The Monge–Ampère equation in transformation theory and an application to 1α$\frac{1}{\alpha }$-probabilities

In this article, we give an application of the theory of Monge–Ampère equations to transformation theory in probability. Before starting our analysis, we recall the definition of 1α$\frac{1}{\alpha }$-characteristic function, introduced in Selvitella1 and Selvitella2 as a natural generalization of the classical characteristic function. The main novelty of this tool is that it permits to extend classical theorems, such as the law of large numbers (LLN) and the central limit theorem (CLT) to basically every distribution, upon the correct choice of a free parameter α. The motivation of this article is to show that the transformation from a pdf to its 1α$\frac{1}{\alpha }$-counterpart is transferred also at the level of the random variables. We treat explicitly the cases of the multivariate normal distribution, multivariate exponential distribution, and Cauchy distribution. Moreover, we prove some rigidity theorems on the possible transformations which send a pdf to its 1α$\frac{1}{\alpha }$-counterpart. In the general case, it is not possible to construct explicitly the transformation between the random variables, despite it is always possible to reduce to quadrature the transformation between the pdfs. We conclude the article by setting our theorems in the context of the Monge–Ampère equations and the optimal transportation theory and by giving some numerics to illustrate the motion of mass, while transforming a pdf to its 1α$\frac{1}{\alpha }$-counterpart.