My question is related to the possibility of stating joint convergence in distribution from marginal weak convergence.
Consider two sequences of random vectors $X_n$ and $Y_n$ defined on the probability space $(\Omega, \mathcal{F}, P)$. Let $X,Y$ be other two random vectors on $(\Omega, \mathcal{F}, P)$. All random vectors take values in $\mathbb{R}^k$.
Let "$\rightarrow_d$" denote convergence in distribution.
General statement: $X_n\rightarrow_dX$ and $Y_n\rightarrow_dY$ does not imply $(X_n,Y_n)\rightarrow_d (X,Y)$
Specific case: Assume $X_n\rightarrow_dX\sim N(0,J)$, $Y_n=h^TX_n$ with $h\in \mathbb{R}^k$ and $J$ is a $k\times k$ matrix. Hence, $Y_n=h^TX_n\rightarrow_d h^TX \sim N(0,h^TJh)$. Is it correct to write down $(X_n,Y_n)\rightarrow_d(X,h^TX)$?
Where I found this specific case: I think it is used in the proof of Theorem 9.4 in van der Vaart "Asymptotic Statistics" p. 128 here where the author says
"By Assumption, the sequence $(\Delta_n,h_0^T\Delta_n)$ converges in distribution under $h_0=0$ to a vector $(\Delta,h^T\Delta)$... "
Did I misinterpret the proof? Any hint would be really appreciated