Convergence in distribution for sum of products of random variables

Question

Given a martingale difference sequence $X_i$ with $\frac{1}{\sqrt n}\sum_{i=1}^n X_i \implies N(0,\Sigma)$, where $\implies$ denotes convergence in distribution, and $\Sigma$ is some covariance matrix. Let $Y_i$ be a sequence of i.i.d. random variables with mean 0, variance 1, independent of $X_i$. Is it true that $\frac{1}{\sqrt n}\sum_{i=1}^n X_i Y_i \implies N(0,\Sigma)$?