On the t-Transformation of Free Convolution

The study of the stability of measure families under measure transformations, as well as the accompanying limit theorems, is motivated by both fundamental and applied probability theory and dynamical systems. Stability analysis tries to uncover invariant or quasi-invariant measures that describe the long-term behavior of stochastic or deterministic systems. Limit theorems, on the other hand, characterize the asymptotic distributional behavior of successively changed measures, which frequently indicate convergence to fixed points or attractors. Together, these studies advance our knowledge of measure development, aid in the categorization of dynamical behavior, and give tools for modeling complicated systems in mathematics and applied sciences. In this paper, the notion of the t-transformation of a measure and convolution is studied from the perspective of families and their relative variance functions (VFs). Using analytical and algebraic approaches, we aim to develop a deeper understanding of how the t-transformation shapes the behavior of probability measures, with possible implications in current probabilistic models. Based on the VF concept, we show that the free Meixner family ($\mathcal{FMF}$) of probability measures (the free equivalent of the Letac Mora class) remains invariant when t-transformation is applied. We also use the VFs to show some new limiting theorems concerning t-deformed free convolution and the combination of free and Boolean additive convolution.