When are Hilbert space valued random variables independent?

Question

I am studying probability theory in infinite dimensional spaces and want to know if things that hold in the usual(?) theory holds in a similar manner.

Fix a real separable Hilbert space $H$ with inner product $\langle\cdot,\cdot\rangle$. Let $X$ be a $H$-valued Gaussian random variable on a probability space$(\Omega,\mathscr{F},P)$ such that $E\langle X,h\rangle\langle X,h'\rangle=\langle Qh,h'\rangle$ (e.g. a $Q$-Wiener process at a fixed time) with a trace class covariance operator $Q\colon H\to H$.

The usual probability theory tells us that for a nice function $f\colon\mathbb{R}\to\mathbb{R}$ we have that $f(\langle X,h\rangle)$ and $f(\langle X,h'\rangle)$ are independent. Also that if $f$ is linear (or affine) $f(\langle X,h\rangle)$ is again Gaussian, but not in general, e.g., $f(\cdot)=|\cdot|$.

My question is,

For nice $T\colon H\to H$ not necessarily linear, can we hope $ \langle TX,h\rangle $ and $ \langle TX,h'\rangle $ are independent, or uncorrelated? When $h,h'$ are eigenfunctions of $Q$ such that they are orthogonal, and $T$ is linear diagonalized by $(h)$, then it should be true, but I have no idea how to prove or disprove the above in general. I wanted to go back to the good old definition of independence of random variables (independence of $\sigma$-algebra generated by them), but didn't seem it would help.

Answer 1

If

• $(\Omega,\mathcal A)$, $(\Omega',\mathcal A')$ and $(\Omega'',\mathcal A'')$ are measurable spaces
• $X:\Omega\to\Omega'$ is measurable with respect to $\mathcal A$-$\mathcal A'$
• $f:\Omega'\to\Omega''$ is measurable with respect to $\mathcal A'$-$\mathcal A''$

then $f\circ X$ is measurable with respect to $\sigma(X)$-$\mathcal A''$ and hence $$\sigma(f\circ X)\subseteq\sigma (X)\tag1\;.$$ Thus, if $\mathcal F\subseteq\mathcal A$ is a $\sigma$-algebra on $\Omega$ and $X$ is independent of $\mathcal F$, then $f\circ X$ is independent of $\mathcal F$ too.

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If

• $H:=\Omega'$ is a separable $\mathbb R$-Hilbert space, $\mathcal A':=\mathcal B(H)$ and $(e_n)_{n\in\mathbb N}$ is an orthonormal basis of $H$
• $E$ is a $\mathbb R$-Banach space, $\Omega''=\mathfrak L(H,E)$ and $\mathcal A''$ is the strong Borel $\sigma$-algebra on $\Omega''$

then $$f(X)=\sum_{n\in\mathbb N}\langle X,e_n\rangle_Uf(e_n)$$ and hence (since $f(e_n)$ is measurable with respect to $\mathcal A'$-$\mathcal B(E)$), $f(X)$ is indepdent of $\mathcal F$ as long as $X$ is independent of $\mathcal F$.