Hilbert-space-valued random variable

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

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

When a covariance operator on a function space is an integral operator, its covariance kernel satisfies $(C_{X}f)(s)=\int c_X(s,t)f(t)dt$. For a centered process, $c_X(s,t)=\mathbb{E}[X(s)X(t)]$ whenever point evaluation is meaningful.

The covariance kernel of standard Brownian motion on $[0,1]$ is $c(s,t)=\min (s,t)$. Its integral operator has eigenfunctions $\sqrt{2}\sin ((k-\frac{1}{2})\pi t)$ and eigenvalues $((k-\frac{1}{2})\pi {)}^{-2}$.

For observations $X_1,\dots ,X_n$ and a chosen center $m$, the empirical covariance operator is ${\hat{C}}_m=n^{-1}{\sum }_i(X_i-m)\otimes (X_i-m)$. Its expectation equals the population covariance plus the rank-one operator formed from the centering error.

A random element $G$ of a Hilbert space is Gaussian when every continuous linear functional of $G$ has a normal distribution. Its mean and covariance operator determine its distribution.