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

Studia Mathematica (1998)

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

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topMarciniak, 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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- [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] M. Bożejko, B. Kümmerer and R. Speicher, q-Gaussian processes: non-commutative and classical aspects, Comm. Math. Phys. 185 (1997), 129-154.
- [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
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- [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] A. Nica, A one-parameter family of transforms linearizing convolution laws for probability distributions, Comm. Math. Phys. 168 (1995), 187-207. Zbl0818.60096
- [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] A. Nica, R-transforms of free joint distributions, and non-crossing partitions, J. Funct. Anal. 135 (1996), 271-296. Zbl0837.60008
- [13] R. Speicher, A new example of 'independence' and 'white noise', Probab. Theory Related Fields 84 (1990), 141-159. Zbl0671.60109
- [14] R. Speicher, Multiplicative functions on the lattice of non-crossing partitions and free convolution, Math. Ann. 298 (1994), 611-628. Zbl0791.06010
- [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] R. Speicher and R. Woroudi, Boolean convolution, ibid., 267-279.
- [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] D. Voiculescu, Addition of certain non-commuting random variables, J. Funct. Anal. 66 (1986), 323-335. Zbl0651.46063
- [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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