Applying Prokhorov's theorem to collection of random variables Suppose we have a collection of random variables $X_{i,n}$ for $i \in [0,1]$ and $n=1,2,3....$ Suppose this collection of random variables is tight. Then, can we construct a subsequence $n'$ such that along this subsequence for every $i$, $X_{i,n}$ converges to some $X_{i}$ in distribution?
 A: It looks like we can create some bad distributions/random variables. There are $2^{\aleph_0}$ subsequences of $\mathbb{N}$. There are $2^{\aleph_0}$ numbers between 0 and 1. So we can associate every number $i$ with one subsequence $(n')$. Then we define $X_{i,n}$ to be whatever (but uniformely bounded) for $n \notin (n')$. For $n \in (n')$ $X_{i,n}$ can be $\delta_0$ and $\delta_1$ by turns (to prevent convergence).
EDIT: For $i$ belonging to a countable set $I$ (instead of $[0,1]$) we can use the diagonal method - we don't even need tightness of the whole family - tightness of each family $(X_{i,n})_{n\in \mathbb{N}}$ ($i$ fixed) would be enough.
A: In Laurent Schwartz's book "Radon measures on arbitrary topological spaces and cylindrical measures" there is an extremely long, complicated and interesting theorem (i.e. p291 Theorem 10) which is essentially Prokhorov's theorem for random variables under a wide range of hypotheses (and a suitable counterexample when hypotheses not met).
This is an outstanding book which is long out of print.
