In this paper, we introduce a fundamental model for independent and
identically distributed sequence with model uncertainty on the canonical space
(RN,B(RN)) via probability
kernels. Thanks to the well-defined upper and lower variances, we obtain a new
functional central limit theorem with mean-uncertainty on the canonical space
by the method based on the martingale central limit theorem and stability of
stochastic integral in the classical probability theory. Then we extend it to
the general sublinear expectation space through a new representation theorem.
Our results generalize Peng's central limit theorem with zero-mean to the case
of mean-uncertainty and provides a purely probabilistic proof instead of the
existing nonlinear partial differential equation approach. As an application,
we consider the two-armed bandit problem and generalize the corresponding
central limit theorem from the case of mean-certainty to mean-uncertainty.Comment: 31 pages. arXiv admin note: substantial text overlap with
arXiv:2203.0017