Strong Laws of Large Numbers for IVM-Events

Nonstandard probability theories have been developed for modeling random systems in complex spaces, such as, quantum systems. One of these theories, the MV-algebraic probability theory, involves the notions of state and observable, which were introduced by abstracting the properties of the Kolmogorovian probability measure and the classical random variable, as well as the notion of independence. Although within these nonstandard probability theories, many important theorems, including the strong law of large numbers (SLLN) for sequences of independent and identically distributed observables, have been considered, some practical applications require their further development. This paper is devoted to the development of the IVM-probability theory for the data described by interval-valued fuzzy random sets (IVM-events). The generalizations of Marcinkiewicz–Zygmund SLLN and Brunk–Prokhorov SLLN for independent IVM-events have been proved within this new theory. Our results open new possibilities in the theoretical analysis of imprecise random events in more complex spaces.

[1]  Beloslav Riecan,et al.  Probability Theory on IF Events , 2006, Algebraic and Proof-theoretic Aspects of Non-classical Logics.

[2]  Olgierd Hryniewicz,et al.  On generalized versions of central limit theorems for IF-events , 2016, Inf. Sci..

[3]  Beloslav Riečan,et al.  Probability on MV algebras , 1997 .

[4]  Philip E. Cheng,et al.  Marcinkiewicz strong laws for linear statistics , 2000 .

[5]  Michel Loève,et al.  Probability Theory I , 1977 .

[6]  D. Mundici,et al.  Algebraic Foundations of Many-Valued Reasoning , 1999 .

[7]  Olgierd Hryniewicz,et al.  Generalized versions of MV-algebraic central limit theorems , 2015, Kybernetika.

[8]  Beloslav Riecan.  On the Atanassov Concept of Fuzziness and One of Its Modification , 2016, Imprecision and Uncertainty in Information Representation and Processing.

[9]  R. Jajte,et al.  On the strong law of large numbers , 2003 .

[10]  Beloslav Riecan.  On limit theorems in fuzzy quantum spaces , 1999, Fuzzy Sets Syst..

[11]  Piotr Nowak.  Monotone measures of intuitionistic fuzzy sets , 2003, EUSFLAT Conf..

[12]  Olgierd Hryniewicz,et al.  On central limit theorems for IV-events , 2018, Soft Comput..

[13]  J. Kacprzyk,et al.  A concept of a probability of an intuitionistic fuzzy event , 1999, FUZZ-IEEE'99. 1999 IEEE International Fuzzy Systems. Conference Proceedings (Cat. No.99CH36315).

[14]  Michael Christoph Paulsen.  Limit theorems for limit order books , 2014 .

[15]  DANIELE MUNDICI,et al.  Averaging the truth-value in Łukasiewicz logic , 1995, Stud Logica.

[16]  B. Riečan.  On the probability theory on MV algebras , 2000, Soft Comput..

[17]  Habib Naderi,et al.  On complete convergence of moving average processes for NSD sequences , 2015 .

[18]  Peter J. Haas,et al.  Large-scale matrix factorization with distributed stochastic gradient descent , 2011, KDD.

[19]  M. Simonnet.  The Strong Law of Large Numbers , 1996 .

[20]  Andrew Rosalsky,et al.  A Refinement of the Kolmogorov–Marcinkiewicz–Zygmund Strong Law of Large Numbers , 2010 .