  Pocock and Simon's marginal procedure (Pocock and Simon, 1975) is often
implemented forbalancing treatment allocation over influential covariates in
clinical trials. However, the theoretical properties of Pocock and Simion's
procedure have remained largely elusive for decades. In this paper, we propose
a general framework for covariate-adaptive designs and establish the
corresponding theory under widely satisfied conditions. As a special case, we
obtain the theoretical properties of Pocock and Simon's marginal procedure: the
marginal imbalances and overall imbalance are bounded in probability, but the
within-stratum imbalances increase with the rate of $\sqrt{n}$ as the sample
size increases. The theoretical results provide new insights about balance
properties of covariate-adaptive randomization procedures and open a door to
study the theoretical properties of statistical inference for clinical trials
based on covariate-adaptive randomization procedures.
