Download PDF - A noncommutative limit theorem for homogeneous correlationsAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

A noncommutative limit theorem for homogeneous correlations

Authors 1

Affiliations

  1. Hugo Steinhaus Center for Stochastic Methods, Institute of Mathematics, Technical University of Wrocław, 50-370 Wrocław, Poland

Abstract

We state and prove a noncommutative limit theorem for correlations which are homogeneous with respect to order-preserving injections. The most interesting examples of central limit theorems in quantum probability (for commuting, anticommuting, and free independence and also various q-qclt's), as well as the limit theorems for the Poisson law and the free Poisson law are special cases of the theorem. In particular, the theorem contains the q-central limit theorem for non-identically distributed variables, derived in our previous work in the context of q-bialgebras and quantum groups. More importantly, new examples of limit theorems of q-Poisson type are derived for both the infinite tensor product algebra and the reduced free product, leading to new q-laws. In the first case the limit as q → 1 is studied in more detail and a connection with partial Bell polynomials is established.

Bibliography

  1. [AFL] L. Accardi, A. Frigerio and J. T. Lewis, Quantum stochastic processes, Publ. RIMS Kyoto Univ. 18 (1982), 97-133.
  2. [A-L] L. Accardi and Y. G. Lu, Quantum central limit theorems for weakly dependent maps, preprint No. 54, Centro Matematico V. Volterra, Universita di Roma II, 1990.
  3. [B-S] M. Bożejko and R. Speicher, Interpolations between bosonic and fermionic relations given by generalized brownian motions, Math. Z. 222 (1996), 135-160.
  4. [C-H] D. D. Cushen and R. L. Hudson, A quantum-mechanical central limit theorem, J. Appl. Probab. 8 (1971), 454-469.
  5. [G-W] N. Giri and W. von Waldenfels, An algebraic version of the central limit theorem, Z. Wahrsch. Verw. Gebiete 42 (1978), 129-134.
  6. [H] R. L. Hudson, A quantum mechanical central limit theorem for anti-commuting observables, J. Appl. Probab. 10 (1973), 502-509.
  7. [L1] R. Lenczewski, On sums of q-independent SUq(2) quantum variables, Comm. Math. Phys. 154 (1993), 127-34.
  8. [L2] R. Lenczewski, Addition of independent variables in quantum groups, Rev. Math. Phys. 6 (1994), 135-147.
  9. [L3] R. Lenczewski, Quantum random walk for Uq(su(2)) and a new example of quantum noise, J. Math. Phys. 37 (1996), 2260-2278.
  10. [L-P] R. Lenczewski and K. Podgórski, A q-analog of the quantum central limit theorem for SUq(2), ibid. 33 (1992), 2768-2778.
  11. [Sch] M. Schürmann, Quantum q-white noise and a q-central limit theorem, Comm. Math. Phys. 140 (1991), 589-615.
  12. [S] R. Speicher, A new example of "independence" and "white noise", Probab. Theory Related Fields 84 (1990), 141-159.
  13. [S-W] R. Speicher and W. von Waldenfels, A general central limit theorem and invariance principle, in: Quantum Probability and Related Topics, Vol. IX, World Scientific, 1994, 371-387.
  14. [T] H. Tamanoi, Higher Schwarzian operators and combinatorics of the Schwarzian derivative, Math. Ann. 305 (1996), 127-151.
  15. [V] D. Voiculescu, Symmetries of some reduced free product C*-algebras, in: Operator Algebras and their Connections with Topology and Ergodic Theory, Lecture Notes in Math. 1132, Springer, Berlin, 1985, 556-588.
  16. [W] W. von Waldenfels, An algebraic central limit theorem in the anticommuting case, Z. Wahrsch. Verw. Gebiete 42 (1979), 135-140.
Studia Mathematica
1998/129/3
Export to PBN, Opens in new tab
Pages:
225-252
Main language of publication
English
Received
1997-02-10
Accepted
1997-11-06
Published
1998
Exact and natural sciences