A generalization of probability theory on MV-algebras to IF-events

Renčová, Magdaléna

doi:10.1016/j.fss.2009.11.009

Cited by 9 publications

(4 citation statements)

References 4 publications

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“…Other results of probability theories concerning IF-events can be found, e.g. in Ciungu and Riečan (2010), Lendelová (2006), Lendelová and Petrovičová (2006), Renčová (2010), Riečan (2004Riečan ( , 2006aRiečan ( , b, 2007a. Generalized versions of central limit theorems within the IF-probability theory and Mprobability theory for non-identically distributed observables were proved in the recent paper by Nowak and Hryniewicz (2016).…”

Section: Introductionmentioning

confidence: 92%

On central limit theorems for IV-events

Nowak

Hryniewicz

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: Introductionmentioning

confidence: 92%

On central limit theorems for IV-events

Nowak

Hryniewicz

2017

Soft Comput

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“…The details can be found in [66]. Many applications of the method has been described in [25], [31], [35], [37], [39] , [52].…”

Section: Theorem 42mentioning

confidence: 99%

Analysis of Fuzzy Logic Models

Riečan¹

2012

Intelligent Systems

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Example 1.1. Let F be the set of all fuzzy subsets of a set Ω.I ff : Ω → [0, 1] then we define A =(f ,1− f), i.e. ν A = 1 − μ A. Example 1.2. Let (Ω, S) be a measurable space, S a σ-algebra, F the family of all pairs such that μ A : Ω → [0, 1], ν A : Ω → [0, 1] are measurable. Then F is closed under the operations ⊕, ⊙, ¬. Example 1.3. Let (Ω, T) be a topological space, F the family of all pairs such that μ A : Ω → [0, 1], ν A : Ω → [0, 1] are continuous. Then F is closed under the operations⊕, ⊙, ¬. Remark. Of course, in any case A ⊕ B, A ⊙ B, ¬A are IF-sets, if A, B are IF-sets. E.g.

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“…The MV-algebraic probability theory was also applied in the Atanassov intuitionistic fuzzy sets and interval-valued fuzzy sets settings (see, e.g., [ 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 ]). The notion of probability for the Atanassov intuitionistic fuzzy sets was introduced by Szmidt and Kacprzyk [ 19 ].…”

Section: Introductionmentioning

confidence: 99%

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

Nowak

Hryniewicz

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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A generalization of probability theory on MV-algebras to IF-events

Cited by 9 publications

References 4 publications

On central limit theorems for IV-events

On central limit theorems for IV-events

Analysis of Fuzzy Logic Models

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

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