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Let $H$ be a real separable Hilbert space. Is it true that there exist a probability space $(\Omega, \mu)$ and a measurable function $\pi\colon \Omega \to H$ such that for any $h \in H$ we have $$ e^{-\lVert h\rVert^2}=\int_\Omega e^{i\langle h,\pi(\omega)\rangle}d\mu(\omega) \ \ ? $$

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    $\begingroup$ Sorry, I deleted my answer since it was incorrect (the sum I defined was not necessarily convergent). The measure you are searching is surely gaussian (you can think in R^N with the multivariate normal measure $N(0, Id)$ and arrive easily to your result). Anyways check again arxiv.org/abs/1003.1649 and the original paper by Ito and Nisio ir.library.osaka-u.ac.jp/metadb/up/LIBOJMK01/ojm05_01_02.pdf $\endgroup$
    – Bunder
    Commented Mar 31, 2013 at 9:51

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In Araujo and Giné book The central limit theorem in Banach spaces, there is an exercise which consists in showing that the map $\phi\colon x\mapsto \exp\left(-\frac{\lVert x\rVert^2}2\right)$ from $H$ to $\Bbb R$ is positive definite (in sense of the statement of Bochner theorem) but it's not the characteristic functional of a random variable.

Indeed, considering $(\eta_j,j\geqslant 1)$ a sequence of i.i.d. Gaussian random variables and $X:=\sum_{j\geqslant 1}\eta_je_j$, where $(e_j,j\geqslant 1)$ is an orthonormal basis of $H$, we have, by uniqueness theorem in separable Hilbert spaces that $\phi$ would be the characteristic function of $X$. However, the sequence $\left(\sum_{j=1}^n\eta_je_j,n\geqslant 1\right)$ is not tight.

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