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!