Consider two sequences of real-valued random variables defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, $X_n:\Omega\rightarrow \mathbb{R}$ and $Y_n:\Omega\rightarrow \mathbb{R}$. Suppose that $X_n\rightarrow_d A$ and $Y_n\rightarrow_d B$, where $A,B$ are real-valued random variables. This does not necessarily imply that $(X,Y)\rightarrow_d(A,B)$. However, I think it implies that there exists a subsequence $(X_{n_j}, Y_{n,j})\rightarrow_d (A,B)$ as $j\rightarrow \infty$ Here my proof:
(1) $X_n\rightarrow_d A$ $\Rightarrow $ $X_n=O_p(1)$
(2) $Y_n\rightarrow_d B$ $\Rightarrow $ $Y_n=O_p(1)$
(3) (1)+(2) $\Rightarrow $ $(X_n,Y_n)=O_p(1)$ $\Rightarrow$ $\exists \{(X_{n_j}, Y_{n,j})\}_j$ such that $(X_{n_j}, Y_{n_j})\rightarrow_d (C,D)$ as $j\rightarrow \infty$ where $C,D$ are real-valued random variables $\Rightarrow$ $X_{n_j}\rightarrow_d C$ and $Y_{n_j}\rightarrow_d D$ as $j\rightarrow \infty$
(4) $X_n\rightarrow_d A$ $\Rightarrow$ every subsequence $\{X_{n_k}\}_k \rightarrow_d A$ as $k\rightarrow \infty$ $\Rightarrow$ $C\sim A$
(5) $Y_n\rightarrow_d B$ $\Rightarrow$ every subsequence $\{Y_{n_k}\}_k \rightarrow_d B$ as $k\rightarrow \infty$ $\Rightarrow$ $D\sim B$
(6) (3)+(5) $\Rightarrow $ $(X_{n_j}, Y_{n_j})\rightarrow_d (A,B)$
Are this proof and its conclusion correct? Any hint would be really appreciated.
Best Answer
In order to go from (5) to (6), you need to prove that $(A,B)$ has the same distribution as $(C,D)$. But this may not be the case. For example, if $X$ is a symmetric non degenerated distribution and $X_n:=X$, $Y_n:=-X$, $A=B=X$, then $X_n\to A$ and $Y_n\to B$ in distribution. But for each $n$, $(X_n,Y_n)$ has the same distribution as $(X,-X)$, hence no subsequence of $\left(\left(X_n,Y_n\right)\right)_{n\geqslant 1}$ can converge in distribution to $(A,B)=(X,X)$.