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

Systems Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, Poland
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Entropy 2019, 21(7), 710; https://doi.org/10.3390/e21070710
Received: 24 May 2019 / Revised: 4 July 2019 / Accepted: 16 July 2019 / Published: 19 July 2019
(This article belongs to the Section Information Theory, Probability and Statistics)
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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. View Full-Text
Keywords: Brunk–Prokhorov SLLN; Korchevsky SLLN; Marcinkiewicz–Zygmund SLLN; MV-algebraic probability Brunk–Prokhorov SLLN; Korchevsky SLLN; Marcinkiewicz–Zygmund SLLN; MV-algebraic probability
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Nowak, P.; Hryniewicz, O. On MV-Algebraic Versions of the Strong Law of Large Numbers. Entropy 2019, 21, 710.

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