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
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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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