Bochner-Minlos for moment-generating functions?

Question

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.

Answer 1

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.

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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)

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, 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.