Asymptotic normality of the principal components of functional time series
Let \(\{X_n,\,n= 1,2,\dots\}\) be a strictly stationary sequence of random functions in the space \(L^2({\mathcal T})\) of square integrable functions on a compact interval \({\mathcal T}\). For the covariance operator \[ C(x)= E[\langle(X- EX),\;x\rangle(X - EX)] \] let \(v_k\), \(k\geq 1\), be the functional principle components (FPC's) which are the eigenfunctions of \(C\), i.e., \(C(v_k)= \lambda_kv_k\), \(k\geq 1\). As the estimator of \(v_k\), it is used the empirical FPC's \(\widehat v_k\) defined as the eigenfunctions of the empirical covariance operator \[ \widehat C(x)= N^{-1} \sum^N_{n=1} (\langle(X_n-\overline X_N), x\rangle(X_n-\overline X_N)],\quad x\in L^2({\mathcal T}), \] where \(\overline X_N\) is the usual sample average. The following asumptions are made: (i) \(E\| X_1\|^4< \infty\) and the first \(p\) eigenvalues of the covariance operator \(C\) are distinct and ordered as \(\lambda_1> \lambda_2>\cdots> \lambda_p\geq \lambda_{p+1},\dots\), (ii) there is a mean zero Gaussian Hilbert-Schmidt operator such that \(Z_N= N^{{1\over 2}}(\widehat C- C)\to^{{\mathcal L}}Z\). For any \(x\) and \(y\) in \(L^2({\mathcal T})\) \(x\otimes y\) is defined to be the integral operator with kernel \(x(t)y(s)\). For \(j= 1,\dots,p\), define \[ T_{j,N}= \sum_{k\neq j} {(Z_N,\,v_j\otimes v_k)v_k\over \lambda_j- \lambda_k}\quad\text{and}\quad T_j= \sum_{k\neq j} {(Z,\,v_j\otimes v_k)v_k\over \lambda_j-\lambda_k}. \] The authors prove, among others, that, under assumptions (i) and (ii), \(T_{j,N}\to^{{\mathcal L}}T_j\), \(N^{{1\over 2}}(\widehat v_j- v_j)\to^{{\mathcal L}}T_j\), each \(\widehat v_j\) is symptotically Gaussian and the \(\widehat v_j\), \(1\leq j\leq p\) are jointly asymptotically Gaussian. As an application of the result, using the \(L^p\)-\(m\)-approximability, one of notions of weak dependence, they show that assumption (ii) holds for every \(L^4\)-\(m\)-approximable sequence in \(L^2\) and that there exist normal elements \(T_j\), \(1\leq j\leq p\), of \(L^2({\mathcal T})\) such that \(N^{{1\over 2}} (\widehat v_j- v_j)\), \(1\leq j\leq p\}\to^{{\mathcal L}}\{T_j\), \(1\leq j\leq p\}\).
scientific article; zbMATH DE number 6156569