What is convergence in distribution of random variables taking values in a non-metrizable product space?

Question

Let $F = (F(x) : x \in \mathbf{R}^n)$ be a family of $\mathbf{R}^k$-valued random variables indexed by $\mathbf{R}^n$ (to be clear there is a single probability space $(\Omega,\Sigma,\mathbf{P})$ such that for each $x \in \mathbf{R}^n$, we have a random variable $F(x) : (\Omega,\Sigma,\mathbf{P}) \to (\mathbf{R}^k,\text{Borel}(\mathbf{R}^k))$.

This is a 'random field in $\mathbf{R}^k$ indexed by $\mathbf{R}^n$' - i.e. we can think of it as a single random variable $F : (\Omega,\Sigma,\mathbf{P}) \to (S, B)$, where $S = (\mathbf{R}^k)^{\mathbf{R}^n}$ is the space of functions from $\mathbf{R}^n$ to $\mathbf{R}^k$ and $$ B = \Pi_{t \in \mathbf{R}^n}\text{Borel}(\mathbf{R}^k) $$ is the product of Borel sigma algebras.

Suppose I have a sequence $F_k$ of such things. What is the correct notion of 'convergence in distribution' $F_k \to F$?

Should I look at $\mu_{F_k} := \mathbf{P}\circ F_k^{-1}$ as a probability measure on $S$ and see if $\int g d\mu_{F_k} \to \int g d\mu_F$ for every bounded continuous function $g$? The reason I'm suspicious is that the topological space $(S,\text{product topology})$ is not metrizable and that throws me off what I usually think weak convergence of measures ought to be...

EDIT: It's possible that this stack exchange answer sort of resolves my confusion but does anyone know of a source? https://math.stackexchange.com/q/4698018/562248

Answer 1

Maybe more a suggestion than an answer.

You could replace the space of all maps $(\mathbb{R}^k)^{\mathbb{R}^d}$ with the space of all continuous maps $C:=C(\mathbb{R}^d,\mathbb{R}^k)$. We can then endow $C$ with the topology of uniform convergence on compacta (aka compact convergence), which coincides in this case with the compact-open topology. According to mathoverflow this topology on $C$ is metrizable, and thus completely regular (see wikipedia).

On the topological space $C$ it is natural to consider $\tau$-additive probability measures and the narrow convergence of measures.

Note that, according to Fremlin Cor. 437L, narrow convergence and the vague convergence coincides for $\tau$-additive probability measures on completely regular topological spaces. So for $C$ there is no ambiguity here.

Then you could model $F$ as a measurable map $F: \Omega \to C$, where $C$ carries the Borel $\sigma$-algebra w.r.t. to the mentioned topology, such that the distribution $\mu$ of $F$ on $C$ is $\tau$-additive.

A sequence $(\mu_n)_{n \in \mathbb{N}}$ of $\tau$-additive probability measures on $C$ convergences to a $\tau$-additive probability measure $\mu$ on $C$ narrowly/weakly/vaguely if and only if for all bounded continuous functions $g: C \to \mathbb{R}$ we have for $n \to \infty$ the convergence: $$ \int g \,d\mu_n \to \int g\,d\mu.$$

Please carefully check for yourself if all arguments made are correct.