Let $S_n(\psi)$ be a uniformly bounded statistic that depends on a real-valued random variable $\psi$, whose the distribution $F_{\psi}$ is known. I established using the Central Limit Theorem that for all $\psi$, $$ S_n(\psi) \overset{d}{\longrightarrow} \mathcal{N}(0,\Sigma)$$ Can I conclude that $$ \int S_n(\psi) dF_{\psi}\overset{d}{\longrightarrow} \mathcal{N}\Bigg(0,\lim_{n \to \infty}\mathbf{V}\Big(\int S_n(\psi) dF_{\psi}\Big)\Bigg)$$ For example, when one uses Monte Carlo approximation for the integral it is possible.
$$ \dfrac{1}{K}\sum_{k=1}^K S_n(\psi_k) \overset{d}{\longrightarrow} \mathcal{N}\Bigg(0,\lim_{n \to \infty}\mathbf{V}\Big(\dfrac{1}{K}\sum_{k=1}^K S_n(\psi_k)\Big)\Bigg)$$ where $\psi_1,\dots,\psi_k$ are independently drawn from $F_{\psi}$. But how can I prove the result for the integral. Or is it wrong?