Suppose $\bar{U}$ and $\bar{V}$ are sample means from a highly correlated or highly dependent random process (e.g., waiting times from a queueing process)
That is, let $$ \bar{U} = \frac{1}{n}\left(X_1+ X_2+ \dots+ X_n \right) $$ and $$ \bar{V} = \frac{1}{m}\left(X_1+ X_2+ \dots+ X_m \right) $$ where $m \gg n$; that is $m$ is much larger than $n$ ($m$ could be infinite),
$\bf{QUESTION:}$Is it the case, in general, that $\bar{U} < \bar{V}$? (since $n < m$).
I have simulated the process and $\bar{U} < \bar{V}$ appears to be the true.
I have no idea how to show this analytically/mathematically...