2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/77519We interpret the Central Limit Theorem as a fixed point theorem for a certain operator, and consider the problem of linearizing this operator. In classical as well as in free probability theory, we consider two methods giving such a linearization, and interpret the result as a weak form of the CLT. In the classical case the analysis involves dilation operators; in the free case more general composition operators appear.13 pages; minor revisions; to be published in Probability Theory and Related FieldsOperator AlgebrasProbability46L50 (primary), 60F05, 47B38 (secondary)text