It is well-known that the Bochner-Minlos theorem characterises measures on duals of nuclear spaces by their characteristic functions. Is there a similar version for moment-generating functions?
I have a sequence of measures admitting moment-generating functions and I wish to prove something like convergence of the moment-generating functions implies convergence of the measures. But it is unclear to me, what kinds of convergences I should look at and in particular what properties I have to demand from the limiting function.



Bochner-Minlos for moment-generating functions?
While waiting for more context around the question, one can already mention the main definitions and tools for this topic.


I will assume the space on which these probability measures live is the space $\mathscr{S}'(\mathbb{R}^d,\mathbb{R})$ of temperate real-valued Schwartz distributions on $\mathbb{R}^d$.
It is the $\mathbb{R}$-vector space given as the topological dual of the
space $\mathscr{S}(\mathbb{R}^d,\mathbb{R})$ of real-valued Schwartz functions.
The latter carries the usual Fréchet topology.
Recall that a subset $A\subset\mathscr{S}(\mathbb{R}^d,\mathbb{R})$ is bounded iff, for all continuous seminorms $\rho$ on $\mathscr{S}(\mathbb{R}^d,\mathbb{R})$
$$
\sup\_{f\in A}\rho(f)\ <\ \infty\ .
$$
The space $\mathscr{S}'(\mathbb{R}^d,\mathbb{R})$ is equipped with its canonical/standard topology, namely the *strong topology* which is the locally convex topology defined by the seminorms
$$
\varphi\ \longmapsto\ \sup\_{f\in A}|\varphi(f)|
$$
where $A$ ranges over all bounded subsets of $\mathscr{S}(\mathbb{R}^d,\mathbb{R})$. Finally, $\mathscr{S}'(\mathbb{R}^d,\mathbb{R})$ is turned into a measurable space thanks to the Borel $\sigma$-algebra for the strong topology.


Let us call a function $\Phi:\mathscr{S}(\mathbb{R}^d,\mathbb{R})\rightarrow \mathbb{C}$ *a* characteristic function iff it satisfies the following three conditions:


1. $\Phi(0)=1$,
2. $\Phi$ is continuous,
3. for all $n\ge 1$ and all $f\_1,\ldots,f\_n$ in $\mathscr{S}(\mathbb{R}^d,\mathbb{R})$, the matrix $(\Phi(f\_i-f\_j))\_{1\le i,j\le n}$ is Hermitian positive semidefinite.


There is no harm in changing 2) to just continuity at the origin.
Now for a Borel probability measure $\mu$ on $\mathscr{S}'(\mathbb{R}^d,\mathbb{R})$, we define *the* characteristic function *of* $\mu$ as the function $\Phi\_{\mu}:\mathscr{S}(\mathbb{R}^d,\mathbb{R})\rightarrow \mathbb{C}$ defined by
$$
\Phi\_{\mu}(f)=\int\_{\mathscr{S}'(\mathbb{R}^d,\mathbb{R})}e^{i\varphi(f)}\ d\mu(\varphi)\ .
$$


**Theorem:** (Bochner-Minlos)
The map $\mu\mapsto\Phi\_{\mu}$ is a bijection from the set of Borel probability measures on $\mathscr{S}'(\mathbb{R}^d,\mathbb{R})$ to the set of characteristic functions.


The weak convergence of probability measures on $\mathscr{S}'(\mathbb{R}^d,\mathbb{R})$ is defined in the same way as for any other topological space.


**Definition:** Let $\mu$ be a Borel probability measure on $\mathscr{S}'(\mathbb{R}^d,\mathbb{R})$ and $(\mu\_n)\_{n\ge 1}$ be a sequence of Borel probability measures on $\mathscr{S}'(\mathbb{R}^d,\mathbb{R})$. We say that $\mu\_n$ *converges weakly to* $\mu$ iff
for all bounded continuous functions $F:\mathscr{S}'(\mathbb{R}^d,\mathbb{R})\rightarrow\mathbb{R}$,
$$
\lim\_{n\rightarrow\infty}\int\_{\mathscr{S}'(\mathbb{R}^d,\mathbb{R})}F(\varphi)\ d\mu\_n(\varphi)
=\int\_{\mathscr{S}'(\mathbb{R}^d,\mathbb{R})}F(\varphi)\ d\mu(\varphi)\ .
$$


We now have an analogue of *Glivenko's Theorem*.


**Theorem:**
The setting being the same as for the previous definition, the weak convergence of $\mu\_n$ to $\mu$ is equivalent to the pointwise convergence of characteristic functions, namely,
$$
\forall f\in\mathscr{S}(\mathbb{R}^d,\mathbb{R}),\ \lim\_{n\rightarrow\infty}\Phi\_{\mu\_n}(f)=\Phi\_{\mu}(f)\ .
$$


Finally, we also have an analogue of the *Lévy Continuity Theorem*, when one does not *a priori* have a candidate for the weak limit.


**Theorem:**
Let $(\mu\_n)\_{n\ge 1}$ be a sequence of Borel probability measures on $\mathscr{S}'(\mathbb{R}^d,\mathbb{R})$. The existence of a Borel probability measure $\mu$ such that $\mu\_n$ converges weakly to $\mu$, is equivalent to requiring that


1. for all $f\in\mathscr{S}(\mathbb{R}^d,\mathbb{R})$, the limit $\lim\_{n\rightarrow\infty}\Phi\_{\mu\_n}(f)$ exists, and
2. the function $\Phi$ defined by $\Phi(f)=\lim\_{n\rightarrow\infty}\Phi\_{\mu\_n}(f)$ is continuous at the origin.


I'll stop here this attempt at a *formulaire raisonné*, for now, but there are also other useful results like $\mathscr{S}'(\mathbb{R}^d,\mathbb{R})$ being a Radon space as well as Prokhorov's Theorem.




---


**Edit: Mar 14, 2021**


Finally, got a little bit of time to explain the explicit example I mentioned.


Consider the Ising model in one dimension with zero magnetic field, nearest-neighbor coupling function $J>0$ and inverse temperature $\beta$, in infinite volume. This is a Borel probability measure $\mu$ on $\{-1,1\}^{\mathbb{Z}}$, with corresponding expectations denoted by $\langle \cdots\rangle$.
Pick, once and for all, your favorite number $L>1$ and for $M\in\mathbb{N}$, define the map
$$
\begin{array}{cccc}
\Gamma\_M: & \{-1,1\}^{\mathbb{Z}} & \longrightarrow & \mathscr{S}'(\mathbb{R},\mathbb{R})\ \\
 & \sigma=(\sigma\_x)\_{x\in\mathbb{Z}} & \longmapsto & 
 L^{-\frac{M}{2}}\sum\limits\_{x\in\mathbb{Z}}\sigma\_x \delta\_{L^{-M}x}
\end{array}
$$
where $\delta\_z$ denotes the Dirac distribution located at $z$.
Similarly to my answer to


[A set of questions on continuous Gaussian Free Fields (GFF)](https://mathoverflow.net/questions/364470/a-set-of-questions-on-continuous-gaussian-free-fields-gff?rq=1)


this map is well defined, continuous, and therefore Borel measurable. This allows you to define the push-forward measure $\mu\_M:=(\Gamma\_M)\_{\ast}\mu$. It turns out that when $M\rightarrow\infty$ the measure $\mu\_M$ converges weakly to a multiple of white noise on $\mathbb{R}$. This is a special case of a theorem of Newman for FKG spin systems, but it is a good exercise to do it using, in a completely explicit way, characteristic functions/moment generating functions via the control of what is happening in a complex neighborhood of the origin.


First recall that the correlation functions vanish for odd number of spins and are otherwise given by
$$
\langle \sigma\_{x\_1}\sigma\_{x\_2}\cdots\sigma\_{x\_{n}}\rangle=e^{-m(|x\_1-x\_2|+|x\_3-x\_4|+\cdots+|x\_{n-1}-x\_n|)}
$$
if $n$ is even and $x\_1<x\_2<\cdots<x\_n$. The mass or rate of exponential decay is $m:=-\log\tanh(\beta J)$.


Now if we have a complex-valued test function $f\in\mathscr{S}(\mathbb{R},\mathbb{C})$, it holds that
$$
\int\_{\mathscr{S}'(\mathbb{R},\mathbb{R})}
e^{i\varphi(f)}\ d\mu\_M(\varphi)
=\left\langle
\exp\left[i\sum\_{x\in\mathbb{Z}}\sigma\_x g\_x\right]
\right\rangle
$$
where $g=(g\_x)\_{x\in\mathbb{Z}}\in \mathcal{s}(\mathbb{Z},\mathbb{C})$ is a discrete test function, with rapid decay on the lattice $\mathbb{Z}$, given by $g\_x=L^{-\frac{M}{2}}f(L^{-M}x)$.
Since we work with complex-valued test functions, the difference between characteristic functions and moment generating function is moot and amounts to deciding to absorb the $i$ into the test function or not. We will keep it out.


Using the cluster expansion techniques from [my article on the Spin-Boson Model](https://link.springer.com/article/10.1007/s00023-011-0103-6), one can show after a bit of work that for all complex function (not necessarily of fast decay) $g:\mathbb{Z}\rightarrow\mathbb{C}$, in the domain $||g||\_{\ell^1}<\frac{1}{2e}$,
$$
\left\langle
\exp\left[i\sum\_{x\in\mathbb{Z}}\sigma\_x g\_x\right]
\right\rangle
=\exp(\mathcal{F}(g))
$$
where
$$
\mathcal{F}(g)=\sum\_{p\ge 1}\frac{1}{p!}
\sum\_{\substack{x\_1,\ldots,x\_p\in\mathbb{Z}\\ y\_1,\ldots,y\_p\in\mathbb{Z}}}
\prod\_{i=1}^{p}\left[
-\mathbf{1}\{x\_i<y\_i\}e^{-m|x\_i-y\_i|}\tan(g\_{x\_i})\tan(g\_{y\_i})
\right]
$$
$$
\times\left(
\sum\_{\substack{H\subset\wedge^2[p]\\ H\ \text{connects}\ [p]}}
\prod\_{\{i,j\}\in H}\left[-\mathbf{1}\left\{
\ [x\_i,y\_i]\cap[x\_j,y\_j]\neq\varnothing\ 
\right\}\right]
\right)\ .
$$
The notation is as follows. $[p]:=\{1,2,\ldots,p\}$, $\wedge^2[p]$ is the set of two-element subsets $\{i,j\}$ of $[p]$. The subset $H$ is a graph (or edge set thereof) required to connect the vertex set $[p]$. $\mathbf{1}\{\cdots\}$ is the indicator function of the enclosed condition. The intervals $[x,y]$ are integer intervals inside $\mathbb{Z}$.
Finally, the sum giving $\mathcal{F}(g)$ converges absolutely provided one keeps the sum over $H$ inside the modulus. One can write this bound in a combinatorially explicit way too.


With the above result one can show that for $f\in\mathscr{S}(\mathbb{R},\mathbb{C})$ such that
$4eK||f||\_{L^{\infty}}||f||\_{L^{1}}<1$,
$$
\lim\limits\_{M\rightarrow \infty}
\int\_{\mathscr{S}'(\mathbb{R},\mathbb{R})}
e^{i\varphi(f)}\ d\mu\_M(\varphi)
=
\exp\left[-\frac{K}{2}\int\_{\mathbb{R}}f(x)^2\ dx\right]\ .
$$
Here the constant $K$ is related to the susceptibility and is given by $K=\frac{1+e^{-m}}{1-e^{-m}}$.


Now take $f\in\mathscr{S}(\mathbb{R},\mathbb{R})$, a real-valued test function but this time with no limitation on its size.
For $z\in\mathbb{C}$, let
$$
G\_M(z)=\int\_{\mathscr{S}'(\mathbb{R},\mathbb{R})}
e^{iz\varphi(f)}\ d\mu\_M(\varphi)\ ,
$$
which is well defined and analytic in an open horizontal strip around the real axis. The bound for $\mathcal{F}$ shows that $G\_M(z)$ is uniformly bounded in $z$ and $M$ on every compact subset of the strip. The Vitali-Porter Theorem then gives pointwise convergence on the real line, and the Lévy Continuity Theorem finishes the job of proving that we weakly converge to $\sqrt{K}$ times white noise.


