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Free probability for probabilists. (English) Zbl 1105.46045
Attal, Stephane (ed.) et al., Quantum probability communications. Vol XII. Lectures from the Grenoble summer school, Grenoble, France, June 1998. River Edge, NJ: World Scientific (ISBN 981-238-427-8/hbk; 981-238-959-8/pbk; 981-238-428-6/set). QP-PQ: Quantum Probability and White Noise Analysis 12, 55-71 (2003).
This survey paper provides an introduction to free probability theory, which deals with noncommutative “random variables” where freeness plays the rôle of independence in classical stochastics. It addresses an audience with a probabilistic background, so the prerequisites on operator algebras are kept to a minimum.
The author stresses the similarities and differences between free probability theory and its classical counterpart. The following topics are treated: free random variables and their relation with random matrices; free convolution of measures; the \(R\)-transform; the free law of large numbers; the free central limit theorem; the free Lévy–Khinchin formula; Speicher’s approach to freeness; eigenvectors of random matrices.
It should be noted that this paper was written in 1998, but published (and reviewed) only much later.
For the entire collection see [Zbl 1073.81003].

MSC:
46L54 Free probability and free operator algebras
60A05 Axioms; other general questions in probability
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
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