Sums of realizations of dependent variables become independent

Question

I have empirically noticed and interesting phenomenon. Suppose we have two continuous random variables $X$ and $Y$ which are dependent but not correlated. For instance:

$X \sim \mathcal{N(0,1)}$

$Y = cos(X) +Z$,$~~~~$ where $Z\sim \mathcal{N(0,0.5)}$

Here you can see some samples generated for these variables. I use this implementation of distance correlation as a measure for dependence:

Now consider two additional random variables $X_{sum}$ and $Y_{sum}$ generated as the summation of $n$ realizations of the original variables respectively. The thing I have noticed is that $X_{sum}$ and $Y_{sum}$ become more and more independent as $n$ grows (click on the image to enlarge):

I know that by the Central Limit Theorem $X_{sum}$ and $Y_{sum}$ will be approximately normal, but why are they becoming independent? Is there a general explanation for this?. Also, I have noticed that this phenomenon only occurs if $X$ and $Y$ are not correlated.

Thanks in advance.

Answer 1

The multidimensional CLT implies that $$\sqrt{n} \begin{bmatrix} \frac{1}{n} X_{sum} - \mathbb{E} X \\ \frac{1}{n} Y_{sum} - \mathbb{E} Y \end{bmatrix}$$ is approximately bivariate normal with mean zero and covariance matrix $$\begin{bmatrix}\text{Var}(X) & \text{Cov}(X,Y) \\ \text{Cov}(X,Y) & \text{Var}(Y)\end{bmatrix}.$$ Since your $X$ and $Y$ are uncorrelated, we have $\text{Cov}(X_{sum}/\sqrt{n}, Y_{sum}/\sqrt{n}) \approx 0$.

I am not sure how to quantify what happens when you scale up by $\sqrt{n}$ to consider $\text{Cov}(X_{sum}, Y_{sum})$; you may need a Berry-Esseen type of non-asymptotic result.