Operator-Valued and Multivariate Free Berry-Esseen Theorems

Mai, Tobias; Speicher, Roland

doi:10.1007/978-3-642-36068-8_7

Cited by 7 publications

(6 citation statements)

References 14 publications

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“…We refer to [16,26,17,27,28,29] for background on operator-valued free independence (a.k.a. free independence with amalgamation), and to [30,31,15,32] for background on operator-valued monotone independence.…”

Section: Operator-valued Non-commutative Probabilitymentioning

confidence: 99%

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Operator-valued chordal Loewner chains and non-commutative probability

Jekel

2020

Journal of Functional Analysis

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We adapt the theory of chordal Loewner chains to the operator-valued matricial upper-half plane over a C * -algebra A. We define an A-valued chordal Loewner chain as a subordination chain of analytic self-maps of the A-valued upper halfplane, such that each F t is the reciprocal Cauchy transform of an A-valued law µ t , such that the mean and variance of µ t are continuous functions of t.We relate A-valued Loewner chains to processes with A-valued free or monotone independent independent increments just as was done in the scalar case by Bauer [1] and Scheißinger [2].We show that the Loewner equation, when interpreted in a certain distributional sense, defines a bijection between Lipschitz mean-zero Loewner chains F t and vector fieldsBased on the Loewner equation, we derive a combinatorial expression for the moments of µ t in terms of ν t . We also construct non-commutative random variables on an operator-valued monotone Fock space which realize the laws µ t . Finally, we prove a version of the monotone central limit theorem which describes the behavior of F t as t → +∞ when ν t has uniformly bounded support.Remark 2.11. Note that by the previous lemma and some basic results on L ∞ Boch , there is an isometric inclusion ι : L ∞ Boch ([0, T ], X ) → L(L 1 [0, T ], X ) given by, so in the sequel we will regard L ∞ Boch ([0, T ], X ) as a subspace of L(L 1 [0, T ], X ).If we had a bounded function R : [0, T ] × [0, T ] → X denoted R(s, t), then could define the diagonal restriction R(t, t). We claim that under appropriate hypotheses, this operation still makes sense when R(s, ·) is an element of L(L 1 [0, T ], X ) rather than a bounded function [0, T ] → X . For this to be rigorous, we must view R as a map [0, T ] → L(L 1 [0, T ], X ). Lemma 2.12 (Diagonal restriction). There exists a unique linear map diag : L ∞ Boch ([0, T ], L(L 1 [0, T ], X )) → L(L 1 [0, T ], X ) such that

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Section: Operator-valued Non-commutative Probabilitymentioning

confidence: 99%

“…Now we will adapt Bauer's observation that processes with freely independent increments give rise to Loewner chains [1] [2, §3.5]. We first review the notion of operator-valued free independence from [26,17,27,28] Definition 4.18. Let (B, E) be an A-valued probability space.…”

Section: Loewner Chains and Free Independencementioning

confidence: 99%

Operator-valued chordal Loewner chains and non-commutative probability

Jekel

2020

Journal of Functional Analysis

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“…This classical result has been successfully extended to random matrices with dependent entries [12] as well to random matrices with exchangeable entries [1]. Finally inspired by application in communication theory operator-valued matrices have been the object of increased interest [5,18].…”

Section: Related Literaturementioning

confidence: 97%

“…Just as in classical probability, Berry-Essen type bounds guarantee a speed of convergence for the free central-limit theorem. They were first established for sums of bounded free operators [14] then later for free unbounded operators [9], and finally to the multivariate and conditionally free case [18]. Semi-circular limits have also been of great interest in the random matrix literature.…”

Section: Related Literaturementioning

confidence: 99%

A free central-limit theorem for dynamical systems

Austern¹

2020

Preprint

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The free central-limit theorem, a fundamental theorem in free probability, states that empirical averages of freely independent random variables are asymptotically semi-circular. We extend this theorem to general dynamical systems of operators that we define using a free random variable X and a group of *-automorphims describing the evolution of X. We introduce free mixing coefficients that measure how far a dynamical system is from being freely independent. Under conditions on those coefficients, we prove that the free central-limit theorem also holds for these processes and provide Berry-Essen bounds. We generalize this to triangular arrays and U-statistics. Finally we draw connections with classical probability and random matrix theory with a series of examples.

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“…For one-dimensional free probability, a sharper rate, without the logarithmic factor, has been obtained in [7]. A rate of convergence at the level of the Cauchy transforms of polynomial evaluations was obtained in [12]; see [1] for an improved result.…”

Section: Introductionmentioning

confidence: 99%

A note on existence of free Stein kernels

Cébron

Fathi

Mai

2019

Proc. Amer. Math. Soc.

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Stein kernels are a way of comparing probability distributions, defined via integration by parts formulas. We provide two constructions of Stein kernels in free probability. One is given by an explicit formula, and the other via free Poincaré inequalities. In particular, we show that unlike in the classical setting, free Stein kernels always exist. As corollaries, we derive new bounds on the rate of convergence in the free CLT, and a strengthening of a characterization of the semicircular law due to Biane. MSC: 46L54; 60F05.

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Operator-Valued and Multivariate Free Berry-Esseen Theorems

Cited by 7 publications

References 14 publications

Operator-valued chordal Loewner chains and non-commutative probability

Operator-valued chordal Loewner chains and non-commutative probability

A free central-limit theorem for dynamical systems

A note on existence of free Stein kernels

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