 Solution

Let $({\lambda }_j,{\varphi }_j)$ be the ordered eigenpairs of the common covariance operator $C_X$, and write $\delta =\mu -{\mu }^{\ast }$. Estimate $C_X$ by the pooled covariance operatorand denote its first $K$ eigenpairs by $({\hat{\lambda }}_j,{\hat{\varphi }}_j)$. The sign ambiguity of each eigenfunction disappears after squaring. Consider the Two-sample FPCA mean statistic

Under $H_0$, the Hilbert-space central limit theorem giveswhere $G$ is a centered Gaussian random element with covariance $C_X$. The assumed eigenvalue gaps give consistency of the estimated eigenvalues and eigenfunctions, so Slutsky's theorem yieldsAn asymptotic level-$\alpha$ test therefore rejects when $T_{n,K}$ exceeds the $(1-\alpha )$-quantile of the chi-squared distribution with $K$ degrees of freedom.

Under a fixed alternative,The test is consequently consistent whenever the mean difference has a nonzero projection onto one of the retained principal components. A difference orthogonal to their span is invisible to this fixed-$K$ test, so $\mu \ne {\mu }^{\ast }$ alone does not guarantee consistency. Increasing $K$ with $n$ can recover such alternatives, but requires additional eigenvalue and approximation control.