Mathematics and Statistics Vol. 11(6), pp. 869 - 873
DOI: 10.13189/ms.2023.110601
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Some Convergence Properties of a Random Closed Set Sequence

Bourakadi Ahssaine ^1,*, Baraka Achraf Chakir ¹, Khalifi Hamid ^2
1 Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Kenitra, Morocco
² Department of Informatics, Faculty of Sciences, Mohammed V University, Rabat, Morocco

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.

KEYWORDS
Random Closed Set, Convergence in Probability, Weak Convergence, Convergence in the Sense of Hausdorff

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Bourakadi Ahssaine , Baraka Achraf Chakir , Khalifi Hamid , "Some Convergence Properties of a Random Closed Set Sequence," Mathematics and Statistics, Vol. 11, No. 6, pp. 869 - 873, 2023. DOI: 10.13189/ms.2023.110601.

(b). APA Format:
Bourakadi Ahssaine , Baraka Achraf Chakir , Khalifi Hamid (2023). Some Convergence Properties of a Random Closed Set Sequence. Mathematics and Statistics, 11(6), 869 - 873. DOI: 10.13189/ms.2023.110601.