scholarly journals Conditional intuitionistic fuzzy probability and martingale convergence theorem using IF-probability

2020 ◽  
Vol 26 (3) ◽  
pp. 13-21
Author(s):  
Katarína Čunderlíková ◽  
Keyword(s):  
Convergence Theorem ◽  
Fuzzy Probability ◽  
Martingale Convergence Theorem ◽  
Intuitionistic Fuzzy ◽  
Martingale Convergence

The aim of this paper is to formulate the conditional intuitionistic fuzzy probability and a version of martingale convergence theorem with respect an intuitionistic fuzzy probability. Since the intuitionistic fuzzy probability can be decomposed to two intuitionistic fuzzy states, we can use the results holding for intuitionistic fuzzy states.

scholarly journals Martingale Convergence Theorem for the Conditional Intuitionistic Fuzzy Probability

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1707
Author(s):  
Katarína Čunderlíková
Keyword(s):  
Fuzzy Sets ◽  
Conditional Probability ◽  
Convergence Theorem ◽  
Intuitionistic Fuzzy Sets ◽  
Fuzzy Probability ◽  
Martingale Convergence Theorem ◽  
Intuitionistic Fuzzy ◽  
Mv Algebra ◽  
Fuzzy State ◽  
Martingale Convergence

For the first time, the concept of conditional probability on intuitionistic fuzzy sets was introduced by K. Lendelová. She defined the conditional intuitionistic fuzzy probability using a separating intuitionistic fuzzy probability. Later in 2009, V. Valenčáková generalized this result and defined the conditional probability for the MV-algebra of inuitionistic fuzzy sets using the state and probability on this MV-algebra. She also proved the properties of conditional intuitionistic fuzzy probability on this MV-algebra. B. Riečan formulated the notion of conditional probability for intuitionistic fuzzy sets using an intuitionistic fuzzy state. We use this definition in our paper. Since the convergence theorems play an important role in classical theory of probability and statistics, we study the martingale convergence theorem for the conditional intuitionistic fuzzy probability. The aim of this contribution is to formulate a version of the martingale convergence theorem for a conditional intuitionistic fuzzy probability induced by an intuitionistic fuzzy state m. We work in the family of intuitionistic fuzzy sets introduced by K. T. Atanassov as an extension of fuzzy sets introduced by L. Zadeh. We proved the properties of the conditional intuitionistic fuzzy probability.

scholarly journals Martingale convergence theorem for a conditional intuitionistic fuzzy mean value

2021 ◽  
Vol 27 (2) ◽  
pp. 94-102
Author(s):  
Katarína Čunderlíková ◽  
Keyword(s):  
Convergence Theorem ◽  
Random Variables ◽  
Mean Value ◽  
Martingale Convergence Theorem ◽  
Conditional Mean ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Mean ◽  
Martingale Convergence

The aim of this contribution is to show a representation of a conditional intuitionistic fuzzy mean value of intuitionistic fuzzy observables by a conditional mean value of random variables. We formulate a martingale convergence theorem for a conditional intuitionistic fuzzy mean value, too.

A Martingale Convergence Theorem in Vector Lattices

1966 ◽  
Vol 18 ◽  
pp. 424-432 ◽  
Author(s):  
Ralph DeMarr
Keyword(s):  
Convergence Theorem ◽  
Vector Lattice ◽  
Random Variables ◽  
Order Convergence ◽  
Arbitrary Vector ◽  
Martingale Convergence Theorem ◽  
Convergence Almost Everywhere ◽  
Vector Lattices ◽  
Almost Everywhere ◽  
Martingale Convergence

The martingale convergence theorem was first proved by Doob (3) who considered a sequence of real-valued random variables. Since various collections of real-valued random variables can be regarded as vector lattices, it seems of interest to prove the martingale convergence theorem in an arbitrary vector lattice. In doing so we use the concept of order convergence that is related to convergence almost everywhere, the type of convergence used in Doob's theorem.

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