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Central limit theorems of a recursive stochastic algorithm with applications to adaptive designs. (English) Zbl 1358.60044

Summary: Stochastic approximation algorithms have been the subject of an enormous body of literature, both theoretical and applied. Recently, S. Laruelle and G. Pagès [Ann. Appl. Probab. 23, No. 4, 1409–1436 (2013; Zbl 1429.62360)] presented a link between the stochastic approximation and response-adaptive designs in clinical trials based on randomized urn models investigated in [Z. D. Bai and F. Hu, Stochastic Processes Appl. 80, No.1, 87–101 (1999; Zbl 0954.62014); Ann. Appl. Probab. 15, No. 1B, 914–940 (2005; Zbl 1059.62111)], 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 [loc. cit.] and those of Bai and Hu [Zbl 1059.62111, loc. cit.] 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 of Bai and Hu [Zbl 0954.62014, loc. cit.; Zbl 1059.62111, loc. cit.] and Laruelle and Pagès [loc. cit.].

MSC:

60F05 Central limit and other weak theorems
62L20 Stochastic approximation
60G15 Gaussian processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60F15 Strong limit theorems
60F17 Functional limit theorems; invariance principles
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