Consider the space $C_c := C_c(\mathbb{R})$ of compactly supported continuous functions with the inductive limit topology and $C_c'$ its topological dual which can be identified with the space of real Radon measures $M := M(\mathbb{R})$. Equip $M$ with the vague topology which is the weak-* topology on $C_c'$, i.e. the measures $\mu_n$ converge vaguely to $\mu$ if $\int f d\mu_n \to \int f d\mu$ for all $f \in C_c$. (This is not to be confused with the vague convergence on the space of finite signed measures by testing against the completion $C_0$ of $C_c$ with respect to the uniform norm.)

I want to consider random Radon measures $X : \Omega \to M$ on some probability space $(\Omega, \mathscr{F}, P)$ and speak of convergence in distribution of such random measures. A sequence $X_n : \Omega_n \to M$ is converging in distribution to $X$ if their image probability measures $P_n \circ X_n^{-1}$ on $M$ converge weakly to $P \circ X^{-1}$, i.e. $\int F d(P_n \circ X_n^{-1}) \to \int F d(P \circ X^{-1})$ for all $F \in C_b(M)$. Note that the $X_n$ might be defined on different probability spaces $(\Omega_n, \mathscr{F}_n, P_n)$.

However, I saw in the broad literature only that weak convergence of (probability) measures is defined for metric spaces.

Q: Is the vague topology on $M$ metrizable?

Bourbaki (Integration INT III, p.58, Exc. 14) and Kallenberg (Random measures, p. 170, 15.7.7.) mention only that the subset of positive Radon measures $M_+ \subseteq M$ is metrizable and Kallenberg gives a precise metric which turns $M_+$ into a Polish space. If $M$ would be metrizable then this would contradict this fact: Weak *-topology of $X^*$ is metrizable if and only if ... So it seems that $M$ is not metrizable.

On the other hand, every real Radon measure is the difference of two positive ones. So it seems that there is not really a big deal with considering weak convergence of probability measures on $M$ (and maybe arbitrary non-metrizable spaces with some additional properties?).

Q: Is there some known theory or reference for weak convergence of (probability) measures on non-metrizable spaces (or even for weak convergence of such random Radon measures)?

EDIT In particular, Kallenberg reduces the check for convergence in distribution of a sequence $X_n : \Omega \to M_+$ of random positive Radon measures to convergence in distribution of all the random variables $\omega \mapsto \int f(u) \, dX_n^\omega(u)$ for $f \in C_c$. I think, there must be a similar check possible for random Radon measures in $M$. The only thing, that I could show so far is that if $\mu_n \to \mu$ vaguely in $M$ iff it holds for the Jordan decomposition $\mu_n = \mu_n^+ - {\mu_n}^-$ that $\mu_n^+ \to \mu_1$ vaguely in $M_+$ and ${\mu_n}^- \to \mu_2$ vaguely in $M_+$ and $\mu = \mu_1 - \mu_2$ (not necessarily the Jordan decomposition of $\mu$).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.