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Frequency domain theory for functional time series: variance decomposition and an invariance principle. (English) Zbl 1441.62931

This paper is a valuable contribution to the theory of inference based on the frequency domain of time series with range in a separable Hilbert space. Some good, extensive and possibly necessary reading prior to the study of this paper can be seen in the book [L. Horváth and P. Kokoszka, Inference for functional data with applications. New York, NY: Springer (2012; Zbl 1279.62017)] one of whose authors is co-author of this paper. From this book a brief introduction to this paper is extracted. Be \(H\) a separable Hilbert space, \(\langle\cdot,\cdot\rangle\) its inner product and \(\Vert\cdot\Vert\) the norm defined by it. Let’s denote by \(L\) the set of bounded linear operators on \(H\). It is said that \(T\in L\) is a compact operator if there are \((v_{j})_{j\geq 1}\) and \((f_{j})_{j\geq 1}\) orthonormal bases of H and a sequence \((\lambda_{j})_{j\geq 1}\) of real numbers such that \(\lambda_{j}\to 0\), and \[ T(x)=\Sigma_{j\geq 1}\lambda_{j}\langle x,v_{j}\rangle f_{j}. \tag{1} \] If \(\Sigma_{j\geq1}\lambda_{j}^2<\infty\), it is said that \(T\) is a Hilbert-Schmidt operator. If \(\Sigma_{j\geq 1}\lambda_{j}<\infty\), it is said that \(T\) is a nuclear operator. Let us denote by \(S\) the space of the Hilbert-Schmidt operators, and by \(N\) that of the nuclear operators. So \(N\subset S\subset L\). Be \(T\) as in (1). If \(T\in N\) is defined \[ \Vert T\Vert_{N}=\operatorname{tr}((T^{*}T)^{(1/2)})=\Sigma_{j\geq1} \langle v_{j},(T^{*}T)^{(1/2)}(v_{j})\rangle \] where \(T^{*}\) is the adjoint operator of \(T\), i.e. \[ \langle T(x),y\rangle = \langle x,T^{*}(y)\rangle,\quad x,y\text{ in }H, \] and \(\operatorname{tr}((T^{*}T)^{(1/2)})\) is the trace of \((T^{*}T)^{(1/2)}\). When \(T\in S\) is defined \[ \Vert T\Vert_{S}^2=\operatorname{tr}(T^{*}T). \] If \(T\in L\) is defined \[ \Vert T\Vert_{L}=\sup_{\Vert x\Vert\leq1}\Vert T(x)\Vert\,. \] It can be proved \(\Vert T\Vert_{L}\), \(\Vert T\Vert_{S}\), and \(\Vert T\Vert_{N}\) are norms on the respective spaces and \[ \Vert T\Vert_{L}\leq\Vert T\Vert_{S}\leq\Vert T\Vert_{N}. \] Hereafter \(H=L^2([0,1])\) is considered, with the Borel’s \(\sigma\)-field and the Lebesgue measure or any measure \(\mu\) with \(\mu([0,1])=1\). An operator \(K\) on \(S\) is said to be a integral operator if there is \(k\in L^2([0,1]\times[0,1])\) (kernel of \(K\)) such that \[ K(f)(t)=\int k(t,s)f(s)ds, \] and \[ \iint |k(t,s)|^2dt\cdot ds=\Vert K\Vert_{S}^2 \] Let \((\omega,F,P)\) be a probability space. For every \(p\geq 1\) be \[ L_{H}^{p}(\omega,F,P):={X:\omega\to H / \mu_{p}(X)=(\int_{\omega}\Vert X(\omega)\Vert^{p}\cdot P(d\omega))^{(1/p)}<\infty}. \] For every \(t\in\mathbb{Z}\) be \(X_{t}\in L_{H}^2(\omega,F,P)\), and \((X_{t})_{t\in\mathbb{Z}}\) is assumed to be a weakly stationary process with \(E(X_{t})=\int_{\omega}X_{t}(\omega)\cdot P(d\omega)=0\), the null point of \(H\). For every \(h\in\mathbb{Z}\) be \(c_{h}\in L^2([0,1]\times [0,1])\) the \(h\)-step autocovariance function of \((X_{t})_{t\in\mathbb{Z}}\), that is \[ c_{h}(u,v)=\int_{\omega}X_{t+h}(\omega,u)X_{t}(u)\cdot P(d\omega). \] If \[ \Sigma_{h\in\mathbb{Z}}\Vert c_{h}\Vert_2<\infty\tag{2} \] where \(\Vert\cdot\Vert_2\) is the norm of \(L^2([0,1]\times[0,1])\), then for every \(\theta\in(-\pi,\pi]\) is defined the function \(f_{\theta}\) on \([0,1]\times [0,1]\) by \[ f_{\theta}(u,v)=(1/(2\pi))\Sigma_{h\in\mathbb{Z}}c_{h}(u,v)\cdot\exp(-ih\theta), \] and the operator \(F_{\theta}\) on \(H\) by \[ F_{\theta}(f)(u)=\int f_{\theta}(u,v)\cdot f(v)\cdot dv. \] \(f_{\theta}\) and \(F_{\Theta}\) are called spectral density kernel and spectral density operator at frequency \(\theta\), respectively. In this paper, under the assumption that (2) is valid, \((f_{\theta})_{\theta\in\Theta}\) and \((F_{\theta})_{\theta\in\Theta}\) estimators are defined and their asymptotic properties are studied. For the authors, the main contribution of this paper is the discussion of an alternative to Karhunen-Loéve decomposition for a stochastic process such as \((X_{t})_{t\in\mathbb{Z}}\), called dynamic functional principal components (DFPCs) decomposition. This in turn allows them to define new \((F_{\theta})_{\theta\in\Theta}\) estimators. The subject studied in this paper requires good knowledge of Functional Analysis in Hilbert’s spaces. Some books from the extensive bibliography of the reference list can help to achieve these requirements. Unfortunately the authors do not prove in this paper the results that they enunciate. The proofs are in a supplementary article that must be purchased. The authors should make this material available free of charge to the readers of this paper.

MSC:

62R10 Functional data analysis
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M15 Inference from stochastic processes and spectral analysis
62H25 Factor analysis and principal components; correspondence analysis

Citations:

Zbl 1279.62017

Software:

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References:

[1] Anderson, T.W. (1971). The Statistical Analysis of Time Series. New York: Wiley. · Zbl 0225.62108
[2] Aue, A. and van Delft, A. (2020). Testing for stationarity of functional time series in the frequency domain. Ann. Statist. To appear. · Zbl 1455.62230
[3] Bardsley, P., Horváth, L., Kokoszka, P. and Young, G. (2017). Change point tests in functional factor models with application to yield curves. Econom. J. 20 86-117. · Zbl 1521.62141
[4] Berkes, I., Hörmann, S. and Schauer, J. (2011). Split invariance principles for stationary processes. Ann. Probab. 39 2441-2473. · Zbl 1236.60037
[5] Berkes, I., Horváth, L. and Rice, G. (2013). Weak invariance principles for sums of dependent random functions. Stochastic Process. Appl. 123 385-403. · Zbl 1269.60040
[6] Billingsley, P. (1968). Convergence of Probability Measures. New York: Wiley. · Zbl 0172.21201
[7] Bosq, D. (2000). Linear Processes in Function Spaces: Theory and Applications. Lecture Notes in Statistics 149. New York: Springer. · Zbl 0962.60004
[8] Brillinger, D.R. (1975). Time Series: Data Analysis and Theory. International Series in Decision Processes. New York: Holt, Rinehart and Winston, Inc. · Zbl 0321.62004
[9] Brockwell, P.J. and Davis, R.A. (1991). Time Series: Theory and Methods. Springer Series in Statistics. New York: Springer. · Zbl 0709.62080
[10] Dauxois, J., Pousse, A. and Romain, Y. (1982). Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference. J. Multivariate Anal. 12 136-154. · Zbl 0539.62064
[11] Garling, D.J.H. (1976). Functional central limit theorems in Banach spaces. Ann. Probab. 4 600-611. · Zbl 0343.60014
[12] Giraitis, L., Kokoszka, P., Leipus, R. and Teyssière, G. (2003). Rescaled variance and related tests for long memory in volatility and levels. J. Econometrics 112 265-294. · Zbl 1027.62064
[13] Górecki, T., Hörmann, S., Horváth, L. and Kokoszka, P. (2018). Testing normality of functional time series. J. Time Series Anal. 39 471-487. · Zbl 1416.62489
[14] Hall, P. and Hosseini-Nasab, M. (2006). On properties of functional principal components analysis. J. R. Stat. Soc. Ser. B. Stat. Methodol. 68 109-126. · Zbl 1141.62048
[15] Hörmann, S., Kidziński, Ł. and Hallin, M. (2015). Dynamic functional principal components. J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 319-348. · Zbl 1414.62133
[16] Hörmann, S., Kidziński, Ł. and Kokoszka, P. (2015). Estimation in functional lagged regression. J. Time Series Anal. 36 541-561. · Zbl 1325.62168
[17] Hörmann, S. and Kokoszka, P. (2010). Weakly dependent functional data. Ann. Statist. 38 1845-1884. · Zbl 1189.62141
[18] Hörmann, S. and Kokoszka, P. (2012). Functional time series. In Time Series (C.R. Rao and T.S. Rao, eds.). Handbook of Statistics 30. Amsterdam: Elsevier.
[19] Hörmann, S., Kokoszka, P. and Nisol, G. (2018). Testing for periodicity in functional time series. Ann. Statist. 46 2960-2984. · Zbl 1416.62496
[20] Horváth, L. and Kokoszka, P. (2012). Inference for Functional Data with Applications. Springer Series in Statistics. New York: Springer. · Zbl 1279.62017
[21] Horváth, L., Kokoszka, P. and Rice, G. (2014). Testing stationarity of functional time series. J. Econometrics 179 66-82. · Zbl 1293.62186
[22] Hsing, T. and Eubank, R. (2015). Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators. Wiley Series in Probability and Statistics. Chichester: Wiley. · Zbl 1338.62009
[23] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Probability and Its Applications (New York). New York: Springer. · Zbl 0996.60001
[24] Kidziński, Ł., Kokoszka, P. and Mohammadi Jouzdani, N. (2018). Principal components analysis of periodically correlated functional time series. J. Time Series Anal. 39 502-522. · Zbl 1416.62503
[25] Kokoszka, P. and Mohammadi Jouzdani, N. (2020). Supplement to “Frequency domain theory for functional time series: Variance decomposition and an invariance principle.” https://doi.org/10.3150/20-BEJ1199SUPP · Zbl 1441.62931
[26] Kokoszka, P. and Reimherr, M. (2013). Asymptotic normality of the principal components of functional time series. Stochastic Process. Appl. 123 1546-1562. · Zbl 1275.62066
[27] Kokoszka, P. and Reimherr, M. (2017). Introduction to Functional Data Analysis. Texts in Statistical Science Series. Boca Raton, FL: CRC Press. · Zbl 1411.62004
[28] Kuelbs, J. (1973). The invariance principle for Banach space valued random variables. J. Multivariate Anal. 3 161-172. · Zbl 0258.60009
[29] Leucht, A., Paparoditis, E. and Sapatinas, T. (2018). Testing equality of spectral density operators for functional linear processes. Preprint, Technische Universität Braunschweig. Available at arXiv:1804.03366. · Zbl 1520.62115
[30] Linde, W. (1986). Probability in Banach Spaces—Stable and Infinitely Divisible Distributions, 2nd ed. Chichester: Wiley. · Zbl 0665.60005
[31] Merlevède, F. (2003). On the central limit theorem and its weak invariance principle for strongly mixing sequences with values in a Hilbert space via martingale approximation. J. Theoret. Probab. 16 625-653. · Zbl 1038.60029
[32] Merlevède, F., Peligrad, M. and Utev, S. (1997). Sharp conditions for the CLT of linear processes in a Hilbert space. J. Theoret. Probab. 10 681-693. · Zbl 0885.60015
[33] Panaretos, V.M. and Tavakoli, S. (2013). Cramér-Karhunen-Loève representation and harmonic principal component analysis of functional time series. Stochastic Process. Appl. 123 2779-2807. · Zbl 1285.62109
[34] Panaretos, V.M. and Tavakoli, S. (2013). Fourier analysis of stationary time series in function space. Ann. Statist. 41 568-603. · Zbl 1267.62094
[35] Pham, T. and Panaretos, V.M. (2018). Methodology and convergence rates for functional time series regression. Statist. Sinica 28 2521-2539. · Zbl 1406.62098
[36] Pourahmadi, M. (2001). Foundations of Time Series Analysis and Prediction Theory. Wiley Series in Probability and Statistics: Applied Probability and Statistics. New York: Wiley. · Zbl 0982.62074
[37] Shao, X. and Wu, W.B. (2007). Asymptotic spectral theory for nonlinear time series. Ann. Statist. 35 1773-1801. · Zbl 1147.62076
[38] van Delft, A. and Eichler, M. (2020). A note on Herglotz’s theorem for time series on function spaces. Stochastic Process. Appl. To appear. · Zbl 1462.60041
[39] Zhang, X. (2016). White noise testing and model diagnostic checking for functional time series. J. Econometrics 194 76-95. · Zbl 1431.62429
[40] Zhou, Z. (2013). Heteroscedasticity and autocorrelation robust structural change detection. J. Amer. Statist. Assoc. 108 726-740. · Zbl 06195974
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