# Generalization of Lévy's continuity theorem for nuclear spaces

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.

## 2 Answers

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.

• This paper is clearly relevant for my question and I thank you for that. One question however. Boulicaut is studying measure on $E$ and defines the characteristic function on $E'$ while I am considering the opposite (measure on $E'$ and the characteristic functional on $E$). This makes no difference for reflexive spaces but what about the general case? Is Boulicaut really approaching the same problem? – Goulifet Jan 7 '14 at 14:16
• Boulicaut is quote in Araujo and Giné book, where characteristic functions on a topological vector space are defined on $E'$ as you said. I don't know what happens in the general case (I'm not used to work on probability measures on topological vector spaces). – Davide Giraudo Jan 7 '14 at 14:26
• Nuclear spaces are reflexive. $\mathscr{D}'$ and $\mathscr{S}'$ are nuclear spaces. – Liviu Nicolaescu Jan 7 '14 at 15:02
• @LiviuNicolaescu A nuclear space is reflexive iff it is quasi-complete and barrelled. So incomplete separable metrizable nuclear spaces are not reflexive. – Robert Furber Jan 4 '19 at 16:23
• See paragraph 1 of this . encyclopediaofmath.org/index.php/Nuclear_space – Liviu Nicolaescu Jan 4 '19 at 17:39

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.

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!)

• Is there another reference which gives the result in Fernique's paper that's in English? – MathNovice Apr 10 '15 at 18:00