Suppose $X_n$ and $Y_n$ are two sequences of random variables, satisfying that $\sqrt{n}(X_n - X)$ and $\sqrt{n}(Y_n - Y)$ both converge to two normal distribution with mean $0$. Will $\sqrt{n}\{(X_n + Y_n) - (X+Y)\}$ converge to normal distribution?
If not, given the facts that $cov(\sqrt{n}(X_n - X), \sqrt{n}(Y_n - Y))\to 0$, can we derive normal for their sum? And if this condition doesn't work, what condition will we need?
I try to prove it using Delta method, that if $$\sqrt{n}(X_n - X, Y_n - Y)^{T}\to Normal((0,0)^{T}, \Sigma),$$ we have $f(X_n, Y_n)$ converges to normal with mean $f(X,Y)$. Is there anything wrong when using Delta method for some counter examples?
Thanks a lot for your help!