Generalized versions of MV-algebraic central limit theorems

Piotr Nowak; Olgierd Hryniewicz

Kybernetika (2015)

  • Volume: 51, Issue: 5, page 765-783
  • ISSN: 0023-5954

Abstract

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MV-algebras can be treated as non-commutative generalizations of boolean algebras. The probability theory of MV-algebras was developed as a generalization of the boolean algebraic probability theory. For both theories the notions of state and observable were introduced by abstracting the properties of the Kolmogorov's probability measure and the classical random variable. Similarly, as in the case of the classical Kolmogorov's probability, the notion of independence is considered. In the framework of the MV-algebraic probability theory many important theorems (as the individual ergodic theorem and the laws of large numbers for observables) were proved. In particular, the central limit theorem (CLT) for sequences of independent and identically distributed observables was considered. In this paper, for triangular arrays of independent, not necessarily identically distributed observables of MV-algebras, we have proved the Lindeberg and the Lyapunov central limit theorems, and the Feller theorem. To show that the generalization proposed by us is essential, we discuss examples of applications of the proved MV-algebraic versions of theorems.

How to cite

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Nowak, Piotr, and Hryniewicz, Olgierd. "Generalized versions of MV-algebraic central limit theorems." Kybernetika 51.5 (2015): 765-783. <http://eudml.org/doc/276226>.

@article{Nowak2015,
abstract = {MV-algebras can be treated as non-commutative generalizations of boolean algebras. The probability theory of MV-algebras was developed as a generalization of the boolean algebraic probability theory. For both theories the notions of state and observable were introduced by abstracting the properties of the Kolmogorov's probability measure and the classical random variable. Similarly, as in the case of the classical Kolmogorov's probability, the notion of independence is considered. In the framework of the MV-algebraic probability theory many important theorems (as the individual ergodic theorem and the laws of large numbers for observables) were proved. In particular, the central limit theorem (CLT) for sequences of independent and identically distributed observables was considered. In this paper, for triangular arrays of independent, not necessarily identically distributed observables of MV-algebras, we have proved the Lindeberg and the Lyapunov central limit theorems, and the Feller theorem. To show that the generalization proposed by us is essential, we discuss examples of applications of the proved MV-algebraic versions of theorems.},
author = {Nowak, Piotr, Hryniewicz, Olgierd},
journal = {Kybernetika},
keywords = {MV-algebra; MV-algebraic probability; central limit theorem},
language = {eng},
number = {5},
pages = {765-783},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Generalized versions of MV-algebraic central limit theorems},
url = {http://eudml.org/doc/276226},
volume = {51},
year = {2015},
}

TY - JOUR
AU - Nowak, Piotr
AU - Hryniewicz, Olgierd
TI - Generalized versions of MV-algebraic central limit theorems
JO - Kybernetika
PY - 2015
PB - Institute of Information Theory and Automation AS CR
VL - 51
IS - 5
SP - 765
EP - 783
AB - MV-algebras can be treated as non-commutative generalizations of boolean algebras. The probability theory of MV-algebras was developed as a generalization of the boolean algebraic probability theory. For both theories the notions of state and observable were introduced by abstracting the properties of the Kolmogorov's probability measure and the classical random variable. Similarly, as in the case of the classical Kolmogorov's probability, the notion of independence is considered. In the framework of the MV-algebraic probability theory many important theorems (as the individual ergodic theorem and the laws of large numbers for observables) were proved. In particular, the central limit theorem (CLT) for sequences of independent and identically distributed observables was considered. In this paper, for triangular arrays of independent, not necessarily identically distributed observables of MV-algebras, we have proved the Lindeberg and the Lyapunov central limit theorems, and the Feller theorem. To show that the generalization proposed by us is essential, we discuss examples of applications of the proved MV-algebraic versions of theorems.
LA - eng
KW - MV-algebra; MV-algebraic probability; central limit theorem
UR - http://eudml.org/doc/276226
ER -

References

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