Suppose that we have two random variables $X$ and $Y$ s.t. for every value of $\left\{i\in\mathbb{Z}:1\leq i\leq n\right\}$, there exists a tuple $\left(X_{i},Y_{i}\right)$.
Let the Monte Carlo estimation for the $\mathbb{E}\left[X\right]$ and $\mathbb{E}\left[Y\right]$ be $\bar{X}$ and $\bar{Y}$ respectively, i.e. $$\bar{X}=\frac{1}{n}\sum_{i=1}^{n}X_{i}$$ $$\bar{Y}=\frac{1}{n}\sum_{i=1}^{n}Y_{i}$$
How do I express $\operatorname{cov}\left(\bar{X},\bar{Y}\right)$ in terms of $\operatorname{cov}\left(X,Y\right)$? Are they equal?
I assume that $X_i$ are iid. Likewise for $Y_i$. Then, $$ \eqalign{ \operatorname{cov}(\bar{X},\bar{Y})&=\mathbb{E}\left[\bar{X}\bar{Y}\right]-\mathbb{E}\left[\bar{X}\right]\mathbb{E}\left[\bar{Y}\right]\\ &=\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^{n}X_{i}\,\frac{1}{n}\sum_{j=1}^{n}Y_{j}\right] -\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^{n}X_{i}\right]\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^{n}Y_{i}\right]\\ &=\mathbb{E}\left[\frac{1}{n^2}\sum_{i=1}^{n}X_{i}Y_{i}\,+\frac{1}{n^2}\sum_{i\ne j}X_{i}Y_{j}\right] -\frac{n}{n}\mathbb{E}\left[X\right]\,\frac{n}{n}\mathbb{E}\left[Y\right]\\ &=\frac{n}{n^2}\mathbb{E}\left[XY\right]+\frac{n^2-n}{n^2}\mathbb{E}\left[X\right]\mathbb{E}\left[Y\right] -\frac{n}{n}\mathbb{E}\left[X\right]\,\frac{n}{n}\mathbb{E}\left[Y\right]\\ &=\frac{1}{n}\mathbb{E}\left[XY\right]-\frac{1}{n}\mathbb{E}\left[X\right]\mathbb{E}\left[Y\right]\\ &=\frac{1}{n}\operatorname{cov}(X,Y) } $$