Suppose there are two i.i.d. sequences $\left\{X_n\right\}$ and $\left\{Y_t\right\}$ that \begin{align*} \sqrt{N}\left(\bar{X}_N-\mathrm{E}\left[X_n\right]\right)\overset{d}{\rightarrow}&\mathcal{N}\left(0,\mathrm{Var}\left[X_n\right]\right)\\ \sqrt{T}\bar{Y}_T\overset{d}{\rightarrow}&\mathcal{N}\left(0,\mathrm{E}\left[Y_t^2\right]\right) \end{align*} and they are independent. Can I infer anything about \begin{align*} \frac{1}{\sqrt{NT}}\sum_{i=1}^{N}{\sum_{t=1}^{T}{X_nY_t}}, \end{align*} i.e. the product of two sequences that converge in distribution? I think I cannot apply Slutsky's Theorem since both converge not in probability, but in distribution.

Here I am assuming the existence of the moments $\mathrm{E}\left[X_n\right]$, $\mathrm{E}\left[X_n^2\right]$, $\mathrm{E}\left[Y_t\right]$ and $\mathrm{E}\left[Y_t^2\right]$, but, if it is necessary, then I can add the assumption that both $X$ and $Y$ are normally distributed.

By the way, I tried Monte Carlo Simulation and it seems the product follows a leptokurtic distribution that is similar to Laplace Distribution.