Quantum central limit theorems for strongly mixing random variables. (English) Zbl 0553.60031

N. Giri and W. von Waldenfels [ibid. 42, 129-134 (1978; Zbl 0362.60043)] extended the classical central limit theorem to a quantum mechanical framework. That approach requires the existence of moments of all orders, but very little on the algebraic structure involved (only the existence of some kind of commutation relations). The authors of the present paper extend the Giri-von Waldenfels method to the quantum analogue of sequences of dependent random variables satisfying a certain mixing condition. This technique applies both to the Bose and to the Fermi case. Hence the results obtained include e.g. those of R. L. Hudson [A quantum-mechanical central limit theorem for anti-commuting observables. J. Appl. Probab. 10, 502-509 (1973)] and of W. von Waldenfels [Z. Wahrscheinlichkeitstheor. Verw. Geb. 42, 135-140 (1978; Zbl 0405.60095)].
Reviewer: K.Schürger


60F05 Central limit and other weak theorems
81P20 Stochastic mechanics (including stochastic electrodynamics)
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