Central limit theorems of a recursive stochastic algorithm with applications to adaptive designs

Stochastic approximation algorithms have been the subject of an enormous body of literature, both theoretical and applied. Recently, Laruelle and Pagès [Ann. Appl. Probab. 23 (2013) 1409.1436] presented a link between the stochastic approximation and response-adaptive designs in clinical trials based on randomized urn models investigated in Bai and Hu [Stochastic Process. Appl. 80 (1999) 87.101; Ann. Appl. Probab. 15 (2005) 914.940], 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 Pagès [Ann. Appl. Probab. 23 (2013) 1409.1436] and those of Bai and Hu [Ann. Appl. Probab. 15 (2005) 914.940] 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 [Stochastic Process. Appl. 80 (1999) 87.101; Ann. Appl. Probab. 15 (2005) 914.940] and Laruelle and Pagès [Ann. Appl. Probab. 23 (2013) 1409.1436].