# Convergence of a subsequence of arrays of random variables

Consider $$X(ij)$$ for $$i = 1, ..., n$$ and $$j = 1, ...,n$$ be random variables.

I proved that for each i, the sequences $$X($$i$$j)$$ converge in distribution to random variable Y as $$n$$ tends to $$\infty$$. More precisely,

$$\begin{bmatrix} X(11) & X(12) & X(13) & X(14) & ... & \rightarrow_d Y \newline X(21) & X(22) & X(23) & X(24) & ... & \rightarrow_d Y \newline X(31) & X(32) & X(33) & X(34) & ... & \rightarrow_d Y \end{bmatrix}$$

I would like to prove that based on this result the following sequence of random variables

$$X(11), X(22), X(33), ...$$

will also converge to $$Y$$ in distribution.

• That's too bad, because the conclusion doesn't follow. Let all $X_{ij}=Y$ for $i\ne j$ and let $X_{ii}=Y+i,$ for instance. Maybe there's more you know about these random variables in your intended application?
– whuber
Commented May 30, 2023 at 18:37
• I know that $X(ij)$ are continuous random variables and that $Y$ is a standard normal random variable. Meaning that I managed to prove a CLT for each row. Commented May 30, 2023 at 18:43
• I updated my comment to accommodate that case. In this example, $(X_{ii})$ doesn't even converge. If you expect a result like the one you want to prove, then there must be some mechanism operating to make the convergences (on each row) "uniform" in some way, so that knowing you are close to the limit in one sequence means you will be reasonably close in almost all other sequences, too.
– whuber
Commented May 30, 2023 at 18:44