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

• Autorzy: Marcin Marciniak
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 46N30: Applications in probability theory and statistics
• 60C05: Combinatorial probability
• 60F05: Central limit and other weak theorems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1998
• Tom: 129
• Numer: 2
• Strony: 113-135

Daty

wydano

1998

otrzymano

1996-10-15

poprawiono

1997-11-24

Twórcy

autor

Marcin Marciniak

• Institute of Mathematics, Gdańsk University, Wita Stwosza 57, 80-952 Gdańsk, Poland

Bibliografia

• [1] R. Askey and M. Ismail, Recurrence relations, continued fractions and orthogonal polynomials, Mem. Amer. Math. Soc. 49 (1984)
• [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.
• [5] M. Bożejko and R. Speicher, An example of a generalized brownian motion, Comm. Math. Phys. 137 (1991), 519-531.
• [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] M. Bożejko and R. Speicher, Interpolations between bosonic and fermionic relations given by generalized brownian motions, SFB-Preprint 691, Heidelberg, 1992.
• [8] W. Feller, An Introduction to Probability Theory and its Applications, Wiley, New York, 1966.
• [9] H. van Leeuwen and H. Maassen, An obstruction for q-deformation of the convolution product, J. Phys. A 29 (1996), 4741-4748.
• [10] A. Nica, A one-parameter family of transforms linearizing convolution laws for probability distributions, Comm. Math. Phys. 168 (1995), 187-207.
• [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.
• [12] A. Nica, R-transforms of free joint distributions, and non-crossing partitions, J. Funct. Anal. 135 (1996), 271-296.
• [13] R. Speicher, A new example of 'independence' and 'white noise', Probab. Theory Related Fields 84 (1990), 141-159.
• [14] R. Speicher, Multiplicative functions on the lattice of non-crossing partitions and free convolution, Math. Ann. 298 (1994), 611-628.
• [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.
• [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.
• [19] D. Voiculescu, K. Dykema and A. Nica, Free Random Variables, CRM Monogr. Ser. 1, Amer. Math. Soc., Providence, R.I., 1993