η-series and a Boolean Bercovici–Pata bijection for bounded k-tuples

Belinschi, Serban; Nica, Alexandru

doi:10.1016/j.aim.2007.06.015

Cited by 36 publications

(138 citation statements)

References 15 publications

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“…The theory of free infinite divisibility parallels the classical one, and in particular, a Lévy-Khitchine formula does exist to characterize infinitely divisible laws, see [BeP00], [BaNT04]. The former paper introduces the Bercovici-Pata bijection between the classical and free infinitely divisible laws (see also the Boolean Bercovici-Pata bijection in [BN08]). Matrix approximations to free infinitely divisible laws are constructed in [BeG05].…”

Section: Bibliographical Notesmentioning

confidence: 99%

An Introduction to Random Matrices

2009

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The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

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Section: Bibliographical Notesmentioning

confidence: 99%

An Introduction to Random Matrices

2009

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“…Regarding equations (45) and (46) below, the reader is referred to the earlier references [2,15]. From [1] we recall the notion of irreducible noncrossing partition, which is a non-crossing partition of the set rns with the first and last element p1, nq being in the same block.…”

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confidence: 99%

Monotone, free, and boolean cumulants: A shuffle algebra approach

Ebrahimi‐Fard

Patras

2018

Advances in Mathematics

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Abstract. The theory of cumulants is revisited in the "Rota way", that is, by following a combinatorial Hopf algebra approach. Monotone, free, and boolean cumulants are considered as infinitesimal characters over a particular combinatorial Hopf algebra. The latter is neither commutative nor cocommutative, and has an underlying unshuffle bialgebra structure which gives rise to a shuffle product on its graded dual. The moment-cumulant relations are encoded in terms of shuffle and half-shuffle exponentials. It is then shown how to express concisely monotone, free, and boolean cumulants in terms of each other using the pre-Lie Magnus expansion together with shuffle and half-shuffle logarithms.

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“…Belinschi and Nica defined a multivariable extension of the Bercovici–Pata bijection , which is a bijection between the set of joint distributions of non‐commutative random variables and the subset of joint distributions which are infinitely divisible. We refer the reader to for background, details and further references. In this section, we show how the algebraic properties of the bijection can be accounted for and studied using the point of view of half‐shuffle logarithms and exponentials.…”

Section: The Bercovici–pata Bijection Revisitedmentioning

confidence: 99%

“…The relations between

normalΦ

and the various cumulants translate into relations between their generating series. For example (compare with ), the identity

Φ = e + Φ ≻ β

translates (using the definition of the right half‐shuffle

≻

and the fact that

β

is an infinitesimal character) into

M_{μ} = η_{μ} + M_{μ} \cdot η_{μ} .

…”

Section: The Bercovici–pata Bijection Revisitedmentioning

confidence: 99%

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Shuffle group laws: applications in free probability

Ebrahimi‐Fard

Patras

2019

Proc. London Math. Soc.

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Commutative shuffle products are known to be intimately related to universal formulas for products, exponentials and logarithms in group theory as well as in the theory of free Lie algebras, such as, for instance, the Baker–Campbell–Hausdorff formula or the analytic expression of a Lie group law in exponential coordinates in the neighbourhood of the identity. Non‐commutative shuffle products happen to have similar properties with respect to pre‐Lie algebras. However, the situation is more complex since in the non‐commutative framework three exponential‐type maps and corresponding logarithms are naturally defined. This results in several new formal group laws together with new operations, for example, a new notion of adjoint action particularly well fitted to the new theory. These developments are largely motivated by various constructions in non‐commutative probability theory. The second part of the article is devoted to exploring and deepening this perspective. We illustrate our approach by revisiting universal products from a group‐theoretical viewpoint, including additive convolution in monotone, free and boolean probability, as well as the Bercovici–Pata bijection and subordination products.

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η-series and a Boolean Bercovici–Pata bijection for bounded k-tuples

Cited by 36 publications

References 15 publications

An Introduction to Random Matrices

An Introduction to Random Matrices

Monotone, free, and boolean cumulants: A shuffle algebra approach

Shuffle group laws: applications in free probability

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