Let
$$
C_X\phi_k=\lambda_k\phi_k,
\qquad
C_Y\psi_l=\gamma_l\psi_l,
$$
and define the <functional principal component scores>
$$
\xi_{ik}=\langle X_i,\phi_k\rangle,
\qquad
\eta_{il}=\langle Y_i,\psi_l\rangle.
$$
Use empirical centered scores $\widehat\xi_{ik},\widehat\eta_{il}$ from the sample covariance eigenfunctions and form
$$
\widehat c_{kl}
=\frac1n\sum_{i=1}^n\widehat\xi_{ik}\widehat\eta_{il}.
$$
The finite-dimensional statistic is
$$
T_{K,L}
=n\sum_{k=1}^K\sum_{l=1}^L
\frac{\widehat c_{kl}^2}{\widehat\lambda_k\widehat\gamma_l}.
$$

Under $H_0$, $Y=\varepsilon$ is independent of $X$. The population cross-covariances vanish, and
$$
\operatorname{Cov}(\xi_k\eta_l,\xi_{k'}\eta_{l'})
=\lambda_k\gamma_l
\mathbf1_{\{k=k'\}}\mathbf1_{\{l=l'\}}.
$$
The multivariate <central limit theorem>, consistency of the empirical eigenpairs, and <Slutsky theorem> therefore give
$$
T_{K,L}\xrightarrow d\chi_{KL}^2.
$$
Rejecting above the $(1-\alpha)$ quantile gives an asymptotic level-$\alpha$ test.

Under a fixed alternative, $\mathbb E[\eta_l\xi_k]$ is the $(l,k)$ coordinate of the <cross-covariance operator> $C_{YX}=BC_X$, where $B$ is the <Hilbert-Schmidt operator> with kernel $\beta$. Thus
$$
\frac{T_{K,L}}n\xrightarrow p
\sum_{k=1}^K\sum_{l=1}^L
\frac{\{\mathbb E(\xi_k\eta_l)\}^2}{\lambda_k\gamma_l}.
$$
The test is consistent whenever this retained block contains a nonzero cross-covariance; fixed truncation can miss alternatives outside the selected principal-component subspaces.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 225 / 4 / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-225/4/solution