On MV-Algebraic Versions of the Strong Law of Large Numbers.

Olgierd Hryniewicz,Piotr Nowak

doi:10.3390/e21070710

Abstract

Many-valued (MV; the many-valued logics considered by Łukasiewicz)-algebras are algebraic systems that generalize Boolean algebras. The MV-algebraic probability theory involves the notions of the state and observable, which abstract the probability measure and the random variable, both considered in the Kolmogorov probability theory. Within the MV-algebraic probability theory, many important theorems (such as various versions of the central limit theorem or the individual ergodic theorem) have been recently studied and proven. In particular, the counterpart of the Kolmogorov strong law of large numbers (SLLN) for sequences of independent observables has been considered. In this paper, we prove generalized MV-algebraic versions of the SLLN, i.e., counterparts of the Marcinkiewicz–Zygmund and Brunk–Prokhorov SLLN for independent observables, as well as the Korchevsky SLLN, where the independence of observables is not assumed. To this end, we apply the classical probability theory and some measure-theoretic methods. We also analyze examples of applications of the proven theorems. Our results open new directions of development of the MV-algebraic probability theory. They can also be applied to the problem of entropy estimation.

Highlights

MV-algebras, being generalizations of Boolean algebras, were introduced by Chang [1] and used in the analysis of many-valued logic
Carathéodory defined the basic notions of point-free probability, replacing Kolmogorovian probability measures on σ-algebras by strictly positive probability measures on σ-complete Boolean algebras and random variables, defined within the Kolmogorov probability theory as measurable functions on the event space Ω, by functions from the σ-algebra of Borel subsets of R into the σ-Boolean algebra of events
As we have mentioned before, Riečan proved the MV-algebraic version of the strong law of large numbers (SLLN) for independent observables satisfying the Kolmogorov condition. This is a counterpart of the classical Kolmogorov theorem for independent square-integrable random variables, which is important for the Kolmogorov probability theory

Summary

Introduction

MV-algebras, being generalizations of Boolean algebras, were introduced by Chang [1] and used in the analysis of many-valued logic. The third MV-algebraic version of the strong law of large numbers concerns the convergence of a -independent sequence { xi }i∈N of square-integrable weak observables, satisfying (K), under the additional assumption that the considered MV-algebra M is weakly σ-distributive. As we have mentioned before, Riečan proved the MV-algebraic version of the SLLN for independent observables satisfying the Kolmogorov condition This is a counterpart of the classical Kolmogorov theorem for independent square-integrable random variables, which is important for the Kolmogorov probability theory. We prove generalized versions of the SLLN, i.e., the Marcinkiewicz–Zygmund, Brunk–Prokhorov, and Korchevsky SLLN, within the MV-algebraic probability theory, applying their classical counterparts and some measure-theoretic methods.

Preliminaries

The Kolmogorov Probability Space of Observables

Generalized SLLN for Independent Observables

Generalized SLLN for Non-Negative Observables

Illustrative Examples

Sequence of Identically Distributed Observables

Sequence of Not Identically Distributed Observables

Sequence of Dependent Observables

Conclusions