Hello,

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

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

Now let's define the bounded function f : , where g is any bounded function, and

Then we have (for any bouned function g)

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

(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