Strong Laws of Large Numbers for IVM-Events

Abstract

Nonstandard probability theories have been developed for modeling random systems in complex spaces, such as, quantum systems. One of these theories, the MV-algebraic probability theory, involves the notions of state and observable, which were introduced by abstracting the properties of the Kolmogorovian probability measure and the classical random variable, as well as the notion of independence. Although within these nonstandard probability theories, many important theorems, including the strong law of large numbers (SLLN) for sequences of independent and identically distributed observables, have been considered, some practical applications require their further development. This paper is devoted to the development of the IVM-probability theory for the data described by interval-valued fuzzy random sets (IVM-events). The generalizations of Marcinkiewicz–Zygmund SLLN and Brunk–Prokhorov SLLN for independent IVM-events have been proved within this new theory. Our results open new possibilities in the theoretical analysis of imprecise random events in more complex spaces.