Suppose $X_n$ is a sequence of random variables such that their second moments are uniformly bounded. I would like to know a compactness criterion for this case.
In analysis, if $K$ is a bounded subset in $L^2(\Omega)$, $\Omega$ is a domain in $R^n$, a compactness criteria is given as following. $K$ is relative compact if and only if for every $\epsilon>0$, there are $\delta>0$, $G$ compact subset of $\Omega$ such that for every $f\in K$, for every $h\in R^n$ with $|h|<\delta$ $$\int_\Omega |f(x+h)-f(x)|^2dx<\epsilon^2 $$ and $$\int_{\Omega- G}|f(x)|^2dx<\epsilon^2 $$ where the functions are extended by 0 outside $\Omega$.
I find out that for random variables, the domain $\Omega$ is very vague, and in general there is no topological structure on it. So one example that I have in mind is $X_n=\int_0^1 f_n(B_s)ds$ where $B$ is Brownian motion and $f_n$ is as nice as possible. Is there a compact criteria for this case?
This notion is called tightness of a sequence of measures.
You can apply it in probability theory with the sequence $P\#{X_n}$ of the image probabilities under the action of $X_n$: $$ P\#{X_n}(A) = P(X_n\in A) $$
You transfer the topology issues to the (metric, often polish) space $\mathcal X$ where $$ X:\Omega \to \mathcal X $$
Note also that in the case of random variables with $2^{\text{nd}}$ moment bounded, you have a uniformly integrable sequence. In this case, the sequence converge in $L^1$ iff it converges in probability.