Let's consider a sequence $(\mu_n)_n$ of random probability measures on $\mathbb R$, and let $C_b$ be the Banach space of bounded continuous functions on $\mathbb R$. I am considering the following types of convergence, where $\mu$ is some non-random probability measure :

(A) $$ \forall f\in C_b, \quad \mathbb P \left(\int f(x)d\mu_n(x)\rightarrow_{n\rightarrow\infty}\int f(x)d\mu(x)\right)=1$$ and

(B) $$ \mathbb P \left(\forall f\in C_b,\quad \int f(x)d\mu_n(x)\rightarrow_{n\rightarrow\infty}\int f(x)d\mu(x)\right)=1.$$ Whereas (B) clearly implies (A), what about the other direction ?

1) Do you have a counter-example where (A) does not implies (B) ?

2) Do we have a sufficient criterion so that (A) implies (B) ? (I have in mind the use of the Borel-Cantelli lemma to improve the probability convergence to the almost sure one)

  • $\begingroup$ What measure is $\mathbb{P}$? $\endgroup$ – T. Eskin Jun 7 '12 at 15:52
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    $\begingroup$ @ThomasE.: $\mu_n$ are random measures. I.e. each $\mu_n$ is a measurable function from some underlying probability space $(\Omega, \mathcal{F}, \mathbb{P})$ into the space of probability measures on $\mathbb{R}$. $\endgroup$ – Nate Eldredge Jun 7 '12 at 16:21
  • $\begingroup$ @NateEldredge. thanks for clearing this:) $\endgroup$ – T. Eskin Jun 7 '12 at 16:23

I believe A and B are equivalent.

First, I believe that when the underlying state space (in this case $\mathbb{R}$) is locally compact, then probability measures $\nu_n$ converge weakly to $\nu$ iff $\int f\, d\nu_n \to \int f\,d\nu$ for all continuous functions $f$ which vanish at infinity, i.e. for all $f \in C_0(\mathbb{R})$. I don't remember the proof off the top of my head, but you can find it in Billingsley's Convergence of Probability Measures.

Now $C_0(\mathbb{R})$ is separable, so one can choose a countable dense subset $\{f_1, f_2,\dots\}$. Then using the triangle inequality and the fact that the measures $\nu_n$ are uniformly bounded in total variation (since they are all probability measures), you can show that $\nu_n \to \nu$ weakly iff $\int f_k \,d\nu_n \to \int f_k \,d\nu$ for all $k$.

Now if A holds, then by countable additivity $$\mathbb{P}\left( \int f_k\,d\mu_n \to \int f_k\,d\mu \text{ for all } k\right) = 1.$$ But we have just argued that on this event, $\mu_n \to \mu$ weakly.

  • $\begingroup$ I believe that's exactly what I was looking for ! Thx $\endgroup$ – Student Jun 7 '12 at 21:57

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