Modifications of the $t$-test are considered which are robust under certain violations of the independence assumption. The additional number of observations these modified tests require in order to obtain under independence the same power as the $t$-test, is obtained asymptotically.
"Testing the Mean of a Normal Population Under Dependence." Ann. Statist. 6 (6) 1337 - 1344, November, 1978. https://doi.org/10.1214/aos/1176344378