Let $X_n$ and $Y_n$ be two sequences of random variables of mean zero and unit variance such that as $n\rightarrow\infty$, the following properties hold:
- $X_n$ has limiting Gaussian law: $X_n\rightarrow N(0,1)$,
- $Y_n$ has limiting Gaussian law $Y_n\rightarrow N(0,1)$,
- The covariance of $X_n$ and $Y_n$ vanishes: $\text{Cov}(X_n,Y_n)\rightarrow 0$.
Is it true that the limiting distribution of $(X_n,Y_n)$ is two-dimensional standard normal $N(0,I_2)$? Thanks in advance for your comments.
No construct a trivial sequence with $X_{n}=X\sim N(0,1)$ and $Y_n=SX\sim N(0,1)$ where $S=1$ with probability $1/2$ and $S=-1$ with probability $1/2$ and $S$ is indpendent of $X$. Then it is easy to see that $SX$ is uncorrelated with $X$ but $X$ and $SX$ are not jointly normally distributed since $X+SX=0$ with probability 1/2