Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
MV-algebra; MV-algebraic probability; central limit theorem
MV-algebra; MV-algebraic probability; central limit theorem
Summary:
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.
References:
[1] Athreya, K. B., Lahiri, S. N.: Measure Theory and Probability Theory. Springer-Verlag, Heidelberg 2006. DOI 10.1007/978-0-387-35434-7 | MR 2247694 | Zbl 1130.60001
[2] Billingsley, P.: Probability and Measure. Second edition. Wiley Press, New York 1986. MR 0830424
[3] Birkhoff, G., Neumann, J. Von: The logic of quantum mechanics. Ann. Math. 37 (1936), 823-843. DOI 10.2307/1968621 | MR 1503312
[4] Carathéodory, C.: Mass und Integral und ihre Algebraisierung. Birkäuser, Boston 1956. DOI 10.1007/978-3-0348-6948-5 | MR 0079628 | Zbl 0074.04003
[5] Chang, C. C.: Algebraic Analysis of Many Valued Logics. Trans. Amer. Math. Soc. 88 (1958), 2, 467-490. DOI 10.2307/1993227 | MR 0094302 | Zbl 0084.00704
[6] Chovanec, F.: States and observables on MV algebras. Tatra Mountains Mathematical Publications 3 (1993), 55-65. MR 1278519 | Zbl 0799.03074
[7] Cignoli, R., D'Ottaviano, I., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning. Kluwer Academic Publishers, Dordrecht 2000. DOI 10.1007/978-94-015-9480-6 | MR 1786097 | Zbl 0937.06009
[8] Dvurečenskij, A., Chovanec, F.: Fuzzy quantum spaces and compatibility. Int. J. Theoret. Physics 27 (1988), 1069-1082. DOI 10.1007/bf00674352 | MR 0967421 | Zbl 0657.60004
[9] Gudder, S.: Stochastic Methods of Quantum Mechanics. Elsevier, North-Holland 1979. MR 0543489
[10] Łukasiewicz, J., Tarski, A.: Untersuchungen über den Aussagenkalkül. Comptes Rendus des séances de la Société des Sciences et des Lettres de Varsovie, Classe III 23 (1930), 30-50.
[11] Mesiar, R.: Fuzzy observables. J. Math. Anal. Appl. 174 (1993), 178-193. DOI 10.1006/jmaa.1993.1109 | MR 1212925 | Zbl 0777.60005
[12] Mesiar, R.: Fuzzy sets, difference posets and MV-algebras. In: Fuzzy Logic and Soft Computing (B. Bouchon-Meunier, R. R. Yager and L. A. Zadeh, eds.), World Scientific, Singapore 1995, pp. 345-352. MR 1391013 | Zbl 0948.03059
[13] Mundici, D.: Interpretation of AFC*-algebras in Lukasiewicz sentential calculus. J. Funct. Anal. 65 (1986), 15-63. DOI 10.1016/0022-1236(86)90015-7 | MR 0819173
[14] Mundici, D.: Logic of infinite quantum systems. Int. J. Theor. Physics 32 (1993), 1941-1955. DOI 10.1007/BF00979516 | MR 1255397 | Zbl 0799.03019
[15] Mundici, D.: Advanced Lukasiewicz Calculus and MV-algebras. Springer, New York 2011. MR 2815182 | Zbl 1235.03002
[16] Nowak, P., Gadomski, J.: Deterministic properties of serially connected distributed lag models. Oper. Res. Decis. 23 (2013), 3, 43-55. MR 3236441
[17] Piasecki, K.: On the Bayes formula for fuzzy probability measures. Fuzzy Sets and Systems 18 (1986), 2, 183-185. DOI 10.1016/0165-0114(86)90020-5 | MR 0828642 | Zbl 0656.60011
[18] Pták, P., Pulmannová, S.: Kvantové logiky (in Slovak). Veda, Bratislava 1989. MR 1176313
[19] Pulmannová, S.: A note on observables on MV-algebras. Soft Computing 4 (2000), 45-48. DOI 10.1007/s005000050081 | Zbl 1005.06006
[20] Pykacz, J.: Quantum logics as families of fuzzy subsets of the set of physical states. In: Preprints of the Second IFSA Congress, Tokyo 1987, pp. 437-440.
[21] Pykacz, J.: Fuzzy set description of physical systems and their dynamics. Busefal 38 (1989), 102-107.
[22] Riečan, B.: A new approach to some notions of statistical quantum mechanics. Busefal 35 (1988), 4-6.
[23] Riečan, B.: Fuzzy connectives and quantum models. In: Cybernetics and Systems Research (R. Trappl, ed.), World Scientific, Singapore 1992, pp. 335-338.
[24] Riečan, B.: On limit theorems in fuzzy quantum spaces. Fuzzy Sets and Systems 101 (199), 79-86. DOI 10.1016/s0165-0114(97)00051-1 | MR 1658964 | Zbl 0966.81011
[25] Riečan, B.: On the conditional expectation of observables in MV algebras of fuzzy sets. Fuzzy Sets and Systems 102 (1999), 445-450. DOI 10.1016/s0165-0114(98)00218-8 | MR 1676911 | Zbl 0930.06009
[26] Riečan, B.: Probability theory on IF events. In: Trends and Progress in System Identification, Papers in Honor of Daniele Mundici on the Occasion of His 60th birthday, Lect. Notes in Computer Sci. 4460 (S. Aguzzoli et al., eds.), Springer, Berlin 2007, pp. 290-308. DOI 10.1007/978-3-540-75939-3_17 | Zbl 1122.60004
[27] Riečan, B.: On the probability theory on MV-algebras. Soft Computing 4 (2000), 49-57. DOI 10.1007/s005000050082 | Zbl 1042.28018
[28] Riečan, V., Mundici, D.: Probability on MV-algebras. In: Handbook of Measure Theory (E. Pap, ed.), Elsevier, Amsterdam 2002, pp. 869-909. DOI 10.1016/b978-044450263-6/50022-1 | MR 1954631 | Zbl 1017.28002
[29] Riečan, B., Neubrunn, T.: Integral, Measure and Ordering. Kluwer Academic Publishers, Bratislava 1997. DOI 10.1007/978-94-015-8919-2 | MR 1489521 | Zbl 0916.28001
[30] Rose, A., Rosser, J. B.: Fragments of many valued statement calculi. Trans. Amer. Math. Soc. 87 (1958), 1-53. DOI 10.2307/1993083 | MR 0094299 | Zbl 0085.24303
[31] Varadarajan, V. C.: Geometry of Quantum Mechanics. van Nostrand, Princeton 1968. DOI 10.1007/978-0-387-49386-2 | MR 0471674
Search
Partner of
