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For a square-integrable centered Hilbert-space-valued random variable $X$, its covariance operator isIt is a positive self-adjoint trace-class operator, and $\mathrm{tr}{C}_X=\mathbb{E}\Vert X{\Vert }^2$.

For Hilbert-space-valued random variables $X_1,\dots ,X_n$ and a fixed deterministic centre $m$, the empirical covariance operator is ${\hat{C}}_m=n^{-1}{\sum }_i(X_i-m)\otimes (X_i-m)$. If the observations share mean $\mu$ and covariance $C$, then $\mathbb{E}{\hat{C}}_m=C+(\mu -m)\otimes (\mu -m)$. This follows by expanding each centred vector and cancelling the mean-zero cross terms. A fitted random centre changes that expectation: for an IID sample centred at its sample mean, $\mathbb{E}{\hat{C}}_{\overline{X}}=(n-1)C/n$. Multiplying by $n/(n-1)$ gives the usual unbiased estimate when $n>1$.

A functional mean test asks whether the expectation $\mu$ of a Hilbert-space-valued random variable is zero. Suppose $X_1,\dots ,X_n$ are independent and identically distributed in a Hilbert space with a countable dense subset $H$, with $\mathbb{E}\Vert X_1{\Vert }^2<\infty$. Write $C$ for their covariance operator and $\overline{X}=n^{-1}{\sum }_i{X}_i$. Under the null hypothesis $\mu =0$, the Hilbert-space central limit theorem giveswhere $G$ is a centered Gaussian random element in $H$, the ${\lambda }_j$ are the eigenvalues of $C$, and the $Z_j$ are independent standard normal random variables. The second assertion follows by the continuous mapping theorem. Since ${\sum }_j{\lambda }_j=\mathbb{E}\Vert X_1-\mu {\Vert }^2<\infty$, the nonnegative series is finite almost surely. Limits given by Gaussian random elements for functional sample means and the corresponding weighted squared-norm laws also extend to suitable dependent observations.^[ref]

The eigenvalues determine the calibration: the limit is generally a weighted sum of random variables with a chi-squared distribution, rather than a single ordinary chi-squared distribution. The squared norm measures deviations across all covariance operator directions. An FPCA mean test instead retains finitely many eigenvectors and divides their squared coordinates by their eigenvalues, producing a different statistic with a simpler limiting calibration.

A Hilbert-space-valued random variable is a measurable map into a Hilbert space, equipped with its Borel sigma-algebra. For a separable real Hilbert space, finite $\mathbb{E}\Vert X\Vert$ defines its mean by the Bochner integral; finite $\mathbb{E}\Vert X{\Vert }^2$ also defines a covariance operator. A random square-integrable function on $[0,1]$, viewed as an element of L2 space, is an example. Its values are whole equivalence classes of functions, so point evaluation is not automatically meaningful. In a nonseparable space, strong measurability or an appropriate separable-valued hypothesis must be checked before using the same integral construction.

Let $X_1,\dots ,X_n$ and $Y_1,\dots ,Y_n$ be two independent samples of independent and identically distributed Hilbert-space-valued random variables, with finite second moments and covariance operators $C_X,C_Y$. To test equality of their expectations, use the pooled covariance operator $C=(C_X+C_Y)/2$. This choice works even when the two population covariance operators differ: under the null hypothesis, the covariance operator of $\sqrt{n/2}(\overline{X}-\overline{Y})$ is exactly $C$.

Let $({\hat{\lambda }}_j,{\hat{\varphi }}_j)$ estimate the eigenvalues and orthonormal eigenvectors of $C$, for example from the average of the two within-sample empirical covariance operators. For fixed $K$, positive retained eigenvalues, a separated retained eigenspace and suitable statistical consistency of these estimates,The proof applies the Hilbert-space central limit theorem to the difference of sample means, then standardizes its first $K$ coordinates and uses Slutsky's theorem. A sufficient simple-eigenvalue condition is ${\lambda }_1>\cdots >{\lambda }_K>{\lambda }_{K+1}$ with ${\lambda }_K>0$.^[ref]

For unequal sample sizes $n,m$, the scaling becomes $nm/(n+m)$ and the relevant population covariance operator is $\{m{C}_X+n{C}_Y\}/(n+m)$. Thus a sample-size-weighted pooled estimate cannot automatically be reused for unequal sizes and unequal covariance operators: its weights must match the variance of the difference being tested.