R-Cyclic Families of Matrices in Free Probability

Nica, Alexandru; Shlyakhtenko, Dimitri; Speicher, Roland

doi:10.1006/jfan.2001.3814

Cited by 33 publications

(38 citation statements)

References 17 publications

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“…, X s from M n (B), but with amalgamation now over the algebra of diagonal matrices with entries in B. This is known in the case B = C from [27], we prove this for general B in Section 3.…”

Section: Theorem 2 Letmentioning

confidence: 90%

Quantum invariant families of matrices in free probability

Curran

Speicher

2011

Journal of Functional Analysis

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We consider (self-adjoint) families of infinite matrices of noncommutative random variables such that the joint distribution of their entries is invariant under conjugation by a free quantum group. For the free orthogonal and hyperoctahedral groups, we obtain complete characterizations of the invariant families in terms of an operator-valued R-cyclicity condition. This is a surprising contrast with the Aldous-Hoover characterization of jointly exchangeable arrays.

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“…, X s from M n (B), but with amalgamation now over the algebra of diagonal matrices with entries in B. This is known in the case B = C from [27], we prove this for general B in Section 3.…”

Section: Theorem 2 Letmentioning

confidence: 90%

Quantum invariant families of matrices in free probability

Curran

Speicher

2011

Journal of Functional Analysis

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“…. , X n ) is free from M n (C) with amalgamation over D n as well [24,Example 2.3]. We have shown above that A • X = ΞAΞ has the same property although A has not.…”

Section: By Part (I) This Ismentioning

confidence: 92%

“…R-cyclic matrices and the distribution of sample variance. The concept of Rcyclicity was introduced by Nica, Shlyakhtenko and Speicher [24]. Our aim is now to exhibit its relation to the sample variance and other quadratic forms.…”

Section: R-cyclic Matrices and Free Infinite Divisibility Of Quadratimentioning

confidence: 99%

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Sample variance in free probability

Ejsmont

Lehner

2017

Journal of Functional Analysis

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Abstract. Let X 1 , X 2 , . . . , X n denote i.i.d. centered standard normal random variables, then the law of the sample variance2 is the χ 2 -distribution with n − 1 degrees of freedom. It is an open problem in classical probability to characterize all distributions with this property and in particular, whether it characterizes the normal law. In this paper we present a solution of the free analogue of this question and show that the only distributions, whose free sample variance is distributed according to a free χ 2 -distribution, are the semicircle law and more generally so-called odd laws, by which we mean laws with vanishing higher order even cumulants. In the way of proof we derive an explicit formula for the free cumulants of Q n which shows that indeed the odd cumulants do not contribute and which exhibits an interesting connection to the concept of R-cyclicity.

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“…(Matrices with noncommutative entries have been studied in [36,39,40,43]. According to [30, Definition 1.2.2] when Σ = I N , B = I M , X i,j form a q-circular system.…”

Section: Introductionmentioning

confidence: 99%

Compound Real Wishart and q-Wishart Matrices

Bryc

2010

International Mathematics Research Notices

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We introduce a family of matrices with non-commutative entries that generalize the classical real Wishart matrices. With the help of the Brauer product, we derive a non-asymptotic expression for the moments of traces of monomials in such matrices; the expression is quite similar to the formula derived in [9, Theorem 2.1] for independent complex Wishart matrices. We then analyze the fluctuations about the Marchenko-Pastur law. We show that after centering by the mean, traces of real symmetric polynomials in q-Wishart matrices converge in distribution, and we identify the asymptotic law as the normal law when q = 1, and as the semicircle law when q = 0.KEY WORDS Compound Wishart matrices, Central Limit Theorem, matrices with noncommutative entries, q-Gaussian random variables, fluctuations about the Marchenko-Pastur law

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R-Cyclic Families of Matrices in Free Probability

Cited by 33 publications

References 17 publications

Quantum invariant families of matrices in free probability

Quantum invariant families of matrices in free probability

Sample variance in free probability

Compound Real Wishart and q-Wishart Matrices

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