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We introduce a new basic model for independent and identical distributedsequence on the canonical space$(\mathbb{R}^\mathbb{N},\mathcal{B}(\mathbb{R}^\mathbb{N}))$ via probabilitykernels with model uncertainty. Thanks to the well-defined upper and lowervariances, we obtain a new functional central limit theorem withmean-uncertainty by the means of martingale central limit theorem and stabilityof stochastic integral in the classical probability theory. Then we extend itfrom the canonical space to the general sublinear expectation space. Thecorresponding proofs are purely probabilistic and do not rely on the nonlinearpartial differential equation.

On the functional central limit theorem with mean-uncertainty