Marginal and joint asymptotic normality

Question

Suppose we have an i.i.d. data set $\{X_{i}:i=1,...,n\}$ where each $X_{i}$ is a real random variable. Let $X=(X_{1},...,X_{n})$. There are $J$ real-valued functions $g_{1},...,g_{J}$ such that for each $j=1,...,J$, $(g_{j}(X)-\theta_{j})$ converges in distribution to $N(0,1)$ for some constants $\theta_{1},...,\theta_{J}$. Now in order to have joint asymptotic normality, i.e., the vector $(g_{1}(X)-\theta_{1},...,g_{J}(X)-\theta_{J})$ converges to a multivariate normal distribution, is it sufficient to show that covariance $cov(g_{k}(X),g_{j}(X))$ converges to some constant for every $k\neq j$?