8
$\begingroup$

I would like to investigate if an analog of the classical Portmanteau theorem holds for vague convergence of Radon measures.

Here are the definitions I'm using.

Let $X$ be a Hausdorff locally compact topological space, and let $\mathcal{B}(X)$ be its Borel $\sigma$-algebra. A positive measure $\mu$ on $(X, \mathcal{B}(X))$ is said to be a Radon measure if : (i) $\mu(K) < \infty$ for all compact subsets $K \subset X$, (ii) $\mu(O) = \sup \, \{\mu(K) \, / \, K \subset O, K \rm{\: is \: compact \:} \}$ for all open subsets $O \subset X$, and (iii) $\mu(A) = \inf \, \{ \mu(O) \, / \, A \subset O, O \rm{\: is \: open \:} \}$ for every Borel subset $A \subset X$.

I will say that a sequence $(\mu_n)_n$ of Radon measures on $X$ converges vaguely to a Radon measure $\mu$ if $\lim_{n \to \infty} \int_X f \, d \mu_n = \int_X f \, d \mu$ for all $f \in C_c(X)$, where $C_c(X)$ denotes the set of all continuous real-valued functions defined on $X$, with a compact support.

Now, consider the following propositions :

$(P_1)$ $(\mu_n)_n$ converges vaguely to $\mu$;

$(P_2)$ $\mu(O) \le \underline{ \lim } \, \mu_n(O)$ for all open subsets $O \subset X$;

$(P_3)$ $\mu(K) \ge \overline{ \lim } \, \mu_n(K)$ for all compact subsets $K \subset X$;

$(P_4)$ $\mu(A) = \lim \, \mu_n(A)$ for all Borel subsets $A \subset X$ with a compact closure, and satisfying $\mu(\partial A) = 0$.

I'm able to prove that $(P_1) \Leftrightarrow ((P_2) + (P_3)) \Rightarrow (P_4)$.

My questions are :

1) Are $(P_2)$ and $(P_3)$ equivalent in full generality ? If $X$ is compact, this is obvious by taking complementary sets.

2) Can one prove that $(P_4) \Rightarrow (P_1)$ ? I succeeded to prove that $(P_3) + (P_4) \Rightarrow (P_1)$. (in fact, instead of $(P_3)$, I only need that $\sup_n \mu_n(K) < \infty$ for all compact subsets $K \subset X$)

Thanks.

$\endgroup$
2
  • $\begingroup$ In Araujo-Ginet's book Probablity measures in Banach spaces, (P_4)$\Rightarrow (P_1)$ when $X$ is a separable locally compact metric space is left as an exercise. $\endgroup$ Nov 2 '12 at 21:51
  • $\begingroup$ have you tried $P_4\Rightarrow (P_3)+(P_2)$? $\endgroup$ Aug 1 '13 at 16:34
3
$\begingroup$

ad 1) No, neither implication holds; let $X = \mathbb{R}$. First take $\mu_n = \delta_1$ and $\mu = 0$. Then $(P_2)$ is satisfied, whereas $(P_3)$ is not. Second, take $\mu_n=0$ and $\mu = \delta_2$. Then $(P_3)$ holds, but $(P_2)$ does not.

ad 2) A statement of the equivalences between $(P_1)$, $((P_2)+(P_3))$ and $(P_4)$ can be found in Theorem 3.2 in Resnick, Heavy-Tail Phenomena, Springer 2007. $(P_2)$ should additionally have the criterion that $O$ is relatively compact. Resnick's book is not explicit about the proof; the space $X$ is required to be locally compact and separable, see page 48.

The book can be found here, https://books.google.de/books?id=p8uq2QFw9PUC&lpg=PP1&dq=resnick%202007%20probability&pg=PA52#v=onepage&q=theorem%203.2&f=false

EDIT: another source which I recently encountered: Lindskog, Resnick and Roy, Regularly varying measures on metric spaces: Hidden regular variation and hidden jumps, available at http://projecteuclid.org/euclid.ps/1413896892. In Section 2 a certain type of convergence is the topic and in Theorem 2.1 you might find a portmanteau theorem.

$\endgroup$
1
  • $\begingroup$ Many thanks for your answer. I actually forgot about this question, but I will take a look at your answer and the references you suggest when time will permit. Thanks again ! $\endgroup$
    – Ahriman
    Mar 11 '15 at 15:28

Your Answer

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.