Abstract
Baoding Liu created uncertainty theory to describe the information represented by human language. In turn, Yuhan Liu founded chance theory for modelling phenomena where both uncertainty and randomness are present. The first theory involves an uncertain measure and variable, whereas the second one introduces the notions of a chance measure and an uncertain random variable. Laws of large numbers (LLNs) are important theorems within both theories. In this paper, we prove a law of large numbers (LLN) for uncertain random variables being continuous functions of pairwise independent, identically distributed random variables and regular, independent, identically distributed uncertain variables, which is a generalisation of a previously proved version of LLN, where the independence of random variables was assumed. Moreover, we prove the Marcinkiewicz–Zygmund type LLN in the case of uncertain random variables. The proved version of the Marcinkiewicz–Zygmund type theorem reflects the difference between probability and chance theory. Furthermore, we obtain the Chow type LLN for delayed sums of uncertain random variables and formulate counterparts of the last two theorems for uncertain variables. Finally, we provide illustrative examples of applications of the proved theorems. All the proved theorems can be applied for uncertain random variables being functions of symmetrically or asymmetrically distributed random variables, and symmetrical or asymmetrical uncertain variables. Furthermore, in some special cases, under the assumption of symmetry of the random and uncertain variables, the limits in the first and the third theorem have forms of symmetrical uncertain variables.
Highlights
The classical probability theory was developed by Kolmogorov
In [28], Laws of large numbers (LLNs) was proved for uncertain random variables being functions of independent, identically distributed random variables and independent, identically distributed regular uncertain variables, in the version corresponding to the classical Kolmogorov Theorem
We prove an LLN for uncertain random variables being functions of pairwise, independent, identically distributed random variables and independent, identically distributed regular uncertain variables
Summary
The classical probability theory was developed by Kolmogorov. It is a field of mathematics that is used to address practical problems in which randomness naturally occurs. The properties of uncertain measures, including normality, self-duality, monotonicity, countable subadditivity, and fulfilling product axiom (see [15]), have made the uncertainty theory a part of mathematics, different from classical probability This theory has provided the notion of an uncertain variable, describing uncertain quantities. Basic versions of the law of large numbers were proved in [17] for fuzzy random variables defined by Kwakernaak and in [18] in the case where the definition proposed by Puri and Ralescu was used. In [28], LLN was proved for uncertain random variables being functions of independent, identically distributed random variables and independent, identically distributed regular uncertain variables, in the version corresponding to the classical Kolmogorov Theorem. Classical probabilistic versions of the proved theorems are presented in Appendix A
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