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An algebraic version of the central limit theorem

  • N. Giri
  • W. von Waldenfels
Article

Summary

A non-commutative analogue of the central limit theorem and the weak law of large numbers has been derived, the analogues of integrable functions being non-commutative polynomials. Without the assumption of positivity higher central limit theorems hold which have no analogy in the classical probabilistic case. The treatment includes this classical case and the convergence to so-called “quasi-free states” in the quantum mechanics of bosons [3, 4].

Keywords

Stochastic Process Quantum Mechanic Probability Theory Limit Theorem Mathematical Biology 
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References

  1. 1.
    Hepp, K., Lieb, E.H.: Phase Transitions in Reservoirdriven Open Systems with Applications to Lasers and Superconductors. Helv. Phys. Acta. 46, 573–603 (1973)Google Scholar
  2. 2.
    Cushen, C.D., Hudson, R.L.: A Quantum Mechanical Central Limit Theorem. J. Appl. Probability 8, 454–469 (1971)Google Scholar
  3. 3.
    Manuceau, J., Sirugue, M., Rocca, F., Verbeure, A.: Etats quasi-libres. Cargèse Lectures in Physics, vol. 4, Ed. D. Kastler. New York: Gordon and Breach 1970Google Scholar
  4. 4.
    Robinson, D.W.: The Ground State of the Boson Gas. Comm. Math. Phys. 1, 159–174 (1964)Google Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • N. Giri
    • 1
  • W. von Waldenfels
    • 2
  1. 1.Dept. des MathématiquesUniversité de MontréalMontréalCanada
  2. 2.Institut für Angewandte Mathematik der Universität HeidelbergHeidelbergBRD

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