A CONNECTION BETWEEN FREE AND CLASSICAL INFINITE DIVISIBILITY

2004 ◽  
Vol 07 (04) ◽  
pp. 573-590 ◽  
Author(s):  
O. E. BARNDORFF-NIELSEN ◽  
S. THORBJØRNSEN
Keyword(s):  
Probability Measure ◽  
Free Probability ◽  
Infinite Divisibility ◽  
Probability Measures ◽  
Stochastic Integration ◽  
Alternative Definition ◽  
Infinitely Divisible ◽  
Definition Of ◽  
Divisible Probability Measure

In this paper we continue our studies, initiated in Refs. 2–4, of the connections between the classes of infinitely divisible probability measures in classical and in free probability. We show that the free cumulant transform of any freely infinitely divisible probability measure equals the classical cumulant transform of a certain classically infinitely divisible probability measure, and we give several characterizations of the latter measure, including an interpretation in terms of stochastic integration. We find, furthermore, an alternative definition of the Bercovici–Pata bijection, which passes directly from the classical to the free cumulant transform, without passing through the Lévy–Khintchine representations (classical and free, respectively).

scholarly journals Interacting Fock Spaces and Gaussianization of Probability Measures

1998 ◽  
Vol 01 (04) ◽  
pp. 663-670 ◽  
Author(s):  
Luigi Accardi ◽  
Marek Bożejko
Keyword(s):  
Probability Measure ◽  
Orthogonal Polynomials ◽  
Fock Space ◽  
Field Operator ◽  
Multiplication Operator ◽  
Probability Measures ◽  
Number Operator ◽  
Infinitely Divisible ◽  
Future Paper ◽  
Interacting Fock Space

We prove that any probability measure on ℝ, with moments of all orders, is the vacuum distribution, in an appropriate interacting Fock space, of the field operator plus (in the nonsymmetric case) a function of the number operator. This follows from a canonical isomorphism between the L2-space of the measure and the interacting Fock space in which the number vectors go into the orthogonal polynomials of the measure and the modified field operator into the multiplication operator by the x-coordinate. A corollary of this is that all the momenta of such a measure are expressible in terms of the Szëgo–Jacobi parameters, associated to its orthogonal polynomials, by means of diagrams involving only noncrossing pair partitions (and singletons, in the nonsymmetric case). This means that, with our construction, the combinatorics of the momenta of any probability measure (with all moments) is reduced to that of a generalized Gaussian. This phenomenon we call Gaussianization. Finally we define, in terms of the Szëgo–Jacobi parameters, a new convolution among probability measures which we call universal because any probability measure (with all moments) is infinitely divisible with respect to this convolution. All these results will be extended to the case of many (in fact infinitely many) variables in a future paper.

INFINITE DIVISIBILITY FOR THE CONDITIONALLY FREE CONVOLUTION

2007 ◽  
Vol 10 (04) ◽  
pp. 499-522 ◽  
Author(s):  
ANNA DOROTA KRYSTEK
Keyword(s):  
Infinite Divisibility ◽  
Weak Limit ◽  
Complete Characterization ◽  
Divisible Distribution ◽  
Probability Measures ◽  
Infinitely Divisible ◽  
Free Convolution ◽  
The Central Limit Theorem ◽  
Free Case

Infinite divisibility for the free additive convolution was studied in Ref. 20. A complete characterization of [Formula: see text]-infinitely divisible distributions was given, and it was explained in Ref. 21 that this characterization is an analogue of the classical Lévy–Khintchine characterization. In fact, the analogue of the Gaussian distribution appeared even earlier, when the central limit theorem for free additive convolution was proven in Ref. 19. In this paper we define the notion of [Formula: see text]-infinitely divisibility and give the description of infinitely divisible compactly supported probability measures relative to the conditionally free convolution. We also show that the Lévy–Khintchine measures associated with a [Formula: see text]-infinitely divisible distribution μ can be calculated, as in the classical or free case, as a weak limit of measures related with the convolution semigroup generated by (μ, φ) for [Formula: see text]-infinitely divisible.

Limits of commutative triangular systems on real and p-adic groups

1996 ◽  
Vol 120 (1) ◽  
pp. 181-192 ◽  
Author(s):  
Riddhi Shah
Keyword(s):  
Symmetric Spaces ◽  
Discrete Subgroup ◽  
Infinite Divisibility ◽  
Probability Measures ◽  
Compact Groups ◽  
Nilpotent Lie Groups ◽  
Infinitely Divisible ◽  
Locally Compact ◽  
Triangular Systems ◽  
Simply Connected

A fundamental theorem of Khinchin says that every limit of an infinitesimal triangular system of probability measures on R is infinitely divisible. This was generalized to all divisible locally compact second countable abelian groups by Parthasarathy et al. (cf. [PRV]). Recently, Ruzsa eliminated the second countability condition and also proved the theorem for all Banach spaces (cf. [R2]). A similar theorem was also proved by Gangolli for certain symmetric spaces (cf. [G]). A result of Carnal shows that infinite divisibility of limits holds for commutative infinitesimal triangular systems on compact groups (cf. [C]). The same was recently proved by Neuenschwander for simply connected step-2 nilpotent Lie groups, provided the system is symmetric or supported on a discrete subgroup (cf. [N]).

scholarly journals The Effect of an Alternative Definition of “Percent Highly Annoyed” on the Exposure–Response Relationship: Comparison of Noise Annoyance Responses Measured by ICBEN 5-Point Verbal and 11-Point Numerical Scales

2021 ◽  
Vol 18 (12) ◽  
pp. 6258
Author(s):  
Makoto Morinaga ◽  
Thu Lan Nguyen ◽  
Shigenori Yokoshima ◽  
Koji Shimoyama ◽  
Takashi Morihara ◽  
...  
Keyword(s):  
Sound Pressure ◽  
Response Relationship ◽  
Alternative Definition ◽  
Exposure Response ◽  
Noise Annoyance ◽  
Numerical Scale ◽  
Verbal Scale ◽  
Sound Pressure Levels ◽  
Acoustic Surveys ◽  
Definition Of

Since the development of the 5-point verbal and 11-point numerical scales for measuring noise annoyance by the ICBEN Team 6, these scales have been widely used in socio-acoustic surveys worldwide, and annoyance responses have been easily compared internationally. However, both the top two categories of the 5–point verbal scale and the top three ones of the 11-point numerical scale are correspond to high annoyance, so it is difficult to precisely compare annoyance responses. Therefore, we calculated differences in day–evening–night-weighted sound pressure levels (Lden) by comparing values corresponding to 10% highly annoyed (HA) on Lden_%HA curves obtained from measurements in 40 datasets regarding surveys conducted in Japan and Vietnam. The results showed that the Lden value corresponding to 10% HA using the 5-point verbal scale was approximately 5 dB lower than that of the 11-point numerical scale. Thus, some correction is required to compare annoyance responses measured by the 5-point verbal and the 11-point numerical scales. The results of this study were also compared with those of a survey in Switzerland.

scholarly journals A note on the idempotent measures on countable semigroups

1970 ◽  
Vol 11 (4) ◽  
pp. 417-420
Author(s):  
Tze-Chien Sun ◽  
N. A. Tserpes
Keyword(s):  
Probability Measure ◽  
Continuous Mapping ◽  
Compact Semigroup ◽  
Probability Measures ◽  
Left Zero ◽  
Product Topology ◽  
Locally Compact ◽  
Locally Compact Semigroup ◽  
Image Position ◽  
Completely Simple

In [6] we announced the following Conjecture: Let S be a locally compact semigroup and let μ be an idempotent regular probability measure on S with support F. Then(a) F is a closed completely simple subsemigroup.(b) F is isomorphic both algebraically and topologically to a paragroup ([2], p.46) X × G × Y where X and Y are locally compact left-zero and right-zero semi-groups respectively and G is a compact group. In X × G × Y the topology is the product topology and the multiplication of any two elements is defined by , x where [y, x′] is continuous mapping from Y × X → G.(c) The induced μ on X × G × Y can be decomposed as a product measure μX × μG× μY where μX and μY are two regular probability measures on X and Y respectively and μG is the normed Haar measure on G.

An alternative definition of sustainable development using stability and chaos theories

2006 ◽  
Vol 14 (1) ◽  
pp. 62-71 ◽  
Author(s):  
Amir Abbas Rassafi ◽  
Hossain Poorzahedy ◽  
Manouchehr Vaziri
Keyword(s):  
Sustainable Development ◽  
Alternative Definition ◽  
Definition Of

Categorical concepts for parameterized partial specifications

1995 ◽  
Vol 5 (2) ◽  
pp. 153-188 ◽  
Author(s):  
Ingo Claßen ◽  
Martin GroßE-Rhode ◽  
Uwe Wolter
Keyword(s):  
Structural Theory ◽  
Specification Languages ◽  
Alternative Definition ◽  
Categorical Approach ◽  
Categorical Constructions ◽  
Definition Of ◽  
Act One

Categorical constructions inherent to a theory of algebras with strict partial operations are presented and exploited to provide a categorical deduction calculus for conditional existence equations and an alternative definition of such algebras based on the notion of syntactic categories. A compact presentation of the structural theory of parameterized (partial) specifications is given using the categorical approach. This theory is shown to be suitable for providing initial semantics as well as the compositionality results necessary for the definition of specification languages like ACT ONE and ACT TWO

On the Very Idea of Civilisation

2021 ◽  
Vol 31 (2) ◽  
pp. 307-321
Author(s):  
Luke O’Sullivan ◽  
Keyword(s):  
Technological Development ◽  
Thomas Hobbes ◽  
Alternative Definition ◽  
Starting Point ◽  
Western Imperialism ◽  
Clash Of Civilisations ◽  
Definition Of ◽  
Contemporary Liberalism ◽  
Relationship Of ◽  
The Relationship

The concept of civilisation is a controversial one because it is unavoidably normative in its implications. Its historical associations with the effort of Western imperialism to impose substantive conditions of life have made it difficult for contemporary liberalism to find a definition of “civilization” that can be reconciled with progressive discourse that seeks to avoid exclusions of various kinds. But because we lack a way of identifying what is peculiar to the relationship of civilisation that avoids the problem of domination, it has tended to be conflated with other ideas. Taking Samuel Huntington's idea of a “Clash of Civilisations” as a starting point, this article argues that we suffer from a widespread confusion of civilisation with “culture,” and that we also confuse it with other ideas including modernity and technological development. Drawing on Thomas Hobbes, the essay proposes an alternative definition of civilisation as the existence of limits on how we may treat others.

Learning Patterns in the Disadvantaged

1967 ◽  
Vol 37 (4) ◽  
pp. 546-593 ◽  
Author(s):  
Susan Stodolsky ◽  
Gerald Lesser
Keyword(s):  
Educational Policy ◽  
Educational Opportunity ◽  
Equal Educational Opportunity ◽  
Alternative Definition ◽  
Future Directions ◽  
Disadvantaged Children ◽  
Learning Patterns ◽  
Definition Of

The authors review evidence and suggest future directions for research on the learning patterns of disadvantaged children. After a detailed description of a specific case of research, some implications for educational policy are discussed. The authors take issue with James S. Coleman's definition of the concept of "equal educational opportunity"and advance an alternative definition. The problem of achieving a useful definition of the term "disadvantaged"is addressed throughout the paper.

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