Nonlinear limit theorems

Motivated by “multi-armed bandit” problem and reinforcement learning, in this paper, we introduce a similar binary model in the context of nonlinear probabilities. This can be viewed as a nonlinear Bernoulli-like model and is motivated in modelling distribution uncertainties. It provides a new probabilistic understanding of the nonlinear probability theory. In one main result we obtain a generalized robust limit theorem for this model with mean-variance uncertainty, and give an explicit formula for the robust limit distribution. The limit is shown to depend heavily on the structure of the events or the integrating functions, which demonstrate the key signature of nonlinear structure. As applications, these limit theorems provide the theoretical foundation for statistical inferences and hypothesis testing.