# what characterizes a characteristic function of a probability measure in separable Hilbert spaces?

As we all know on real line $\mathbb{R}$, the following is valid

A $\mathbb{C}$-valued function $\varphi$ is a characteristic function of a probability measure on $\mathbb{R}$ if and only if $\varphi$ is a continuous, positive definite function such that $\varphi(0)=1$.

But on separable Hilbert space, Theorem 2.13 in Stochastic Equations in infinite Dimensions (by Da Prato) says that

A $\mathbb{C}$-valued function on $H$ $\varphi$ is a characteristic function of a probability measure on $H$ if and only if

(i) $\varphi$ is a continuous, positive definite function such that $\varphi(0)=1$.

(ii) $\forall \varepsilon >0,$ $\exists$ a nonnegative nuclear operator $S$ such that $$1-\mathrm{Re}\ \varphi(\lambda) \leq \varepsilon, \mathrm{for\ all}\ \lambda \text{ satisfying }\langle S\lambda,\lambda \rangle \leq1$$

What confused me is that why (ii) condition is added in the infinite case? Is there any intuitive or intelligible explanation for it?

• alternatively if you want the theorems like Bochner and Levy continuity to read exactly like in finitely-many dimensions, ie., without extra hypotheses like (ii), then you need to abandon Hilbert and Banach spaces and work with duals of nuclear spaces, like $S'(\mathbb{R}^d)$ or $\mathbb{R}^{\mathbb{N}}$. – Abdelmalek Abdesselam Dec 8 '14 at 20:26

Define $\phi(x):=\exp(-\lVert x\rVert^2_H)$: it satisfies (i). Now we shall see that (ii) does not hold with $\varepsilon:=1/2$. Suppose that there exists a non-negative nuclear operator $S$ such that if $\langle Sx,x\rangle\leqslant 1$, then $\phi(x)\geqslant 1/2$. Notice that $S$ has the form $$S(x)=\sum_{j=1}^{+\infty}\lambda_j\langle x,a_j\rangle b_j$$ where $\lambda_j\geqslant 0$, $\sum_{j=1}^{+\infty}\lambda_j<+\infty$, $a_j,b_j\in H$ and $M:=\sup_j\lVert a_j\rVert+\lVert b_j\rVert$ is finite. Fix an integer $N\geqslant 1$. If $x$ belongs to the orthogonal of the vector space generated by $a_j,j\leqslant N$, then the condition $\langle Sx,x\rangle$ will be satisfied once $M^2\lVert x\rVert^2\sum_{j\geqslant N}\lambda_j\leqslant 1$. If we pick $x$ such that the equality holds, then we should have $$\exp\left(-\frac 1{M^2\sum_{j\geqslant N}\lambda_j}\right)\geqslant \frac 12.$$ If we choose $N$ large enough we get a contradiction.
The problem is that for infinite dimensional spaces $\langle Sx,x\rangle$ may be smaller than $1$ for $x$ having a large norm.
• To slightly expand on Davide's answer: you can embed $H$ into a larger Hilbert space $H'$ such that, after making the natural identifications, $\phi$ is the characteristic function of a probability measure on $H'$. The corresponding random variable is of the form $\sum_n e_n \xi_n$ with $\xi_n$ an iid $N(0,1)$ sequence and $e_n$ an o.n.b. of $H$. This of course doesn't converge in $H$ (all coefficients are of about the same size). – Martin Hairer May 1 '14 at 15:22