Quantum central limit theorems for weakly dependent maps. I
The authors consider non-commutative (n.c.) laws of large numbers and central limit theorems. In the n.c. setting the usual classical series of random variables \(X_ 1, X_ 2, \dots\) is replaced by a series \(j_ 1, j_ 2,\dots\) of homomorphisms \(j_ k : {\mathcal B} \to {\mathcal A}\) between two not necessarily commutative \(*\)-algebras \(\mathcal B\) and \(\mathcal A\). One considers sums of the forms \({1\over N} \sum^ N_ 1 j_ k(b)\) and \({1\over \sqrt{N}} \sum^ N_ 1 (j_ k(b) - \alpha_ k (b))\) in the law of large numbers and in the central limit theorem (CLT) case, respectively. (Here \(\alpha_ k(b) = \Phi(j_ k(b))\) where \(\Phi\) is a state on \(\mathcal A\) replacing the classical probability measure.) The authors generalize the usual conditions under which n.c. CLTs hold in assuming 1) commutation relations between \(j_ k(b)\) and \(j_ l(b')\) using commutation factors \(\sigma(k, l, b, b')\) and translation terms \(\varepsilon(k, l, b, b')\) and 2) weak dependence rather than independence. On the other hand, the authors' assumption \(\sigma(k, l, b, b') = \sigma(b, b')\) in the CLT case means that some of the known n.c. CLTs are excluded. There is also a continuous parameter version of the discrete results.
scientific article; zbMATH DE number 556300
scientific article