On Independence of Events in Noncommutative Probability Theory

2021 ◽  
Vol 42 (10) ◽  
pp. 2306-2314
Author(s):  
A. M. Bikchentaev ◽  
P. N. Ivanshin
Keyword(s):  
Probability Theory ◽  
Noncommutative Probability ◽  
Independence Of Events ◽  
Noncommutative Probability Theory

THE FIVE INDEPENDENCES AS NATURAL PRODUCTS

2003 ◽  
Vol 06 (03) ◽  
pp. 337-371 ◽  
Author(s):  
NAOFUMI MURAKI
Keyword(s):  
Probability Theory ◽  
Natural Products ◽  
Tensor Product ◽  
Natural Product ◽  
Free Product ◽  
Noncommutative Probability ◽  
Stochastic Independence ◽  
Boolean Product ◽  
Probability Spaces ◽  
Noncommutative Probability Theory

Let [Formula: see text] be the class of all algebraic probability spaces. A "natural product" is, by definition, a map [Formula: see text] which is required to satisfy all the canonical axioms of Ben Ghorbal and Schürmann for "universal product" except for the commutativity axiom. We show that there exist only five natural products, namely tensor product, free product, Boolean product, monotone product and anti-monotone product. This means that, in a sense, there exist only five universal notions of stochastic independence in noncommutative probability theory.

scholarly journals The chordal Loewner equation and monotone probability theory

2017 ◽  
Vol 20 (03) ◽  
pp. 1750016 ◽  
Author(s):  
Sebastian Schleißinger
Keyword(s):  
Probability Theory ◽  
Evolution Equations ◽  
Free Probability ◽  
Noncommutative Probability ◽  
Loewner Equation ◽  
Free Probability Theory ◽  
Chordal Loewner Equation ◽  
Monotone Probability ◽  
Noncommutative Probability Theory

In Ref. 5, O. Bauer interpreted the chordal Loewner equation in terms of noncommutative probability theory. We follow this perspective and identify the chordal Loewner equations as the non-autonomous versions of evolution equations for semigroups in monotone and anti-monotone probability theory. We also look at the corresponding equation for free probability theory.

scholarly journals Wick polynomials in noncommutative probability: a group-theoretical approach

2021 ◽  
pp. 1-27
Author(s):  
Kurusch Ebrahimi-Fard ◽  
Frédéric Patras ◽  
Nikolas Tapia ◽  
Lorenzo Zambotti
Keyword(s):  
Probability Theory ◽  
Hopf Algebra ◽  
Theoretical Approach ◽  
Algebraic Approach ◽  
Noncommutative Probability ◽  
Classical Probability ◽  
Classical Probability Theory ◽  
Wick Products ◽  
Noncommutative Probability Theory

Abstract Wick polynomials and Wick products are studied in the context of noncommutative probability theory. It is shown that free, Boolean, and conditionally free Wick polynomials can be defined and related through the action of the group of characters over a particular Hopf algebra. These results generalize our previous developments of a Hopf-algebraic approach to cumulants and Wick products in classical probability theory.

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