MULTILINEAR FUNCTION SERIES AND TRANSFORMS IN FREE PROBABILITY THEORY
By K. Dykema. 2005
52 citations recorded by the source corpus.
Abstract The algebra Mul 〚 B 〛 of formal multilinear function series over an algebra B and its quotient SymMul 〚 B 〛 are introduced, as well as corresponding operations of formal composition. In the setting of Mul 〚 B 〛 , the unsymmetrized R- and T-transforms of random variables in B-valued noncommutative probability spaces are introduced. These satisfy properties analogous to the usual R- and T-transforms (the latter being just the reciprocal of the S-transform), but describe all moments of a random variable, not only the symmetric moments. The partially ordered set of noncrossing linked partitions is introduced and is used to prove properties of the unsymmetrized T-transform.
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/dcf355c61921dbfcc9bb26339727fd3bc4d0ad73
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In Algebra and Number Theory
In Discrete Mathematics and Combinatorics
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Citations in the corpus
- Lectures on the Combinatorics of Free Probability (2006, methodology): “[27] K. Dykema and U. Haagerup: Invariant subspaces of Voiculescu’s circular operator, Geometric and Functional Analysis 11 (2001), 693–741.”
- Free analysis questions II: The Grassmannian completion and the series expansions at the origin (2008, background): “n how the B-valued R-transform ([13]) gives rise naturally to a fully matricial R-transform and this in turn to a noncommutative series R-transform which is precisely the unsymmetrized R-transform of [5]. In this way all B-moments, not only the symmetric ones, are encoded in a generalization of the ”
- Free Probability and Random Matrices (2014, background): “For a more thorough discussion of this point of view (and more results about operator-valued free probability) one should consult the work of Dykema [68].”