I have two dependent samples $\mathbf{X}$ and $\mathbf{Y}$ of $s$ and $t$ observations, respectively, that is $$ \mathbf{X} = \{X_1, \dots, X_s\} \;, \\ \mathbf{Y} = \{Y_1, \dots, Y_t\} \;. $$ For example, you could imagine that there is a random variable Z and functions $f, g$ such that $\mathbf{X} = f(Z), \mathbf{Y} = g(Z)$.
My quantities of interest are the sample means $\bar{X}$ and $\bar{Y}$ of $\mathbf{X}$ and $\mathbf{Y}$, respectively.
Is there a standard estimate for the covariance $\mathrm{Cov}[\bar{X}, \bar{Y}]$ of the sample means?
