scholarly journals Remarks on continuously distributed sequences

2021 ◽  
Vol 13 (1) ◽  
pp. 89-97
Author(s):  
M. Paštéka
Keyword(s):  
Distribution Function ◽  
Probability Measure ◽  
Finitely Additive Probability ◽  
Additive Probability

In the first part of the paper we define the notion of the density as certain type of finitely additive probability measure and the distribution function of sequences with respect to the density. Then we derive some simple criterions providing the continuity of the distribution function of given sequence. These criterions we apply to the van der Corput's sequences. The Weyl's type criterions of continuity of the distribution function are proven.

PROBABILITY MEASURES ON PROJECTIONS IN VON NEUMANN ALGEBRAS

1989 ◽  
Vol 01 (02n03) ◽  
pp. 235-290 ◽  
Author(s):  
SHUICHIRO MAEDA
Keyword(s):  
Probability Measure ◽  
Linear Functional ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Probability Measures ◽  
Type I ◽  
Von Neumann ◽  
Positive Linear Functional ◽  
Finitely Additive Probability ◽  
Additive Probability

A state ϕ on a von Neumann algebra A is a positive linear functional on A with ϕ(1) = 1, and the restriction of ϕ to the set of projections in A is a finitely additive probability measure. Recently it was proved that if A has no type I 2 summand then every finitely additive probability measure on projections can be extended to a state on A. Here we give precise and complete arguments for proving this result.

scholarly journals Translation invariance and finite additivity in a probability measure on the natural numbers

2002 ◽  
Vol 29 (10) ◽  
pp. 585-589 ◽  
Author(s):  
Robert Gardner ◽  
Robert Price
Keyword(s):  
Probability Measure ◽  
Translation Invariance ◽  
Finite Additivity ◽  
Natural Numbers ◽  
Translation Invariant ◽  
Finitely Additive Probability ◽  
Power Set ◽  
Additive Probability ◽  
Two Envelopes

Inspired by the “two envelopes exchange paradox,” a finitely additive probability measuremon the natural numbers is introduced. The measure is uniform in the sense thatm({i})=m({j})for alli,j∈ℕ. The measure is shown to be translation invariant and has such desirable properties asm({i∈ℕ|i≡0(mod2)})=1/2. For anyr∈[0,1], a setAis constructed such thatm(A)=r; however,mis not defined on the power set ofℕ. Finally, a resolution to the two envelopes exchange paradox is presented in terms ofm.

scholarly journals Probability logic: A model-theoretic perspective

2020 ◽  
Author(s):  
M Pourmahdian ◽  
R Zoghifard
Keyword(s):  
Characterization Theorem ◽  
Expressive Power ◽  
Probability Logic ◽  
Theoretic Analysis ◽  
Compactness Property ◽  
Probability Models ◽  
Finitely Additive Probability ◽  
Additive Probability ◽  
Basic Probability ◽  
Abstract Logics

Abstract This paper provides some model-theoretic analysis for probability (modal) logic ($PL$). It is known that this logic does not enjoy the compactness property. However, by passing into the sublogic of $PL$, namely basic probability logic ($BPL$), it is shown that this logic satisfies the compactness property. Furthermore, by drawing some special attention to some essential model-theoretic properties of $PL$, a version of Lindström characterization theorem is investigated. In fact, it is verified that probability logic has the maximal expressive power among those abstract logics extending $PL$ and satisfying both the filtration and disjoint unions properties. Finally, by alternating the semantics to the finitely additive probability models ($\mathcal{F}\mathcal{P}\mathcal{M}$) and introducing positive sublogic of $PL$ including $BPL$, it is proved that this sublogic possesses the compactness property with respect to $\mathcal{F}\mathcal{P}\mathcal{M}$.

scholarly journals Bounds of modes and unimodal processes with independent increments

1986 ◽  
Vol 104 ◽  
pp. 29-42 ◽  
Author(s):  
Ken-Iti Sato
Keyword(s):  
Distribution Function ◽  
Probability Measure ◽  
Closed Interval ◽  
Finite Variance ◽  
Unimodal Distribution ◽  
Independent Increments ◽  
Mean And Variance ◽  
Point Set ◽  
Processes With Independent Increments

A probability measure μ is called unimodal if there is a point α such that the distribution function of μ is convex on (− ∞, α) and concave on (α, ∞). The point α is called a mode of μ. When μ is unimodal, the mode of μ is not always unique; the set of modes is a one point set or a closed interval. If μ is a unimodal distribution with finite variance, Johnson and Rogers [6] give a boundwhere m and v are mean and variance of μ (see also [11]).

Analytic Sets, Baire Sets and the Standard Part Map

1979 ◽  
Vol 31 (3) ◽  
pp. 663-672 ◽  
Author(s):  
C. Ward Henson
Keyword(s):  
Hausdorff Space ◽  
Original Problem ◽  
Nonstandard Analysis ◽  
Topological Spaces ◽  
Theoretical Structure ◽  
Standard Part ◽  
Finitely Additive Probability ◽  
The Family ◽  
Additive Probability ◽  
Analytic Sets

The problems considered here arose in connection with the interesting use by Loeb [8] and Anderson [1], [2] of Loeb's measure construction [7] to define measures on certain topological spaces. The original problem, from which the results given here developed, was to identify precisely the family of sets on which these measures are defined.To be precise, let be a set theoretical structure and * a nonstandard extension of , as in the usual framework for nonstandard analysis (see [10]). Let X be a Hausdorff space in and stx the standard part map for X, defined on the set of nearstandard points in *X. Suppose, for example, µ is an internal, finitely additive probability measure defined on the internal subsets of *X.

scholarly journals The Fréchet transform

1993 ◽  
Vol 16 (1) ◽  
pp. 155-164
Author(s):  
Piotor Mikusiński ◽  
Morgan Phillips ◽  
Howard Sherwood ◽  
Michael D. Taylor
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Probability Measure ◽  
Probability Distribution Function ◽  
Random Variables ◽  
Distribution Functions ◽  
Probability Distribution Functions

LetF1,…,FNbe1-dimensional probability distribution functions andCbe anN-copula. Define anN-dimensional probability distribution functionGbyG(x1,…,xN)=C(F1(x1),…,FN(xN)). Letν, be the probability measure induced onℝNbyGandμbe the probability measure induced on[0,1]NbyC. We construct a certain transformationΦof subsets ofℝNto subsets of[0,1]Nwhich we call the Fréchet transform and prove that it is measure-preserving. It is intended that this transform be used as a tool to study the types of dependence which can exist between pairs orN-tuples of random variables, but no applications are presented in this paper.

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