Let $(X_n(\theta))_{n \geq 1}$ be a sequence of random variables with value in $\mathbb{R}^q$ indexed by a parameter $\theta \in \Theta \subset \mathbb{R}^q$. Suppose that for all $\theta \in \Theta$: \begin{align*} X_n(\theta) \longrightarrow \mathcal{N}(0,1) \quad \text{($\star$)} \end{align*} in distribution as $n \rightarrow \infty$.
Now let $\nu$ be a probability measure such that $\nu(\Theta) = 1$. If we let $\theta \sim \nu$, does ($\star$) still hold?