Infinite Product Measures and Infinite Convolutions

1940 ◽  
Vol 62 (1/4) ◽  
pp. 417 ◽  
Author(s):  
E. R. Van Kampen
Keyword(s):  
Infinite Product ◽  
Product Measures

On Equivalence of Infinite Product Measures

1948 ◽  
Vol 49 (1) ◽  
pp. 214 ◽  
Author(s):  
Shizuo Kakutani
Keyword(s):  
Infinite Product ◽  
Product Measures

scholarly journals Γ-convergence of Onsager–Machlup functionals: II. Infinite product measures on Banach spaces

2021 ◽  
Vol 38 (2) ◽  
pp. 025006 ◽  
Author(s):  
Birzhan Ayanbayev ◽  
Ilja Klebanov ◽  
Han Cheng Lie ◽  
T J Sullivan
Keyword(s):  
Hilbert Spaces ◽  
Infinite Product ◽  
A Posteriori ◽  
Transition Path Theory ◽  
Bayesian Inverse Problems ◽  
Countable Product ◽  
Convergence Of Sequences ◽  
Product Measures ◽  
Γ Convergence ◽  
Convergent Sequences

Abstract We derive Onsager–Machlup functionals for countable product measures on weighted ℓ p subspaces of the sequence space R N . Each measure in the product is a shifted and scaled copy of a reference probability measure on R that admits a sufficiently regular Lebesgue density. We study the equicoercivity and Γ-convergence of sequences of Onsager–Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussian, Cauchy, and Besov measures with summability parameter 1 ⩽ p ⩽ 2. Together with part I of this paper, this provides a basis for analysis of the convergence of maximum a posteriori estimators in Bayesian inverse problems and most likely paths in transition path theory.

A Class of Singular Functions

1968 ◽  
Vol 20 ◽  
pp. 1425-1431 ◽  
Author(s):  
Harold S. Shapiro
Keyword(s):  
Infinite Product ◽  
General Theorem ◽  
Sufficient Conditions ◽  
Measurable Space ◽  
Absolutely Continuous ◽  
Borel Field ◽  
Singular Functions ◽  
Product Measures ◽  
Necessary And Sufficient ◽  
Set Function

Kakutani (2) has proved a very general theorem, giving necessary and sufficient conditions for two infinite product measures to be mutually absolutely continuous. To formulate Kakutani's result, let us first recall that a measurable space is a pair (E, B), where B denotes a Borel field (also called σ-ring) of subsets of E, and a measure m on this space is a countably additive set function on B (see Halmos (1)).

scholarly journals Infinite Matrix Products and the Representation of the Matrix Gamma Function

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
J.-C. Cortés ◽  
L. Jódar ◽  
Francisco J. Solís ◽  
Roberto Ku-Carrillo
Keyword(s):  
Gamma Function ◽  
Infinite Product ◽  
Infinite Matrix ◽  
Convergence Results ◽  
Matrix Products ◽  
The Matrix ◽  
Definition Of

We introduce infinite matrix products including some of their main properties and convergence results. We apply them in order to extend to the matrix scenario the definition of the scalar gamma function given by an infinite product due to Weierstrass. A limit representation of the matrix gamma function is also provided.

Inequalities and infinite product formula for Ramanujan generalized modular equation function

2017 ◽  
Vol 46 (1) ◽  
pp. 189-200 ◽  
Author(s):  
Miao-Kun Wang ◽  
Yong-Min Li ◽  
Yu-Ming Chu
Keyword(s):  
Infinite Product ◽  
Product Formula ◽  
Modular Equation

scholarly journals To what extent and why adolescents do or do not support future tobacco control measures: a multimethod study in the Netherlands

2017 ◽  
Vol 27 (5) ◽  
pp. 596-599 ◽  
Author(s):  
Michael Schreuders ◽  
Naomi A Lagerweij ◽  
Bas van den Putte ◽  
Anton E Kunst
Keyword(s):  
The Netherlands ◽  
Tobacco Control ◽  
Control Measures ◽  
Price Increases ◽  
Mixed Support ◽  
Mixed Method Design ◽  
Product Measures ◽  
Group Interviews ◽  
Political Leverage ◽  
High Support

BackgroundIn the Netherlands, the adoption of new tobacco control measures is needed to further reduce rates of adolescent smoking. Adolescents’ support for future measures could increase the likelihood of adoption as this provides political leverage for tobacco control advocates. There is, however, scant evidence about to what extent and why adolescents support future measures. We therefore assessed adolescents’ support for a range of future measures and explored the criteria that adolescents use to underpin their support.MethodsA mixed-method design involved surveys and group interviews with fourth-year students (predominantly 15–16 years). The survey, completed by 345 adolescents, included statements about future tobacco control measures and a smoke-free future where nobody starts or continues smoking. Thereafter, 15 adolescents participated in five group interviews to discuss their support for future measures.ResultsThe survey showed that adolescents generally support a smoke-free future. They expressed most support for product measures, mixed support for smoke-free areas, ambivalent support for price increases and least support for sales restrictions. The group interviews revealed that differences in support were explained by adolescents’ criteria that future measures should: have the potential to be effective, not violate individuals’ right to smoke, protect children from pro-smoking social influences and protect non-smokers from secondhand smoke.ConclusionAdolescents’ high support for a smoke-free future does not lead to categorical support for any measure. Addressing the underlying criteria may increase adolescents’ support and therewith provide political leverage for the adoption of future measures.

Sign in / Sign up

Export Citation Format

Share Document