{title2=$C_X$}
{wiki}

For a square-integrable centered <Hilbert-space-valued random variable> $X$, its covariance operator is
$$
C_Xh=\mathbb E[\langle X,h\rangle X]=\mathbb E[X\otimes X]h.
$$
It is a positive self-adjoint <trace-class operator>, and $\operatorname{tr}C_X=\mathbb E\lVert X\rVert^2$.


Covariance operator

A functional mean test tests whether the mean of a <Hilbert-space-valued random variable> is zero. The squared-norm statistic $n\lVert\overline X\rVert^2$ has a weighted chi-squared limit under the null, while an FPCA test standardizes and truncates the coordinates.


Functional mean test

{c}

A Hilbert-space-valued random variable is a measurable map into a <Hilbert space>. It is square-integrable when $\mathbb E\lVert X\rVert^2<\infty$, in which case its mean is defined by the <Bochner integral>.

= Random element of a Hilbert space
{synonym}


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 Incoming links: Hilbert-space-valued random variable