Lévy laws in free probability
- †Department of Mathematical Sciences, University of Aarhus, DK-8000 Aarhus C, Denmark; and ‡Department of Mathematics and Computer Science, University of Southern Denmark, 5230 Odense M, Denmark
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Edited by Richard V. Kadison, University of Pennsylvania, Philadelphia, PA, and approved October 7, 2002 (received for review January 29, 2002)
Abstract
This article and its sequel outline recent developments in the theory of infinite divisibility and Lévy processes in free probability, a subject area belonging to noncommutative (or quantum) probability. The present paper discusses the classes of infinitely divisible probability measures in classical and free probability, respectively, via a study of the Bercovici–Pata bijection between these classes.
Footnotes
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↵ § To whom correspondence should be addressed. E-mail: steenth{at}imada.sdu.dk.
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This paper was submitted directly (Track II) to the PNAS office.
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↵ ¶ In quantum physics, τ is of the form τ(a) = tr(ρa), where ρ is a trace class self-adjoint operator on ℋ with trace 1, that expresses the state of a quantum system, and a would be an observable, i.e., a self-adjoint operator on ℋ, the mean value of the outcome of observing a being τ (a) =tr(ρa).
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↵ ∥ In the classical sense, at the level of the entries.
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↵ ** The symbol “=d” means “has the same distribution as.”
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↵ †† Bercovici and Pata actually proved an even stronger result, namely that Λ preserves the so-called partial domain of attraction.
- Abbreviations:
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w.r.t., with respect to
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- Copyright © 2002, The National Academy of Sciences
