The Lévy Continuity Theorem for random Schwartz distributions due to Fernique:
First let me recall the well known Lévy continuity theorem for Borel probability measures on a finite-dimensional vector space $V$ like $\mathbb{R}^d$.
To a probability measure $\mu$ one can associate the characteristic function $$ \begin{array}{llll} \Phi_{\mu}: & V' & \longrightarrow & \mathbb{C}\\ & \ell & \longmapsto & \Phi_{\mu}(\ell)= \int_V e^{i \ell(v)}\ d\mu(v) \end{array} $$ which is defined on the dual space. By definition, a sequence of probability measures $(\mu_n)$ converges weakly to a probability measure $\mu$, iff for all bounded continuous functions $F:V\rightarrow \mathbb{R}$ (or $\mathbb{C}$), $$ \lim_{n\rightarrow \infty} \int_V F(v)\ d\mu_n(v) = \int_V F(v)\ d\mu(v)\ . $$ For $V$-valued random variables, this corresponds to convergence in law or in distribution.
We now have the following well-known result say for $V=\mathbb{R}^d$.
Lévy Continuity Theorem:
A sequence of Borel probability measures $(\mu_n)$ on $V$ converges weakly to some (unspecified) Borel probability measure iff the corresponding characteristic functions $\Phi_{\mu_n}$ converge pointwise on $V'$ to some function which is continuous at the origin.
Now the not well known result I propose in this answer, is the analogue for $V=\mathcal{S}'(\mathbb{R}^d)$ the space of temperate Schwartz distributions on $\mathbb{R}^d$.
Lévy-Fernique Continuity Theorem:
A sequence of Borel probability measures $(\mu_n)$ on $V$ converges weakly to some (unspecified) Borel probability measure iff the corresponding characteristic functions $\Phi_{\mu_n}$ converge pointwise on $V'$ to some function which is continuous at the origin.
To clarify, here $V=\mathcal{S}'(\mathbb{R}^d)$ equipped with the strong topology. Also $V'$ is the topological dual equipped with the strong topology. One has $V'\simeq \mathcal{S}(\mathbb{R}^d)$ with its usual topology, i.e., these spaces are reflexive. The definitions of characteristic functions and weak convergence of Borel probability measures are the same as in the finite-dimensional case above.
Comments:
This is important because just about any random object/process can be seen as living inside a space of distributions like $\mathcal{S}'$ or $\mathcal{D}'$.
(Will add more comments later when I find time).
References:
- X. Fernique, Processus linéaires, processus généralisés, Annales de l'Institut Fourier, Volume 17 (1967) no. 1, p. 1-92.
- H. Biermé, O. Durieu, Y. Wang, Generalized random fields and Lévy's continuity theorem on the space of tempered distributions, arXiv 2017.