1. ## Probability

Hello,

I'm needing some help here, please. Suppose that

$\displaystyle P_n \rightarrow P$

weakly, where both measures are defined in $\displaystyle R^2$, and that $\displaystyle P_1$ is the marginal of $\displaystyle P_n$.

By Portmanteau's theorem, we have

$\displaystyle P(A X R) \leq \liminf P_n(A X R) = P_1(A)$

for all A open. So far, ok. The problem arises now. I've been told that all the above implies

$\displaystyle P(A X R) = P_1(A)$

Can someone help me to prove that statement?

*******************

The real problem is the following:
If

$\displaystyle P_n \rightarrow P$

weakly, and if

$\displaystyle P_1$ and $\displaystyle P_2$ are the marginals of $\displaystyle P_n$,

the limit $\displaystyle P$ must have the same marginals.

***************

Thank you very much and happy new year!

2. Hello,

What about that ? (I'm not sure at all)

We know that $\displaystyle P_n$ converges weakly to $\displaystyle P$ iff for any bounded function f, $\displaystyle E[f(X_{n,1},X_{n,2})] \to E[f(X_1,X_2)]$, where [tex](X_{n,1},X_{n,2}) \sim P_n[tex] and $\displaystyle (X_1,X_2) \sim P$

Now let's define the bounded function f : $\displaystyle f=g\circ \pi_1$, where g is any bounded function, and $\displaystyle \pi_1 ~:~ (x_1,x_2)\mapsto x_1$

Then we have (for any bouned function g) $\displaystyle E[g(X_{n,1})]=E[f(X_{n,1},X_{n,2})] \to E[f(X_1,X_2)]=E[g(X_1)]$

which proves that the marginal of $\displaystyle P_n$ converges to the marginal of $\displaystyle P$, so since the marginal of $\displaystyle P_n$ doesn't depend on n, the marginal of P is $\displaystyle P_1$ ?

(I hope you don't get confused with the index of P, because one designs the sequence, the other designs the marginal, but I guess you'll be able to make the difference).

I didn't use the porte-manteau theorem, because the version you have don't seem to get you anywhere...
There are many points that are available on the internet (apart from the liminf stuff).
I have something that says that if, for an open O set of R (more generally, a borelian of R), P( frontier of O )=0, then P_n(O) converges to P(O)
It deals with R, but it should work with Rē.
Then I'm not too sure... You may be able to apply the porte-manteau theorem in its "closed" version : for any closed set C, limsup P_n(C) < P(C)
and consider (R\A)\D, where D is the frontier of A (which is a closed set contained in R\A, by definition)

But I don't know if it's a valuable idea...

again, I'm not sure at all whether my method is correct or not. So just get inspired by it, and if you find something false, tell me

3. Thank you, Moo.