  We introduce a new basic model for independent and identical distributed
sequence on the canonical space
$(\mathbb{R}^\mathbb{N},\mathcal{B}(\mathbb{R}^\mathbb{N}))$ via probability
kernels with model uncertainty. Thanks to the well-defined upper and lower
variances, we obtain a new functional central limit theorem with
mean-uncertainty by the means of martingale central limit theorem and stability
of stochastic integral in the classical probability theory. Then we extend it
from the canonical space to the general sublinear expectation space. The
corresponding proofs are purely probabilistic and do not rely on the nonlinear
partial differential equation.
