Free random variables in noncommutative probability theory

1992 ◽  
pp. 11-20 ◽  
Keyword(s):  
Probability Theory ◽  
Random Variables ◽  
Noncommutative Probability ◽  
Free Random Variables ◽  
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 CUMULANTS IN NONCOMMUTATIVE PROBABILITY THEORY III: CREATION AND ANNIHILATION OPERATORS ON FOCK SPACES

2005 ◽  
Vol 08 (03) ◽  
pp. 407-437 ◽  
Author(s):  
FRANZ LEHNER
Keyword(s):  
Probability Theory ◽  
Symmetric Group ◽  
Toeplitz Operators ◽  
Fock Space ◽  
Random Variables ◽  
Fock Spaces ◽  
Noncommutative Probability ◽  
Infinite Symmetric Group ◽  
Fock States ◽  
Cycle Indicator

Exchangeability systems arising from Fock space constructions are considered and the corresponding cumulants are computed for generalized Toeplitz operators and similar noncommutative random variables. In particular, simplified calculations are given for the two known examples of q-cumulants. In the second half of the paper we consider in detail the Fock states associated to characters of the infinite symmetric group recently constructed by Bożejko and Guta. We express moments of multidimensional Dyck words in terms of the so-called cycle indicator polynomials of certain digraphs.

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