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].
Article PDF
Similar content being viewed by others
References
Hepp, K., Lieb, E.H.: Phase Transitions in Reservoirdriven Open Systems with Applications to Lasers and Superconductors. Helv. Phys. Acta. 46, 573–603 (1973)
Cushen, C.D., Hudson, R.L.: A Quantum Mechanical Central Limit Theorem. J. Appl. Probability 8, 454–469 (1971)
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 1970
Robinson, D.W.: The Ground State of the Boson Gas. Comm. Math. Phys. 1, 159–174 (1964)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Giri, N., von Waldenfels, W. An algebraic version of the central limit theorem. Z. Wahrscheinlichkeitstheorie verw Gebiete 42, 129–134 (1978). https://doi.org/10.1007/BF00536048
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00536048