Singleton independence

Luigi Accardi; Yukihiro Hashimoto; Nobuaki Obata

Banach Center Publications (1998)

  • Volume: 43, Issue: 1, page 9-24
  • ISSN: 0137-6934

Abstract

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Motivated by the central limit problem for algebraic probability spaces arising from the Haagerup states on the free group with countably infinite generators, we introduce a new notion of statistical independence in terms of inequalities rather than of usual algebraic identities. In the case of the Haagerup states the role of the Gaussian law is played by the Ullman distribution. The limit process is realized explicitly on the finite temperature Boltzmannian Fock space. Furthermore, a functional central limit theorem associated with the Haagerup states is proved and the limit white noise is investigated.

How to cite

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Accardi, Luigi, Hashimoto, Yukihiro, and Obata, Nobuaki. "Singleton independence." Banach Center Publications 43.1 (1998): 9-24. <http://eudml.org/doc/208869>.

@article{Accardi1998,
abstract = {Motivated by the central limit problem for algebraic probability spaces arising from the Haagerup states on the free group with countably infinite generators, we introduce a new notion of statistical independence in terms of inequalities rather than of usual algebraic identities. In the case of the Haagerup states the role of the Gaussian law is played by the Ullman distribution. The limit process is realized explicitly on the finite temperature Boltzmannian Fock space. Furthermore, a functional central limit theorem associated with the Haagerup states is proved and the limit white noise is investigated.},
author = {Accardi, Luigi, Hashimoto, Yukihiro, Obata, Nobuaki},
journal = {Banach Center Publications},
keywords = {Ullman distribution; Boltzmannian Fock space; functional central limit theorem; Haagerup states},
language = {eng},
number = {1},
pages = {9-24},
title = {Singleton independence},
url = {http://eudml.org/doc/208869},
volume = {43},
year = {1998},
}

TY - JOUR
AU - Accardi, Luigi
AU - Hashimoto, Yukihiro
AU - Obata, Nobuaki
TI - Singleton independence
JO - Banach Center Publications
PY - 1998
VL - 43
IS - 1
SP - 9
EP - 24
AB - Motivated by the central limit problem for algebraic probability spaces arising from the Haagerup states on the free group with countably infinite generators, we introduce a new notion of statistical independence in terms of inequalities rather than of usual algebraic identities. In the case of the Haagerup states the role of the Gaussian law is played by the Ullman distribution. The limit process is realized explicitly on the finite temperature Boltzmannian Fock space. Furthermore, a functional central limit theorem associated with the Haagerup states is proved and the limit white noise is investigated.
LA - eng
KW - Ullman distribution; Boltzmannian Fock space; functional central limit theorem; Haagerup states
UR - http://eudml.org/doc/208869
ER -

References

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  1. [1] L. Accardi, I. Ya. Aref'eva and I. V. Volovich, The master field for half-planar diagrams and free non-commutative random variables, to appear in Quarks `96 (V. Matveev and V. Rubakov, eds.), HEP-TH/9502092. 
  2. [2] L. Accardi, A. Frigerio and J. Lewis, Quantum stochastic processes, Publ. RIMS Kyoto University 18 (1982), 97-133. Zbl0498.60099
  3. [3] L. Accardi, S. V. Kozyrev and I. V. Volovich, Dynamics of dissipative two-state systems in the stochastic approximation, Phys. Rev. A 56 (1997), 1-7. 
  4. [4] L. Accardi, Y. Hashimoto and N. Obata, Notions of independence related to the free group, Infinite Dimen. Anal. Quantum Probab. 1 (1998), 221-246. Zbl0913.46057
  5. [5] M. Bożejko, Uniformly bounded representations of free groups, J. Reine Angew. Math. 377 (1987), 170-186. Zbl0604.43004
  6. [6] M. Bożejko, Positive definite kernels, length functions on groups and noncommutative von Neumann inequality, Studia Math. 95 (1989), 107-118. Zbl0714.43007
  7. [7] M. Bożejko, Harmonic analysis on discrete groups and noncommutative probability, Volterra preprint series No. 93, 1992. 
  8. [8] M. Bożejko, private communication, November, 1997. 
  9. [9] M. Bożejko, B. Kümmerer and R. Speicher, q-Gaussian processes: Non-commutative and classical aspects, Commun. Math. Phys. 185 (1997), 129-154. 
  10. [10] M. Bożejko, M. Leinert and R. Speicher, Convolution and limit theorems for conditionally free random variables, Pacific J. Math. 175 (1996), 357-388. Zbl0874.60010
  11. [11] M. Bożejko and R. Speicher, ψ-Independent and symmetrized white noises, in: Quantum Probability and Related Fields VI, pp. 219-236, World Scientific, 1991. 
  12. [12] I. Chiswell, Abstract length functions in groups, Math. Proc. Camb. Phil. Soc. 80 (1976), 451-463. Zbl0351.20024
  13. [13] F. Fagnola, A Lévy theorem for free noises, Probab. Th. Rel. Fields 90 (1991), 491-504. %Preprint,1991, Rendiconti Accademia dei Lincei (1992). Zbl0729.60074
  14. [14] A. Figà-Talamanca and M. Picardello, Harmonic Analysis on Free Groups, Marcel Dekker, New York and Basel, 1983. Zbl0536.43001
  15. [15] M. de Giosa and Y. G. Lu, From quantum Bernoulli process to creation and annihilation operators on interacting q-Fock space, to appear in Nagoya Math. J. Zbl0916.60082
  16. [16] N. Giri and W. von Waldenfels, An algebraic version of the central limit theorem, ZW 42 (1978), 129-134. Zbl0362.60043
  17. [17] U. Haagerup, An example of a non-nuclear C*-algebra which has the metric approximation property, Invent. Math. 50 (1979), 279-293. Zbl0408.46046
  18. [18] Y. Hashimoto, Deformations of the semi-circle law derived from random walks on free groups, to appear in Prob. Math. Stat. 18 (1998). 
  19. [19] F. Hiai and D. Petz, Maximizing free entropy, Preprint No.17, Mathematical Institute, Hungarian Academy of Sciences, Budapest, 1996. Zbl0913.94003
  20. [20] A. Hora, Central limit theorems and asymptotic spectral analysis on large graphs, submitted to Infinite Dimensional Analysis and Quantum Probability, 1997. 
  21. [21] R. Lenczewski, Quantum central limit theorems, in: Symmetries in Sciences VIII (B. Gruber, ed.), pp. 299-314, Plenum, 1995. Zbl0952.60021
  22. [22] V. Liebscher, Note on entangled ergodic theorems, preprint, 1997. 
  23. [23] R. Lyndon, Length functions in groups, Math. Scand. 12 (1963), 209-234. Zbl0119.26402
  24. [24] N. Muraki, A new example of noncommutative 'de Moivre-Laplace theorem', in: Probability Theory and Mathematical Statistics (S. Watanabe et al., eds.), pp. 353-362, World Scientific, 1996. Zbl1147.81307
  25. [25] M. Schürmann, White Noise on Bialgebras, Lect. Notes in Math. Vol. 1544, Springer-Verlag, 1993. Zbl0773.60100
  26. [26] R. Speicher and W. von Waldenfels, A general central limit theorem and invariance principle, in: Quantum Probability and Related Topics IX, pp. 371-387, World Scientific, 1994. 
  27. [27] D. Voiculescu, Free noncommutative random variables, random matrices and the I I 1 factors of free groups, in: Quantum Probability and Related Fields VI, pp. 473-487, World Scientific, 1991. Zbl0928.46045
  28. [28] W. von Waldenfels, An approach to the theory of pressure broadening of spectral lines, in: Probability and Information Theory II (M. Behara et al, eds.), pp. 19-69, Lect. Notes in Math. Vol. 296, Springer-Verlag, 1973. 
  29. [29] W. von Waldenfels, Interval partitions and pair interactions, in: Séminaire de Probabilités IX (P. A. Meyer, ed.), pp. 565-588, Lect. Notes in Math. Vol. 465, Springer-Verlag, 1975. 

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