# Covariance between two sample means of correlated data

I have two sets of random data $X=\{x_1,...,x_N\}$ and $Y\{y_1,...,y_N\}$ both of length $N$. The sets are autocorrelated such that the correlation between $x_i$ and $x_j$ depends only on $|i-j|$. From both of these I can find the sample mean, $$\bar{X} = \frac{1}{N}\sum_{i=1}^N x_i$$ and similarly for $Y$. I believe I can find the variance of each mean as follows, $$\text{var}(\bar{X}) = \frac{S^2}{N}\frac{N-1}{\frac{N}{\gamma_2}-1}$$ where $S^2 = \frac{1}{N}\sum_{i=1}^N(x_i-\bar{X})^2$ and $\gamma_2 = 1+2\sum_{j=1}^{N-1}(1-j/N)\rho_j$, with $\rho_j$ being the autocorrelation function at a lag $j$. My question is this. What is $\text{cov}(\bar{X},\bar{Y})$ if the data sets are also correlated? The correlation between the sets should be the same for all pairs i.e. $\text{cov}(x_i,y_i) = \text{cov}(x_j,y_j)$. Will it be some generalisation of $\text{var}(\bar{X})$ or can I calculate it from the variances of the means and the correlation alone?