Consider you have some distributions $Z_1$ and $Z_2$ of which you select $n_1$ samples from $Z_1$ and $n_2$ samples from $Z_2$.
We now add up these samples and take the mean. The question becomes, what is the distribution of $\mu$? Can this distribution be approximated via the CLT for sufficiently large values if $n$?
Normally this would be a standard CLT question if it involved just one distribution, but I am not sure how two different distributions with different sample sizes can effect this.
In case you want more specific examples of the problem, $Z_1$ and $Z_2$ are both uniform distributions on integers, just different ranges of integers. For example, $Z_1$ can be uniform selection of $\{1,2,3\}$ and $Z_2$ uniform selection of $\{4,5,6\}$