Asymptotic normality of the difference between the number of subjects assigned to a treatment and the desired number to be assigned is established for allocation rules which use Eisele's biased coin design. Subject responses are assumed to be independent random variables from standard exponential families. In the proof, it is shown that the difference may be magnified by appropriate constants so that the magnified difference is nearly a martingale. An application to the Behrens-Fisher problem in the normal case is described briefly.
"Central Limit Theorems for Doubly Adaptive Biased Coin Designs." Ann. Statist. 23 (1) 234 - 254, February, 1995. https://doi.org/10.1214/aos/1176324465