I have a question involving the a sort of central limit theorem which has two index. Suppose that $\{X_{n,m}\}$ is a sequence of random variables which satisfies $$X_{n,m} \overset{\mathscr{L}}{\longrightarrow} Y_m \ \ \text{as}\ n \to \infty$$ where $\{Y_m\}$ is another sequence of random variables. Moreover, we have $$Y_m \overset{\mathscr{L}}{\longrightarrow} Y_m \ \ \text{as}\ m \to \infty$$ for some random variable $Y$.
I'm just wondering in this case, what conclusion can I draw about the original sequence $\{X_{n,m}\}$? Is there any result of the form like $$\lim_{m\to \infty} \lim_{n \to \infty}X_{n,m}\overset{\mathscr{L}}{\longrightarrow} Y?$$ Or it is inconclusive for $X_{n,m}$ at all?
A side question is how should I prove a sort of central limit theorem involving two index? I know triangular array (Lindeberg CLT) is one of the kind, but it is different from the situation we consider here, I think.