  In this paper, we provide some general convergence results for adaptive
designs for treatment comparison, both in the absence and presence of
covariates. In particular, we demonstrate the almost sure convergence of the
treatment allocation proportion for a vast class of adaptive procedures, also
including designs that have not been formally investigated but mainly explored
through simulations, such as Atkinson's optimum biased coin design, Pocock and
Simon's minimization method and some of its generalizations. Even if the large
majority of the proposals in the literature rely on continuous allocation
rules, our results allow to prove via a unique mathematical framework the
convergence of adaptive allocation methods based on both continuous and
discontinuous randomization functions. Although several examples of earlier
works are included in order to enhance the applicability, our approach provides
substantial insight for future suggestions, especially in the absence of a
prefixed target and for designs characterized by sequences of allocation rules.
