Quantum central limit theorems for weakly dependent maps. I
type: Item
instance of: scholarly article
title: Quantum central limit theorems for weakly dependent maps. I
zbMATH Open document ID: 0823.60017
DOI: 10.1007/BF01874948
author: L. Accardi
author: Yun Gang Lu
published in: Acta Mathematica Hungarica
publication date: 1995-10-23T00:00:00Z
review text: The authors consider non-commutative (n.c.) laws of large numbers and central limit theorems. In the n.c. setting the usual classical series of random variables \(X_ 1, X_ 2, \dots\) is replaced by a series \(j_ 1, j_ 2,\dots\) of homomorphisms \(j_ k : {\mathcal B} \to {\mathcal A}\) between two not necessarily commutative \(*\)-algebras \(\mathcal B\) and \(\mathcal A\). One considers sums of the forms \({1\over N} \sum^ N_ 1 j_ k(b)\) and \({1\over \sqrt{N}} \sum^ N_ 1 (j_ k(b) - \alpha_ k (b))\) in the law of large numbers and in the central limit theorem (CLT) case, respectively. (Here \(\alpha_ k(b) = \Phi(j_ k(b))\) where \(\Phi\) is a state on \(\mathcal A\) replacing the classical probability measure.) The authors generalize the usual conditions under which n.c. CLTs hold in assuming 1) commutation relations between \(j_ k(b)\) and \(j_ l(b')\) using commutation factors \(\sigma(k, l, b, b')\) and translation terms \(\varepsilon(k, l, b, b')\) and 2) weak dependence rather than independence. On the other hand, the authors' assumption \(\sigma(k, l, b, b') = \sigma(b, b')\) in the CLT case means that some of the known n.c. CLTs are excluded. There is also a continuous parameter version of the discrete results.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 81S05
Mathematics Subject Classification ID: 60A99
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
zbMATH DE Number: 556300
zbMATH Keywords: non-commutative probability
zbMATH Keywords: central limit theorems
zbMATH Keywords: laws of large numbers
zbMATH Keywords: weak dependence
MaRDI profile type: Publication
cites work: Quantum central limit theorems for strongly mixing random variables
cites work: The weak coupling limit as a quantum functional central limit
cites work: The number process as low density limit of Hamiltonian models
cites work: Some semigroups of completely positive maps on the CCR algebra
cites work: Completely positive quasi-free maps on the CAR algebra
cites work: A Lévy theorem for free noises
cites work: Central limits of product mappings between CAR-algebras
cites work: An algebraic version of the central limit theorem
cites work: Non-commutative central limits
cites work: Dynamics of fluctuations for quantum lattice systems
cites work: Theory of fluctuations and small oscillations for quantum lattice systems
cites work: Central limit theorem for mixing quantum systems and the CCR-algebra of fluctuations
cites work: A quantum-mechanical functional central limit theorem
cites work: Quantum mechanical Wiener processes
cites work: Q3996804
cites work: A noncommutative central limit theorem for CCR-algebras
cites work: A new example of `independence' and `white noise'
cites work: Limit laws for random matrices and free products
cites work: An algebraic central limit theorem in the anti-commuting case
cites work: Noncommutative analogs of probabilistic notions and results
cites work: Completely positive maps on the CCR-algebra
full work available at URL: https://doi.org/10.1007/bf01874948
OpenAlex ID: W4238765665
Recommended article: Quantum central limit theorems for weakly dependent maps. II
Recommended article: Quantum central limit theorems for strongly mixing random variables
Recommended article: Q4241820
Recommended article: A central limit theorem in many-body quantum dynamics
Recommended article: Q5089567
Recommended article: A quantum central limit theorem for sums of independent identically distributed random variables
Recommended article: A tracial quantum central limit theorem
Recommended article: Q5439571
Recommended article: A q-analog of the quantum central limit theorem for SUq(2)
Recommended article: A CENTRAL LIMIT THEOREM FOR BOSE ${\mathcal Z}$-INDEPENDENT QUANTUM RANDOM VARIABLES
description: scientific article; zbMATH DE number 556300