Some Convergence Properties of a Random Closed Set Sequence

Abstract

In this article, we have discussed the properties of the probability law "T" called functional capacity and other closely related functionals "Q and C" pertaining to random closed sets.We are interested in the most widely used functional in random set theory "T".We have established the belonging of "T" to the interval [0,1], and proven that it is increasing in the sense of inclusion, and its sub-additivity property through probability techniques.Moreover, we have explored the various types of convergences of a sequence of random closed sets, such as weak convergence, strong convergence (almost surely in the sense of Hausdorff), convergence in the sense of Painlevé -Kuratowski and Wijsman-Mosco, as well as convergence in probability.In the second part of our work, we have proven a new corollary which states that the strong convergence in the sense of Hausdorff implies the convergence in probability of a sequence of random closed sets at infinity.Our proof involves the definition of mathematical expectation for a discrete variable and the indicator variable, which is a random variable that takes two possible values, 0 or 1.