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.


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


Zbl 1279.62017


Full Text: DOI Euclid


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