Distribuciones de probabilidad de elementos aleatorios imprecisos

Abstract

Fuzzy random variables are used to model situations in which randomness is accompanied by imprecision. Regarded as random elements of the space of fuzzy sets, it is possible to study their properties within more general frameworks, among which are spaces of convex combinations, structures composed of a metric space and a convex combination operation. The first chapters of this work are dedicated to the study of convex combination spaces and, in particular, random elements that take values in them. We will extend the Bartels-Pallaschke metric to metric convex cones and study its properties, focusing on the cases of compact and fuzzy sets, where we will also analyze its relationship with other metrics defined in them. We will develop general versions of Vitali's Theorem and the Dominated Convergence Theorem and apply them to random sets and fuzzy random variables. Next, we will study the continuity and measurability of mappings induced by fuzzy sets and fuzzy random variables. We will also study the embedding of the space of pairs of fuzzy sets in the higher-dimensional space of fuzzy sets, a result that will be useful to study the properties of pairs of fuzzy random variables. Furthermore, we will use the properties of the BartelsPallaschke metric to obtain a strong law of large numbers for fuzzy random numbers. We will analyze the properties of convergence in the distribution of fuzzy random variables through the general definition for metric spaces. From these results, we will derive two versions of Slutsky's Theorem. Finally, we will introduce two independence tests for fuzzy random variables based on both distance correlation statistics and perform simulation studies to analyze their suitability.