$M$-structure: A nonlinear probabilistic structure with symmetric independence under model uncertainty

Model uncertainty is a long-existing fundamental concern in various areas. The nonlinear expectation framework offers a profound theoretical view for discussing model uncertainty, presenting a non-trivial generalization of the classical probability system. However, when discussing the connection with real-world problems, we need to be more cautious about the interpretation of many basic concepts under nonlinear expectation, such as normal distribution and independence. Our initial motivation to introduce the M-structure is to have a structure that preserves statistical flexibility, so that it can serve as a bridge to make nonlinear expectation more accessible to a broader audience and also provide a fresh perspective on model uncertainty. The flexibility of this structure can be elaborated as follows. Regarding independence, M-independence preserves symmetry, capturing model uncertainty with spatial or temporal symmetry, and is related to G-independence, which primarily addresses model uncertainty with temporal asymmetry. We establish a central limit 定理 under the M-structure, describing the asymptotic distribution under spatiotemporal symmetry, which is the M-normal distribution. Notably, the M-normal distribution preserves a direct link between univariate and multivariate distributions under independence, a 性质 lacking in the G-normal distribution. More importantly, the M-structure uncovers the implicit connection and the source of distinctions between concepts under classical and nonlinear expectations. This foundational insight brings a far-reaching advantage for future developments in both theoretical and practical contexts.