Independent random partial matching with general types

Sun, Xiang

doi:10.1016/j.aim.2015.08.030

Cited by 8 publications

(4 citation statements)

References 24 publications

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“…The Loeb measure construction ( [Loe75]) has many fruitful applications in various areas in mathematics such as probability theory (see [And76], [AR78], [Per81], [Kei84], [Sto86], [Lin90], [Lin04], [DRW18], [ADS18] etc), statistical decision theory (see [DR18]), potential theory ( [Loe76]), mathematical physics (see [AHKFL86]) and mathematical economics (see [BR74], [Kha74], [BR75], [KR75], [Kha76], [Ras78], [Emm84], [And85], [And88], [And91], [Sun96], [AKRS97], [Sun99], [KS99], [AKS03], [Rau07], [DS07], [AR08], [Sun16], [DQS18], [CHLS19], among others). 1 These applications are made possible by a well-developed theory of integration (see [Loe75] and [And76]), representation of measures (see [And82] and [Sti97]) and a Fubini theorem for Loeb measures (see [Kei84]).…”

Section: Introductionmentioning

confidence: 99%

Loeb extension and Loeb equivalence

Anderson

Duanmu

Schrittesser

et al. 2021

Proc. Amer. Math. Soc. Ser. B

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In [J. London Math. Soc. 69 (2004), pp. 258-272] Keisler and Sun leave open several questions regarding Loeb equivalence between internal probability spaces; specifically, whether under certain conditions, the Loeb measure construction applied to two such spaces gives rise to the same measure. We present answers to two of these questions, by giving two examples of probability spaces. Moreover, we reduce their third question to the following: Is the internal algebra generated by the union of two Loeb equivalent internal algebras a subset of their common Loeb extension? We also present a sufficient condition for a positive answer to this question.

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Section: Introductionmentioning

confidence: 99%

Loeb extension and Loeb equivalence

Anderson

Duanmu

Schrittesser

et al. 2021

Proc. Amer. Math. Soc. Ser. B

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show abstract

“…The Loeb measure construction ( [Loe75]) has provided many fruitful applications in various areas in mathematics such as probability theory (see [And76], [AR78], [Per81], [Kei84], [Sto86], [Lin90], [Lin04], [DRW18], [ADS18] etc), statistical decision theory (see [DR18]), potential theory ( [Loe76]), mathematical physics (see [Alb+86]) and mathematical economics (see [BR74], [Kha74], [BR75], [KR75], [Kha76], [Ras78], [Emm84], [And85], [And88], [And91], [Sun96], [KRS97], [Sun99], [KS99], [AKS03], [Rau07], [DS07], [AR08], [Sun16], [DQS18], [Che+19] etc). 1 These applications are supported by the development of mathematical infrastructures such as integration theory (see [Loe75] and [And76]), representation of measures (see [And82] and [Sti97]) and Fubini theorem (see [Kei84]).…”

Section: Introductionmentioning

confidence: 99%

Loeb Extension and Loeb Equivalence

Anderson¹,

Duanmu²,

Schrittesser³

et al. 2020

Preprint

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In [KS04], the authors raise several open problems on Loeb equivalences between various internal probability spaces. We provide counter-examples for the first two open problems. Moreover, we reduce the third open problem to the following question: Is the internal algebra generated by the union of two Loeb equivalent internal algebras a subset of the Loeb extension of any one of the internal algebra?

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“…These measures, together with the Loeb measure construction [38,39,40], have become the main tool of nonstandard measure theory and can be applied to the study of a variety of mathematical objects. Some examples include generalized functions (see for instance Bottazzi [6] and Cutland [12]), stochastic processes (examples include Anderson [1], Duanmu, Rosenthal and William [13], Keisler [29], Perkins [42]), statistical decision theory (Duanmu and Roy [14]), and mathematical economics (Anderson and Raimondo [2], Yeneng Sun and collaborators [15,16,31,32,49,50], Khan [30] and Xiang Sun [48]).…”

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confidence: 99%

Integration with filters

Bottazzi

Eskew

2022

J. Log. Anal.

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We introduce a notion of integration defined from filters over families of finite sets. This procedure corresponds to determining the average value of functions whose range lies in any algebraic structure in which finite averages make sense. The most relevant scenario involves algebraic structures that extend the field of rational numbers; hence, it is possible to associate to the filter integral an upper and lower standard part, which can be interpreted as upper and lower bounds on the average value of the function that one expects to observe empirically. We discuss the main properties of the filter integral and we show that it is expressive enough to represent every real integral. As an application, we define a geometric measure on an infinite-dimensional vector space that overcomes some of the known limitations of real-valued measures. We also discuss how the filter integral can be applied to the problem of non-Archimedean integration, and we develop the iteration theory for these integrals.

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Independent random partial matching with general types

Cited by 8 publications

References 24 publications

Loeb extension and Loeb equivalence

Loeb extension and Loeb equivalence

Loeb Extension and Loeb Equivalence

Integration with filters

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