A noncommutative limit theorem for homogeneous correlations

Romuald Lenczewski

Studia Mathematica (1998)

  • Volume: 129, Issue: 3, page 225-252
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Lenczewski, Romuald. "A noncommutative limit theorem for homogeneous correlations." Studia Mathematica 129.3 (1998): 225-252. <http://eudml.org/doc/216502>.

@article{Lenczewski1998,
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.},
author = {Lenczewski, Romuald},
journal = {Studia Mathematica},
keywords = {limit theorem; quantum probability; convergence of Poisson type},
language = {eng},
number = {3},
pages = {225-252},
title = {A noncommutative limit theorem for homogeneous correlations},
url = {http://eudml.org/doc/216502},
volume = {129},
year = {1998},
}

TY - JOUR
AU - Lenczewski, Romuald
TI - A noncommutative limit theorem for homogeneous correlations
JO - Studia Mathematica
PY - 1998
VL - 129
IS - 3
SP - 225
EP - 252
AB - 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.
LA - eng
KW - limit theorem; quantum probability; convergence of Poisson type
UR - http://eudml.org/doc/216502
ER -

References

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