An Approach to the Construction of Inequivalent Models of Central Limit Theorem for Gaussianization of a Symmetric Probability Measure (Information and mathematics of non-additivity and non-extensivity : contacts with convex analysis)

For any symmetric probability measure $\nu$ on the real line $\mathbb{R}$ with finite moments of all orders such that $\nu$ is not finitely supported, we construct a large family of models of central limit theorem related to the 'Gaussianiza- tion' of measure $\nu$ in the sense of L. Accardi and M. Bozejko.The models are parametrized with infinitely many parameters $q=\{q_{n}\}_{n=2}^{\infty},$ $q_{n}\in(-1,1)$ , and constructed so that, for each $q$ , the central limit distribution of the model realizes the same measure $\nu$ , but that, for each pair of different values $q\neq q'$ , the two limit processes arising from the functional central limit (i.e.Brow- nian motions) are not stochastically equivalent to each other.Although the models do not explicitly contain the notion of 'independence' (for example, 'independence' as a universal calculation rule in the sense of R. Speicher), our result suggests that, in non-commutative probability theory, the correspondence from 'independence' to 'central limit distribution' is highly 'many to one'.Our result looks similar to the result of T. Cabanal-Duvillard and V. Ionescu, but our approach is different from theirs.