Generalization of Lévy's continuity theorem for nuclear spaces

Question

I am interested in a generalization of the following finite-dimensional results in infinite dimensional vector-space with nuclear structure, especially for the cases of the spaces of distributions $\mathcal{D}'(\mathrm{R}^N)$ and $\mathcal{S}'(\mathrm{R}^N)$.

Theorem 1: Let $X_n$, $n\in\mathbb{N}$ and $X$ be random variables in $\mathbb{R}^N$ and $\Phi_n$ and $\Phi$ their characteristic functions. Then, $$\left( X_n \overset{\mathcal{L}}{\rightarrow} X \right) \Leftrightarrow \left( \forall \omega, \Phi_n(\omega) \rightarrow \Phi(\omega) \right).$$

Theorem 2: (Lévy's continuity theorem) Again, the $X_n$'s are random variable and the $\Phi_n$'s are their characteristic functions. Assume that the limit $\lim \Phi_n(\omega)$ exists pointwise and is denoted by $\Phi(\omega)$. We then have the equivalence:

• $(X_n)$ converges in law to some random variable $X$.

• $\Phi$ is continuous at $0$.

Now I precise my question. Given a probability measure $\mu$ on $\mathcal{N}' = \mathcal{D}'(\mathrm{R}^N)$ or $\mathcal{S}'(\mathrm{R}^N)$, we can define its characteristic functional on $\mathcal{N}$ by $$\hat{\mu}(\varphi) = \int_{\mathcal{N}'} \mathrm{e}^{\mathrm{j} \langle u , \varphi \rangle} \mathrm{d}\mu (u).$$ This generalizes the concept characteristic function of a random variable (Bochner's theorem being generalized by Minlos's theorem). Is the following result true?

Possible generalization of theorem 1: Let $\mu_n$, $n\in\mathbb{N}$ and $\mu$ be probability measures on $\mathcal{N}'$ and $\hat{\mu}_n$ and $\mu$ their characteristic functionals. Then, $$\left( \mu_n \overset{\mathrm{weakly}}{\rightarrow} \mu \right) \Leftrightarrow \left( \forall \varphi, \hat{\mu}_n(\varphi) \rightarrow \mu(\varphi) \right).$$

Similarly, can we generalize the Lévy's continuity theorem?

Thanks for attention.

Answer 1

There is a partial result due to Boulicaut (1973), which states

Theorem: Let $E$ be a separable metrizable Hausdorff locally convex topological vector space. Then $E$ is nuclear if and only if for every sequence $\{\mu_n\}$ of tight probability measures, weak convergence to a tight probability measure $\mu$ is equivalent to the pointwise convergence of the characteristic functions of $\mu_n$ to the ch. f. of $\mu$.

This makes characteristic function(al)s more useful than in separable Banach spaces, where they are only used for uniqueness but not weak convergence.

Answer 2

In fact both theorems are true for probability measures on $\mathcal{D}'$ and $\mathcal{S}'$ and they were proved before in the thesis of Xavier Fernique. The paper that came out of it is: "Processus linéaires, processus généralisés". Annales de l'institut Fourier, 17 no. 1 (1967), p. 1-92.

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Update: A new reference on the Lévy-Fernique continuity theorem for $\mathcal{S}'$ is https://arxiv.org/abs/1706.09326 (re MathNovice's comment below: this one is English!)

Answer 3

A more abstract version of this theorem can be found in Le théorème de continuité de P. Lévy sur les espaces nucléaires where it is proven for Borel measures on strict inductive limits of nuclear Fréchet spaces. The method of proof appears to be the one from Processus linéaires, processus généralisés as mentioned in another answer.