Suppose $\{X\_n\}$ and $\{Y\_n\}$ are two sequences of random variables and we know that $X\_n \overset{L^2}{\to} X$ and $Y\_n \overset{L^2}{\to} Y$, where $\overset{L^2}{\to}$ means converge in mean square sense. In addition, we know that $X$ and $Y$ are **independent** random variables, and both have finite mean and variance. I'm interested in showing
$$\lim\_{n \to +\infty} Cov(X\_n, Y\_n) = 0$$


I know that the above is not always true (e.g., see [this Math StackExchange thread](https://math.stackexchange.com/questions/1291086/independence-of-random-variables-and-covariance-in-the-limit)). However, I wonder **what other conditions can one impose that will make it true?**



Covariance in the limit of random variables
First off $\mathbf{E}[X\_n] \to \mathbf{E}[X]$ and $\mathbf{E}[Y\_n] \to \mathbf{E}[Y]$ as $n \to \infty$, whence also $\mathbf{E}[X\_n] \mathbf{E}[Y\_n] \to \mathbf{E}[X] \mathbf{E}[Y]$. Via Holder's inequality one additionally has $\mathbf{E}[X\_n Y\_n] - \mathbf{E}[XY] = \mathbf{E}[X\_n (Y\_n - Y)] + \mathbf{E}[Y(X\_n - X)] \to 0$.


