  Stochastic approximation algorithms have been the subject of an enormous body
of literature, both theoretical and applied. Recently, Laruelle and Pag\`es
(2013) presented a link between the stochastic approximation and
response-adaptive designs in clinical trials based on randomized urn models
investigated in Bai and Hu (1999, 2005), and derived the asymptotic normality
or central limit theorem for the normalized procedure using a central limit
theorem for the stochastic approximation algorithm. However, the classical
central limit theorem for the stochastic approximation algorithm does not
include all cases of its regression function, creating a gap between the
results of Laruelle and Pag\`es (2013) and those of Bai and Hu (2005) for
randomized urn models. In this paper, we establish new central limit theorems
of the stochastic approximation algorithm under the popular Lindeberg condition
to fill this gap. Moreover, we prove that the process of the algorithms can be
approximated by a Gaussian process that is a solution of a stochastic
differential equation. In our application, we investigate a more involved
family of urn models and related adaptive designs in which it is possible to
remove the balls from the urn, and the expectation of the total number of balls
updated at each stage is not necessary a constant. The asymptotic properties
are derived under much less stringent assumptions than those in Bai and Hu
(1999, 2005) and Laruelle and Pag\`es (2013).
