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Non-commutative algebraic central limit theorems

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1210)

Abstract

We want to generalize some algebraic aspects of the weak law of large numbers and the central limit theorem to the non-commutative case. Let and two 2-graded algebras and a linear even mapping preserving 1. Let (ai)iGI a family of homogeneous elements of and f a polynomial in the non-commutative indeterminates xi, iGI. Assume a fixed integral number s s≧ 1. Assuming that for i1, ..., ilGI and 1≦ℓ≦s−1 we study for N→∞. At first it can be shown that it is sufficient to consider to be equal to the free algebra generated by xi, iGI and to be the free graded commutative algebra generated by the ξw, wGW′(I), where W′(I) denotes the set of nonempty words of alphabet I. Then it is proved that CN (S)(f)→μ · γS(f), where γS is higher Gaussian mapping and μ is a homomorphism containing all special informations on the ai and ω. Finally a structure theorem for γS is proved. In the case s=1 we obtain convergence to Grassmann numbers for the averages of odd ai. The structure theorem in the case s=2 induces some commutation relations, how they are known from quantum mechanics, the commutator between odd and even quantities being a Grassmann number.

Keywords

  • Central Limit Theorem
  • Commutative Algebra
  • Algebra Homomorphism
  • Homogeneous Element
  • Finite Dimensional Subspace

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Literature

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© 1986 Springer-Verlag

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von Waldenfels, W. (1986). Non-commutative algebraic central limit theorems. In: Heyer, H. (eds) Probability Measures on Groups VIII. Lecture Notes in Mathematics, vol 1210. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0077184

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  • DOI: https://doi.org/10.1007/BFb0077184

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16806-5

  • Online ISBN: 978-3-540-44852-5

  • eBook Packages: Springer Book Archive