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Letand estimate it by the average of the two within-sample empirical covariance operators. Let $({\hat{\lambda }}_k,{\hat{\varphi }}_k)$ be its leading empirical eigenpairs and let ${\overline{X}}_n,{\overline{X}}_n^{\ast }$ be the sample means. Use 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 $\overline{C}$. Distinct eigenvalues give consistent empirical eigenpairs, up to signs, and the standardized leading scores are independent standard normal variables. HenceRejecting above the $(1-\alpha )$ quantile gives an asymptotic level-$\alpha$ test.

Under a fixed alternative $\delta =\mu -{\mu }^{\ast }$,The test is consistent whenever one retained projection is nonzero. Alternatives orthogonal to the first $K$ eigenfunctions are invisible at fixed $K$. Under local alternatives $\delta =h/\sqrt{n}$, the limit is noncentral chi-squared with noncentrality $\frac{1}{2}{\sum }_{k\le K}⟨h,{\varphi }_k{⟩}^2/{\lambda }_k$.

Write $x\otimes y$ for the rank-one operator $h\mapsto ⟨h,y⟩x$. The estimator is the kernel representation of the empirical covariance operatorSince $\mathbb{E}X=0$,Thus the estimator has the fixed bias of an estimator ${\mu }_0\otimes {\mu }_0$ for $C_X$.

The fourth-moment assumption makes $(X_i-{\mu }_0)\otimes (X_i-{\mu }_0)$ square-integrable in the Hilbert space of Hilbert-Schmidt operators. The weak law of large numbers therefore givesConsequently consistency for $C_X$ holds exactly when ${\mu }_0=0$. More precisely, if $Y=(X-{\mu }_0)\otimes (X-{\mu }_0)$, thenThe Hilbert-space central limit theorem also yieldswhere $G$ is a centered Gaussian random element in the Hilbert-Schmidt operator space with covariance determined by $Y$. Relative to $C_X$, the same fluctuation is displaced by $\sqrt{n}({\mu }_0\otimes {\mu }_0)$ and hence does not have a finite centered limit when ${\mu }_0\ne 0$.