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.

THE FIVE INDEPENDENCES AS QUASI-UNIVERSAL PRODUCTS

2002 ◽  
Vol 05 (01) ◽  
pp. 113-134 ◽  
Author(s):  
NAOFUMI MURAKI
Keyword(s):  
Tensor Product ◽  
Free Product ◽  
Boolean Product ◽  
Probability Spaces

A notion of "quasi-universal product" for algebraic probability spaces is introduced as a generalization of Speicher's "universal product". It is proved that there exist only five quasi-universal products, namely, tensor product, free product, Boolean product, monotone product and anti-monotone product. This result means that, in a sense, there exist only five independences which have nice properties of "associativity" and "(quasi-)universality".

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 Unification of Independence in Quantum Probability

1998 ◽  
Vol 01 (03) ◽  
pp. 383-405 ◽  
Author(s):  
Romuald Lenczewski
Keyword(s):  
Free Product ◽  
Tensor Products ◽  
Quantum Probability ◽  
Product Formula ◽  
Free Algebras ◽  
Noncommutative Probability ◽  
Free Variables ◽  
Probability Spaces

Let [Formula: see text], be the conditionally free product of unital free *-algebras [Formula: see text], where ϕl, ψl are states on [Formula: see text], l∈I. We construct a sequence of noncommutative probability spaces [Formula: see text], m∈N, where [Formula: see text] and [Formula: see text], m∈N, [Formula: see text], and the states [Formula: see text], ϕl are Boolean extensions of ϕl, ψl, l∈I, respectively. We define unital *-homomorphisms [Formula: see text] such that [Formula: see text] converges pointwise to *l∈I(ϕl,ψl). Thus, the variables j(m)(w), where w is a word in [Formula: see text], converge in law to the conditionally free variables. The sequence of noncommutative probability spaces [Formula: see text], where [Formula: see text] and Φ(m) is the restriction of [Formula: see text] to [Formula: see text], is called a hierarchy of freeness. Since all finite joint correlations for known examples of independence can be obtained from tensor products of appropriate *-algebras, this approach can be viewed as a unification of independence. Finally, we show how to make the m-fold free product [Formula: see text] into a cocommutative *-bialgebra associated with m-freeness.

scholarly journals CONDITIONALLY MONOTONE INDEPENDENCE I: INDEPENDENCE, ADDITIVE CONVOLUTIONS AND RELATED CONVOLUTIONS

2011 ◽  
Vol 14 (03) ◽  
pp. 465-516 ◽  
Author(s):  
TAKAHIRO HASEBE
Keyword(s):  
Probability Theory ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Infinite Divisibility ◽  
Noncommutative Probability ◽  
Limit Distributions ◽  
Boolean Products ◽  
Probability Spaces ◽  
Orthogonal Product

We define a product of algebraic probability spaces equipped with two states. This product is called a conditionally monotone product. This product is a new example of independence in noncommutative probability theory and unifies the monotone and Boolean products, and moreover, the orthogonal product. Then we define the associated cumulants and calculate the limit distributions in central limit theorem and Poisson's law of small numbers. We also prove a combinatorial moment-cumulant formula using monotone partitions. We investigate some other topics such as infinite divisibility for the additive convolution and deformations of the monotone convolution. We define cumulants for a general convolution to analyze the deformed convolutions.

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