On Q-independence, limit theorems and q-Gaussian distribution

Marcin Marciniak

Studia Mathematica (1998)

  • Volume: 129, Issue: 2, page 113-135
  • ISSN: 0039-3223

Abstract

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We formulate the notion of Q-independence which generalizes the classical independence of random variables and free independence introduced by Voiculescu. Here Q stands for a family of polynomials indexed by tiny partitions of finite sets. The analogs of the central limit theorem and Poisson limit theorem are proved. Moreover, it is shown that in some special cases this kind of independence leads to the q-probability theory of Bożejko and Speicher.

How to cite

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Marciniak, Marcin. "On Q-independence, limit theorems and q-Gaussian distribution." Studia Mathematica 129.2 (1998): 113-135. <http://eudml.org/doc/216494>.

@article{Marciniak1998,
abstract = {We formulate the notion of Q-independence which generalizes the classical independence of random variables and free independence introduced by Voiculescu. Here Q stands for a family of polynomials indexed by tiny partitions of finite sets. The analogs of the central limit theorem and Poisson limit theorem are proved. Moreover, it is shown that in some special cases this kind of independence leads to the q-probability theory of Bożejko and Speicher.},
author = {Marciniak, Marcin},
journal = {Studia Mathematica},
keywords = {-independence; probability system; independence of random variables; central limit theorem; Poisson limit theorem},
language = {eng},
number = {2},
pages = {113-135},
title = {On Q-independence, limit theorems and q-Gaussian distribution},
url = {http://eudml.org/doc/216494},
volume = {129},
year = {1998},
}

TY - JOUR
AU - Marciniak, Marcin
TI - On Q-independence, limit theorems and q-Gaussian distribution
JO - Studia Mathematica
PY - 1998
VL - 129
IS - 2
SP - 113
EP - 135
AB - We formulate the notion of Q-independence which generalizes the classical independence of random variables and free independence introduced by Voiculescu. Here Q stands for a family of polynomials indexed by tiny partitions of finite sets. The analogs of the central limit theorem and Poisson limit theorem are proved. Moreover, it is shown that in some special cases this kind of independence leads to the q-probability theory of Bożejko and Speicher.
LA - eng
KW - -independence; probability system; independence of random variables; central limit theorem; Poisson limit theorem
UR - http://eudml.org/doc/216494
ER -

References

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  1. [1] R. Askey and M. Ismail, Recurrence relations, continued fractions and orthogonal polynomials, Mem. Amer. Math. Soc. 49 (1984) Zbl0548.33001
  2. [2] M. Bożejko, A q-deformed probability, Nelson's inequality and central limit theorems, in: Non-linear Fields, Classical, Random, Semiclassical, P. Garbaczewski and Z. Popowicz (eds.), World Sci., Singapore, 1991, 312-335. 
  3. [3] M. Bożejko, B. Kümmerer and R. Speicher, q-Gaussian processes: non-commutative and classical aspects, Comm. Math. Phys. 185 (1997), 129-154. 
  4. [4] 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
  5. [5] M. Bożejko and R. Speicher, An example of a generalized brownian motion, Comm. Math. Phys. 137 (1991), 519-531. Zbl0722.60033
  6. [6] M. Bożejko and R. Speicher, ψ-independent and symmetrized white noises, in: Quantum Probability and Related Topics VII, World Sci., Singapore, 1992, 219-235. 
  7. [7] M. Bożejko and R. Speicher, Interpolations between bosonic and fermionic relations given by generalized brownian motions, SFB-Preprint 691, Heidelberg, 1992. 
  8. [8] W. Feller, An Introduction to Probability Theory and its Applications, Wiley, New York, 1966. 
  9. [9] H. van Leeuwen and H. Maassen, An obstruction for q-deformation of the convolution product, J. Phys. A 29 (1996), 4741-4748. Zbl0905.60008
  10. [10] A. Nica, A one-parameter family of transforms linearizing convolution laws for probability distributions, Comm. Math. Phys. 168 (1995), 187-207. Zbl0818.60096
  11. [11] A. Nica, Crossings and embracings of set-partitions, and q-analogues of the logarithm of the Fourier transform, Discrete Math. 157 (1996), 285-309. Zbl0878.05009
  12. [12] A. Nica, R-transforms of free joint distributions, and non-crossing partitions, J. Funct. Anal. 135 (1996), 271-296. Zbl0837.60008
  13. [13] R. Speicher, A new example of 'independence' and 'white noise', Probab. Theory Related Fields 84 (1990), 141-159. Zbl0671.60109
  14. [14] R. Speicher, Multiplicative functions on the lattice of non-crossing partitions and free convolution, Math. Ann. 298 (1994), 611-628. Zbl0791.06010
  15. [15] R. Speicher, On universal products, in: Free Probability Theory, D. Voiculescu (ed.), Fields Inst. Commun. 12, Amer. Math. Soc., Providence, R.I., 1997, 257-266. Zbl0877.46044
  16. [16] R. Speicher and R. Woroudi, Boolean convolution, ibid., 267-279. 
  17. [17] D. Voiculescu, Symmetries of some reduced free products of C*-algebras, in: H. Araki et al. (eds.), Operator Algebras and their Connection with Topology and Ergodic Theory (Romania, 1983), Lecture Notes in Math. 1132, Springer, Berlin, 1985, 556-588. 
  18. [18] D. Voiculescu, Addition of certain non-commuting random variables, J. Funct. Anal. 66 (1986), 323-335. Zbl0651.46063
  19. [19] D. Voiculescu, K. Dykema and A. Nica, Free Random Variables, CRM Monogr. Ser. 1, Amer. Math. Soc., Providence, R.I., 1993 

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