Accardi, Luigi; Hashimoto, Yukihiro; Obata, Nobuaki Notions of independence related to the free group. (English) Zbl 0913.46057 Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1, No. 2, 201-220 (1998). Summary: The central limit problem for algebraic probability spaces associated with the Haagerup states on the free group with countably many generators leads to a new form of statistical independence in which the singleton condition is not satisfied. This circumstance allows us to obtain nonsymmetric distributions from the central limit theorems deduced from this notion of independence. In the particular case of the Haagerup states, the role of the Gaussian law is played by the Ullman distribution. The limit process is explicitly realized on the finite temperature Boltzmannian Fock space. The role of entangled ergodic theorems in the proof of the central limit theorems is discussed. Cited in 2 ReviewsCited in 23 Documents MSC: 46N30 Applications of functional analysis in probability theory and statistics 81S25 Quantum stochastic calculus Keywords:central limit problem; algebraic probability spaces associated with the Haagerup states; free group with countably many generators; statistical independence; nonsymmetric distributions; Gaussian law; Ullman distribution; finite temperature Boltzmannian Fock space PDFBibTeX XMLCite \textit{L. Accardi} et al., Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1, No. 2, 201--220 (1998; Zbl 0913.46057) Full Text: DOI References: [1] DOI: 10.2977/prims/1195184017 · Zbl 0498.60099 · doi:10.2977/prims/1195184017 [2] DOI: 10.1007/s004400050119 · Zbl 0954.60029 · doi:10.1007/s004400050119 [3] Bo\.zejko M., Studia Math. 95 pp 107– (1989) [4] DOI: 10.2140/pjm.1996.175.357 · Zbl 0874.60010 · doi:10.2140/pjm.1996.175.357 [5] DOI: 10.1017/S0305004100053093 · Zbl 0351.20024 · doi:10.1017/S0305004100053093 [6] DOI: 10.1007/BF01192140 · Zbl 0729.60074 · doi:10.1007/BF01192140 [7] DOI: 10.1007/BF00536048 · Zbl 0362.60043 · doi:10.1007/BF00536048 [8] DOI: 10.1007/BF01410082 · Zbl 0408.46046 · doi:10.1007/BF01410082 [9] DOI: 10.1142/S0219025798000144 · Zbl 0922.05036 · doi:10.1142/S0219025798000144 [10] DOI: 10.1007/BF01393251 · Zbl 0379.46060 · doi:10.1007/BF01393251 [11] Lenczewski R., Plenum 199 pp 299– [12] Lyndon R., Math. Scand. 12 pp 209– (1963) · Zbl 0119.26402 · doi:10.7146/math.scand.a-10684 [13] DOI: 10.1007/BF01197843 · Zbl 0671.60109 · doi:10.1007/BF01197843 [14] Voiculescu D., World Scientific 199 pp 473– [15] DOI: 10.1007/BF02100050 · Zbl 0781.60006 · doi:10.1007/BF02100050 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.