On central limit theorems for IV-events

Interval-valued fuzzy sets were introduced in 1970s as an extension of Zadeh’s fuzzy sets. For interval-valued 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.

[1]  Przemysław Grzegorzewski,et al.  Probability of Intuitionistic Fuzzy Events , 2002 .

[2]  J. Deng,et al.  Introduction to Grey system theory , 1989 .

[3]  C. Chang,et al.  Algebraic analysis of many valued logics , 1958 .

[4]  Didier Dubois,et al.  Gradualness, uncertainty and bipolarity: Making sense of fuzzy sets , 2012, Fuzzy Sets Syst..

[5]  Lotfi A. Zadeh,et al.  The concept of a linguistic variable and its application to approximate reasoning-III , 1975, Inf. Sci..

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

[7]  T. Fine.  Theories of Probability: An Examination of Foundations , 1973 .

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

[9]  Binod Chandra Tripathy,et al.  On I-Convergent Double Sequences of Fuzzy Real Numbers , 2012 .

[10]  D. Mundici.  Advanced Łukasiewicz calculus and MV-algebras , 2011 .

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

[12]  Huibert Kwakernaak,et al.  Fuzzy random variables - I. definitions and theorems , 1978, Inf. Sci..

[13]  Lavinia Corina Ciungu,et al.  Representation theorem for probabilities on IFS-events , 2010, Inf. Sci..

[14]  Olgierd Hryniewicz,et al.  Statistics with Imprecise Data , 2009, Encyclopedia of Complexity and Systems Science.

[15]  K. Jahn.  Intervall‐wertige Mengen , 1975 .

[16]  Etienne E. Kerre,et al.  On the relationship between some extensions of fuzzy set theory , 2003, Fuzzy Sets Syst..

[17]  J. Neumann,et al.  The Logic of Quantum Mechanics , 1936 .

[18]  Ana Colubi,et al.  SMIRE Research Group at the University of Oviedo: A distance-based statistical analysis of fuzzy number-valued data , 2014, Int. J. Approx. Reason..

[19]  Krassimir T. Atanassov,et al.  Intuitionistic Fuzzy Sets - Theory and Applications , 1999, Studies in Fuzziness and Soft Computing.

[20]  C. Carathéodory,et al.  Mass und Integral und ihre Algebraisierung , 1956 .

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

[22]  Katarína Lendelová,et al.  Conditional IF-probability , 2006, SMPS.

[23]  Binod Chandra Tripathy,et al.  On mixed fuzzy topological spaces and countability , 2012, Soft Computing.

[24]  Janusz Kacprzyk,et al.  Probability of intuitionistic fuzzy events and their applications in decision making , 1999, EUSFLAT-ESTYLF Joint Conf..

[25]  Yian-Kui Liu,et al.  Fuzzy Random Variables: A Scalar Expected Value Operator , 2003, Fuzzy Optim. Decis. Mak..

[26]  Patrick Billingsley,et al.  Probability and Measure. , 1986 .

[27]  B. Riecan,et al.  Integral, Measure, and Ordering , 1997 .

[28]  Magdaléna Rencová.  A generalization of probability theory on MV-algebras to IF-events , 2010, Fuzzy Sets Syst..

[29]  Katarína Lendelová,et al.  Representation of IF-probability on MV-algebras , 2006, Soft Comput..

[30]  R. Kruse,et al.  Statistics with vague data , 1987 .

[31]  D. Mundici.  Interpretation of AF -algebras in ukasiewicz sentential calculus , 1986 .

[32]  Beloslav Riečan,et al.  On the Law of Large Numbers on IFS Events , 2004, Fuzzy Days.

[33]  Beloslav Riecan,et al.  On Two Ways for the Probability Theory on IF-sets , 2006, SMPS.

[34]  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).

[35]  Binod Chandra Tripathy,et al.  ON ALMOST STATISTICAL CONVERGENCE OF NEW TYPE OF GENERALIZED DIFFERENCE SEQUENCE OF FUZZY NUMBERS , 2012 .

[36]  Ivor Grattan-Guinness,et al.  Fuzzy Membership Mapped onto Intervals and Many-Valued Quantities , 1976, Math. Log. Q..

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

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

[39]  M. Puri,et al.  Fuzzy Random Variables , 1986 .

[40]  Beloslav Riecan,et al.  On a problem of Radko Mesiar: General form of IF-probabilities , 2006, Fuzzy Sets Syst..

[41]  Didier Dubois,et al.  Terminological difficulties in fuzzy set theory - The case of "Intuitionistic Fuzzy Sets" , 2005, Fuzzy Sets Syst..

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

[43]  Didier Dubois,et al.  Random Sets and Random Fuzzy Sets as Ill-Perceived Random Variables: An Introduction for Ph.D. Students and Practitioners , 2014 .

[44]  I. Turksen.  Interval valued fuzzy sets based on normal forms , 1986 .

[45]  Ana Colubi,et al.  Rejoinder on "A distance-based statistical analysis of fuzzy number-valued data" , 2014, Int. J. Approx. Reason..