Generalized versions of MV-algebraic central limit theorems

2017

Soft Comput

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Interval-valued fuzzy sets were introduced in 1970s as an extension of Zadeh's fuzzy sets. For intervalvalued fuzzy events, (IV-events for short) IV-probability theory has been developed. In this paper, we prove central limit theorems for triangular arrays of IV-observables within this theory. We prove the Lindeberg CLT and the Lyapunov CLT, assuming that IV-observables are not necessary identically distributed. We also prove the Feller theorem for null arrays of IV-observables. Furthermore, we present examples of applications of the aforementioned theorems. In particular, we study the convergence in distribution of scaled sums of identically distributed IV-observables.

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Section: A Function F (Not Necessarily Non-negative) Is Integrable Wimentioning

confidence: 99%

“…We recall generalized versions of MV-algebraic central limit theorems and the Feller theorem proved in Nowak and Hryniewicz (2015).…”

Section: A Function F (Not Necessarily Non-negative) Is Integrable Wimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On central limit theorems for IV-events

2017

Soft Comput

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“…The third MV-algebraic version of the strong law of large numbers concerns the convergence of a ⊙-independent sequence

of square-integrable weak observables, satisfying (K), under the additional assumption that the considered MV-algebra M is weakly

-distributive. Generalized versions of the MV-algebraic central limit theorem, i.e., the Lindeberg and Lyapunov CLT, as well as the Feller theorem were proven by Nowak and Hryniewicz [ 10 ]. It is important to underline that, similar to in the Kolmogorov probability theory, the MV-algebraic versions of central limit theorem and strong law of large numbers are different types of theorems, since they concern different types of convergence of scaled sums of observables, i.e., the convergence in distribution in the first case and the convergence m-almost everywhere in the second case.…”

Section: Introductionmentioning

confidence: 99%

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

2019

Entropy

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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.

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Strong Laws of Large Numbers for IVM-Events

IEEE Trans. Fuzzy Syst.

2019