Limit Theorems for Non-Commutative Random Variables

A. Coja-Oghlan () and J. Michaliček ()
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A. Coja-Oghlan: Humboldt-Universität zu Berlin
J. Michaliček: Universität Hamburg

Journal of Theoretical Probability, 2005, vol. 18, issue 3, 595-614

Abstract: Abstract We study the weak law of large numbers and the central limit theorem for non-commutative random variables. We first define the concepts of variance and expectation for probability measures on homogeneous spaces, and formulate the weak law of large numbers and the central limit theorem for probability measures on locally compact groups. Then, we consider the non-commutative case, where the homogeneous space is replaced by a C*-algebra $${\cal A}$$ that is equipped with a locally compact group G of automorphisms. We define the concepts of variance and expectation in the non-commutative situation. Furthermore, we prove that the weak law of large numbers and the central limit theorem hold for non-commutative random variables on $${\cal A}$$ if they hold on the group G of automorphisms.

Keywords: C*-algebras; non-commutative probability theory; central limit theorem; law of large numbers (search for similar items in EconPapers)
Date: 2005
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DOI: 10.1007/s10959-005-7251-x

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