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Eta-diagonal distributions and infinite divisibility for R-diagonals
Eta-diagonal distributions and infinite divisibility for R-diagonals
- Source :
- Ann. Inst. H. Poincaré Probab. Statist. 54, no. 2 (2018), 907-937
- Publication Year :
- 2016
- Publisher :
- arXiv, 2016.
-
Abstract
- The class of R-diagonal *-distributions is fairly well understood in free probability. In this class, we consider the concept of infinite divisibility with respect to the operation $\boxplus$ of free additive convolution. We exploit the relation between free probability and the parallel (and simpler) world of Boolean probability. It is natural to introduce the concept of an eta-diagonal distribution that is the Boolean counterpart of an R-diagonal distribution. We establish a number of properties of eta-diagonal distributions, then we examine the canonical bijection relating eta-diagonal distributions to infinitely divisible R-diagonal ones. The overall result is a parametrization of an arbitrary $\boxplus$-infinitely divisible R-diagonal distribution that can arise in a C*-probability space, by a pair of compactly supported Borel probability measures on $[ 0, \infty )$. Among the applications of this parametrization, we prove that the set of $\boxplus$-infinitely divisible R-diagonal distributions is closed under the operation $\boxtimes$ of free multiplicative convolution.<br />Comment: 33 pages
- Subjects :
- Statistics and Probability
Diagonal
46L54, 60E07, 05A18
01 natural sciences
Combinatorics
010104 statistics & probability
60E07
Infinite divisibility
FOS: Mathematics
Mathematics - Combinatorics
46L53
0101 mathematics
Operator Algebras (math.OA)
Mathematics
46L54
010102 general mathematics
Probability (math.PR)
Mathematics - Operator Algebras
$\eta$-diagonal distribution
16. Peace & justice
Free additive convolution
R-diagonal distribution
Combinatorics (math.CO)
Statistics, Probability and Uncertainty
60E10
Mathematics - Probability
Subjects
Details
- Database :
- OpenAIRE
- Journal :
- Ann. Inst. H. Poincaré Probab. Statist. 54, no. 2 (2018), 907-937
- Accession number :
- edsair.doi.dedup.....d56fd6d983a77ca63b72b182f0b33769
- Full Text :
- https://doi.org/10.48550/arxiv.1608.03515