scholarly journals A general principle for limit theorems in finitely additive probability — the dependent case

1985 ◽  
Vol 21 (1) ◽  
pp. 49
Keyword(s):  
General Principle ◽  
Limit Theorems ◽  
Dependent Case ◽  
Finitely Additive Probability ◽  
Additive Probability

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 $\mathcal{F}\mathcal{P}\mathcal{M}$.

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 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.

Functional limit theorems for power series with rapid decay of moving averages of Hermite processes

2021 ◽  
pp. 2150043
Author(s):  
Johann Rudolf Gehringer
Keyword(s):  
Power Series ◽  
Long Range ◽  
Limit Theorems ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Rapid Decay ◽  
Fast System ◽  
Dependent Case ◽  
Non Gaussian ◽  
Do So

We aim to obtain a homogenization theorem for a passive tracer interacting with a fractional, possibly non-Gaussian, noise. To do so, we analyze limit theorems for normalized functionals of Hermite–Volterra processes and extend existing results to cover power series with fast decaying coefficients. We obtain either convergence to a Wiener process, in the short-range dependent case, or to a Hermite process, in the long-range dependent case. Furthermore, we prove convergence in the multivariate case with both, short- and long-range dependent components. Applying this theorem, we obtain a homogenization result for a slow/fast system driven by such Hermite noises.

Automorphism–invariant measures on ℵ0-categorical structures without the independence property

1996 ◽  
Vol 61 (2) ◽  
pp. 640-652
Author(s):  
Douglas E. Ensley
Keyword(s):  
Boolean Algebra ◽  
Invariant Measures ◽  
Measure Zero ◽  
Probability Measures ◽  
Independence Property ◽  
Natural Action ◽  
Zero Sets ◽  
Finitely Additive Probability ◽  
Additive Probability

AbstractWe address the classification of the possible finitely-additive probability measures on the Boolean algebra of definable subsets of M which are invariant under the natural action of Aut(M). This pursuit requires a generalization of Shelah's forking formulas [8] to “essentially measure zero” sets and an application of Myer's “rank diagram” [5] of the Boolean algebra under consideration. The classification is completed for a large class of ℵ0-categorical structures without the independence property including those which are stable.

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