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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?

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Can we regard this double sequence as a sequence in the space of the product space of random variables and apply Tychonoff's theorem? – user12586 Dec 25 '11 at 21:16
Do you mean sth like this: $X_{i,n}: \Omega \to \mathbb{X}$ and treat the family of $X_{i,n}$ as a sequence of $Y_n: \Omega \times [0,1] \to \mathbb{X} \times [0,1]$? An interesting approach, but even if it may lead us to convergence of the joint distribution, I'm afraid that the condition "from every $i$" from the question will not be satisfied. – savick01 Dec 26 '11 at 0:16
But will it be that weak convergence of joint distribution will lead to weak convergence of every $i$ with at most a countable number of exceptions? – webster Dec 26 '11 at 1:24
No. In my answer I wrote "There are $2^{\aleph_0}$ numbers between 0 and 1.". I could have written "There are $2^{\aleph_0} \cdot 2^{\aleph_0}$ numbers between 0 and 1." and then the number of exceptions would be at least $2^{\aleph_0}$. – savick01 Dec 26 '11 at 11:08
But will it be still measure zero? – webster Dec 28 '11 at 16:30
up vote 2 down vote accepted

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.

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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.

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