theorem.
- 17.
Hence if the right-hand side is a finite sum,
$$\displaystyle \begin{aligned}\xi\bigg(\bigcup_{j=1}^p S_j, \omega\bigg)=\sum_{j=1}^p \xi (S_j, \omega)\;\; \text{a.e.}\end{aligned}$$ - 18.
According to Itô [10] p. 255, Kolmogorov’s important article was published in C.R. Acad. Sci. URSS, 26 (1940), 115–118. However, I have never read it, very regrettably. That is why I dropped it from the reference list.
- 19.
See Maruyama [21] pp. 74–76.
- 20.
\(\nu _\xi (S)=\|\xi (S,\omega )\|{ }_2^2\) is a special case of Theorem 7.8(i) (p. 177).
- 21.
The last equality is verified as follows. We consider a simple function \(\varphi (\lambda )=\displaystyle \sum _{i=1}^n \alpha _i \chi _{S_i}(\lambda )\quad (S_i\cap S_j=\emptyset \quad \text{if}\quad i\neq j)\).
$$\displaystyle \begin{aligned} \int_{\mathbb{R}}\varphi(\lambda)dE(\lambda)X(0,\omega)=\sum_{i=1}^n\alpha_i E(S_i)X(0,\omega) =\sum_{i=1}^n \alpha_i \xi(S_i,\omega)=\int_{\mathbb{R}} \varphi(\lambda)\xi(d\lambda,\omega). \end{aligned}$$The integration of e −iλt is a limit of such integrations of simple functions.
- 22.
- 23.
See Doob [4] Chap. II, §2.
- 24.
- 25.
In the case \(\mathbb {T}_0=\emptyset \) (and so α(λ) never vanishes), the discussion becomes much easier, since it is enough to define γ(S, ω) simply by
$$\displaystyle \begin{aligned} \gamma(S,\omega)=\int_S\frac{1}{\alpha(\lambda)}\xi(d\lambda,\omega) \end{aligned}$$for any \(S\in \mathbb {B}(\mathbb {T})\). Clearly, ν γ(S) = m(S).
- 26.$$\displaystyle \begin{aligned} \mathbb{E}_{(\omega,\omega')}&|\gamma'(S,\omega,\omega')|{}^2 \\ =&\; \mathbb{E}_\omega\Big|\int_S\alpha_1(\lambda)\xi(d\lambda,\omega)\Big|{}^2 \\ &+\mathbb{E}_{\omega'}\Big|\int_S\alpha_2(\lambda)\eta(d\lambda,\omega')\Big|{}^2 \\ &+2{\mathbb Re}\mathbb{E}_{(\omega,\omega')}\int_S\alpha_1(\lambda)\xi(d\lambda,\omega)1(\omega')\int_S \overline{\alpha_2(\lambda)\eta(d\lambda,\omega')1(\omega)} \\ =&\int_S|\alpha_1(\lambda)|{}^2\nu_\xi(d\lambda)+\int_S|\alpha_2(\lambda)|{}^2\nu_\eta(d\lambda) \\ &+2{\mathbb Re}\mathbb{E}_{(\omega,\omega')}\int_{S\cap\mathbb{T}_+}\alpha_1(\lambda)\xi(d\lambda,\omega) \int_{S\cap\mathbb{T}_0}\overline{\alpha_2(\lambda)\eta(d\lambda,\omega')} \\ =&\; m(S\cap\mathbb{T}_+)+m(S\cap\mathbb{T}_0)+0. \end{aligned} $$
- 27.$$\displaystyle \begin{aligned} \int_{\mathbb{T}}&e^{-in\lambda}\alpha(\lambda)\gamma(d\lambda,\omega) \\ &=\int_{\mathbb{T}_+}e^{-in\lambda}\alpha(\lambda)\cdot\frac{1}{\alpha(\lambda)}\xi(d\lambda,\omega) +\mathbb{E}_{\omega'}\int_{\mathbb{T}_0}e^{-in\lambda}\alpha(\lambda)\cdot\alpha_2(\lambda)\eta(d\lambda,\omega') \\ &=\int_{\mathbb{T}_+}e^{-in\lambda}\xi(d\lambda,\omega)=\int_{\mathbb{T}}e^{-in\lambda}\xi(d\lambda,\omega). \end{aligned} $$
The final equality is justified by
$$\displaystyle \begin{aligned} \mathbb{E}_\omega&\Big|\int_{\mathbb{T}_0}e^{-in\lambda}\xi(d\lambda,\omega)\Big|{}^2=\int_{\mathbb{T}_0}\nu_\xi(d\lambda)\quad \text{(by D-K formula)} \\ &=\int_{\mathbb{T}_0}p(\lambda)dm=0\quad (p(\lambda)=0\; \text{on}\; \mathbb{T}_0). \end{aligned} $$ - 28.
Since Z n(ω) is a white noise, the covariance is given by
$$\displaystyle \begin{aligned}\mathbb{E}Z_{n+u}(\omega)\overline{Z_n(\omega)}= \begin{cases} 1\quad \text{if} \quad u=0,\\ 0\quad \text{if} \quad u\neq0. \end{cases} \end{aligned}$$ - 29.
The convergence of \((1/\sqrt {2\pi })\displaystyle \sum _{k=-p}^qc_ke^{-ik\lambda }\) to C(λ) also holds good “almost everywhere” thanks to the Carleson theorem. Hence c k is the Fourier coefficient of C(λ) corresponding to \((1/\sqrt {2\pi })e^{-ik\lambda }\).
- 30.
The orthogonality, for instance, can be verified as follows. If S and \(S'\in \mathbb {B}(\mathbb {T})\) are disjoint,
$$\displaystyle \begin{aligned} \mathbb{E}\theta(S,\omega)\overline{\theta(S',\omega)} &=\mathbb{E}\int_{\mathbb{T}}C(\lambda)\chi_S(\lambda)\xi(d\lambda,\omega) \int_{\mathbb{T}}\overline{C(\lambda)\chi_{S'}(\lambda)\xi(d\lambda,\omega)} \\ &=\int_{\mathbb{T}}|C(\lambda)|{}^2\chi_S(\lambda)\chi_{S'}(\lambda)d\nu_\xi=0\quad (\text{by D-K formula}). \end{aligned} $$ - 31.
We can also establish
$$\displaystyle \begin{aligned} \frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}f(\lambda)g(\lambda)e^{-iz\lambda}dm(\lambda) =\frac{1}{\sqrt{2\pi}}(\mathbb{F}_2f\ast \mathbb{F}_2g)(z).\end{aligned}$$(cf. Kawata [15] pp. 282–283.)
- 32.
The third line of (8.56)
$$\displaystyle \begin{aligned} &=\frac{1}{\sqrt{2\pi}}\Big\{ \int_{\mathbb{R}_+} \alpha(\lambda-(t+u))\alpha(\lambda-t)dm(\lambda) +\int_{\mathbb{R}_0}\underbrace{\alpha(\lambda-(t+u))\alpha(\lambda-t)}_{(\dagger)}d\nu_\gamma\Big\} \\ &=\frac{1}{\sqrt{2\pi}}\Big\{\int_{\mathbb{R}}(\dagger)dm(\lambda)-\int_{\mathbb{R}_0}(\dagger)dm(\lambda) +\int_{\mathbb{R}_0}(\dagger)d\nu_\gamma\Big\}\\ &=\frac{1}{\sqrt{2\pi}}\Big\{ \int_{\mathbb{R}}(\dagger)dm(\lambda) -\int_{\begin{subarray}{l}\lambda \in \mathbb{R}_0 \\ \lambda-t\in \mathbb{R}_+ \end{subarray}} (\dagger)dm(\lambda) -\int_{\begin{subarray}{l}\lambda \in \mathbb{R}_0 \\ \lambda-t\in \mathbb{R}_0 \end{subarray}} (\dagger)dm(\lambda)\\ &\hspace{25mm}+\int_{\begin{subarray}{l}\lambda\in \mathbb{R}_0 \\ \lambda-t\in \mathbb{R}_+ \end{subarray}} (\dagger)d\nu_\gamma(\lambda) +\int_{\begin{subarray}{l}\lambda\in \mathbb{R}_0 \\ \lambda-t\in \mathbb{R}_0 \end{subarray}} (\dagger)d\nu_\gamma(\lambda)\Big\}\\ &=\frac{1}{\sqrt{2\pi}}\Big\{\int_{\mathbb{R}}(\dagger)dm(\lambda) -\int_{(\mathbb{R}_0-t)\cap\mathbb{R}_+} \alpha(\lambda'-u)\alpha(\lambda')dm(\lambda') \\ &\hspace{25mm} +\int_{(\mathbb{R}_0-t)\cap\mathbb{R}_+} \alpha(\lambda'-u) \alpha(\lambda')d\nu_\gamma(\lambda')\Big\}. \end{aligned} $$The last two terms cancel out. So we obtain (8.56).
- 33.
This section was added according to Professor S. Kusuoka’s suggestion. It is not included in the Japanese edition of this monograph.
- 34.
For backward operators, see Granger and Newbold [7] Chap. 1.
- 35.
Loève [19] p. 11.
- 36.
References
Brémaud, P.: Fourier Analysis and Stochastic Processes. Springer, Cham (2014)
Carleson, L.: On the convergence and growth of partial sums of fourier series. Acta Math. 116, 135–157 (1966)
Crum, M.M.: On positive definite functions. Proc. Lond. Math. Soc. 6, 548–560 (1956)
Doob, J.L.: Stochastic Processes. Wiley, New York (1953)
Dudley, R.M.: Real Analysis and Probability. Wadsworth and Brooks, Pacific Grove (1988)
Frisch, R.: Propagation problems and inpulse problems in dynamic economics. In: Economic Essays in Honor of Gustav Cassel. Allen and Unwin, London (1933)
Granger, C.W.J., Newbold., P.: Forecasting Economic Time Series, 2nd edn. Academic, Orlando (1986)
Hamilton, J.D.: Time Series Analysis. Princeton University Press, Princeton (1994)
Hicks, J.R.: A Contribution to the Theory of the Trade Cycle. Oxford University Press, London (1950)
Itô, K.: Kakuritsu-ron (Probability Theory). Iwanami Shoten, Tokyo (1953) (Originally published in Japanese)
Kaldor, N.: A model of the trade cycle. Econ. J. 50, 78–92 (1940)
Kawata, T.: Ohyo Sugaku Gairon (Elements of Applied Mathematics), I, II. Iwanami Shoten, Tokyo (1950, 1952) (Originally published in Japanese)
Kawata, T.: Fourier Henkan to Laplace Henkan (Fourier and Laplace Transforms). Iwanami Shoten, Tokyo (1957) (Originally published in Japanese)
Kawata, T.: On the Fourier series of a stationary stochastic process, I, II. Z. Wahrsch. Verw. Gebiete 6, 224–245 (1966); 13, 25–38 (1969)
Kawata, T.: Fourier Kaiseki (Fourier Analysis). Sangyo Tosho, Tokyo (1975) (Originally published in Japanese)
Kawata, T.: Fourier Kaiseki to Tokei (Fourier Analysis and Statistics). Kyoritsu Shuppan, Tokyo (1985) (Originally published in Japanese)
Kawata, T.: Teijyo Kakuritsu Katei (Stationary Stochastic Processes). Kyoritsu Shuppan, Tokyo (1985) (Originally published in Japanese)
Keynes, J.M.: The General Theory of Employment, Interest and Money. Macmillan, London (1936)
Loève, M.: Probability Theory, 3rd edn. van Nostrand, Princeton (1963)
Maruyama, G.: Harmonic Analysis of Stationary Stochastic Processes. Mem. Fac. Sci. Kyushu Univ. Ser. A, 4, 45–106 (1949)
Maruyama, T.: Suri-keizaigaku no Hoho (Methods in Mathematical Economics). Sobunsha, Tokyo (1995) (Originally published in Japanese)
Maruyama, T.: Shinko Keizai Genron (Principles of Economics), 3rd edn. Iwanami Shoten, Tokyo (2013) (Originally published in Japanese)
Maruyama, T.: Fourier Analysis of Periodic Weakly Stationary Processes. A Note on Slutsky’s Observation. Adv. Math. Eco. 20, 151–180 (2016)
Samuelson, P.A.: Interaction between the multiplier analysis and the principle of acceleration. Rev. Econ. Stud. 21, 75–78 (1939)
Samuelson, P.A.: A synthesis of the principle of acceleration and the multiplier. J. Polit. Econ. 47, 786–797 (1939)
Sargent, T.J.: Macroeconomic Theory. Academic, New York (1979)
Shinkai, Y.: Keizai-hendo no Riron (Theory of Economic Fluctuations). Iwanami Shoten, Tokyo (1967) (Originally published in Japanese)
Slutsky, E.: Alcune proposizioni sulla teoria delle funzioni aleatorie. Giorn. Inst. Ital. degli Attuari 8, 193–199 (1937)
Slutsky, E.: The summation of random causes as the source of cyclic processes. Econometrica 5, 105–146 (1937)
Slutsky, E.: Sur les fonctions aléatoires presque-periodiques et sur la décomposition des fonctions aléatoires stationnaires en composantes. Actualités Sci. Ind. 738, 38–55 (1938)
Wold, H.: A Study in the Analysis of Stationary Time Series, 2nd edn. Almquist and Wicksell, Uppsala (1953)
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Maruyama, T. (2018). Fourier Analysis of Periodic Weakly Stationary Processes. In: Fourier Analysis of Economic Phenomena. Monographs in Mathematical Economics, vol 2. Springer, Singapore. https://doi.org/10.1007/978-981-13-2730-8_8
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