scholarly journals Topological Stationarity and Precompactness of Probability Measures

2020 ◽  
Author(s):  
Yu-Lin Chou
Keyword(s):  
Weak Convergence ◽  
Metric Space ◽  
Convergence Theory ◽  
Probability Measures ◽  
Direct Part ◽  
Sequential Behavior ◽  
Borel Probability

We prove the precompactness of a collection of Borel probability measures over an arbitrary metric space precisely under a new legitimate notion, which we term $topological$ $stationarity$, regulating the sequential behavior of Borel probability measures directly in terms of the open sets. Thus the important direct part of Prokhorov's theorem, which permeates the weak convergence theory, admits a new version with the original and sole assumption --- tightness --- replaced by topological stationarity. Since, as will be justified, our new condition is not vacuous and is logically independent of tightness, our result deepens the understanding of the connection between precompactness of Borel probability measures and metric topologies.

On the set of expansive measures

2018 ◽  
Vol 20 (07) ◽  
pp. 1750086 ◽  
Author(s):  
Keonhee Lee ◽  
C. A. Morales ◽  
Bomi Shin
Keyword(s):  
Metric Space ◽  
Weak Topology ◽  
Hausdorff Metric ◽  
Probability Measures ◽  
Compact Metric Space ◽  
Whole Space ◽  
Isolated Points ◽  
Borel Probability ◽  
Heteroclinic Points

We prove that the set of expansive measures of a homeomorphism of a compact metric space is a [Formula: see text] subset of the space of Borel probability measures equipped with the weak* topology. Next that every expansive measure of a homeomorphism of a compact metric space can be weak* approximated by expansive measures with invariant support. In addition, if the expansive measures of a homeomorphism of a compact metric space are dense in the space of Borel probability measures, then there is an expansive measure whose support is both invariant and close to the whole space with respect to the Hausdorff metric. Henceforth, if the expansive measures are dense in the space of Borel probability measures, the set of heteroclinic points has no interior and the space has no isolated points.

scholarly journals Uniformly positive entropy of induced transformations

2020 ◽  
pp. 1-10
Author(s):  
NILSON C. BERNARDES ◽  
UDAYAN B. DARJI ◽  
RÔMULO M. VERMERSCH
Keyword(s):  
Dynamical System ◽  
Metric Space ◽  
Weak Topology ◽  
Probability Measures ◽  
Entropy Theory ◽  
Positive Entropy ◽  
Local Entropy ◽  
Topological Dynamical System ◽  
Compact Metric Space ◽  
Borel Probability

Abstract Let $(X,T)$ be a topological dynamical system consisting of a compact metric space X and a continuous surjective map $T : X \to X$ . By using local entropy theory, we prove that $(X,T)$ has uniformly positive entropy if and only if so does the induced system $({\mathcal {M}}(X),\widetilde {T})$ on the space of Borel probability measures endowed with the weak* topology. This result can be seen as a version for the notion of uniformly positive entropy of the corresponding result for topological entropy due to Glasner and Weiss.

scholarly journals Weak convergence on non-separable metric spaces

1979 ◽  
Vol 28 (2) ◽  
pp. 197-204 ◽  
Author(s):  
David Pollard
Keyword(s):  
Weak Convergence ◽  
Metric Spaces ◽  
Subject Classification ◽  
Convergence Theory ◽  
Probability Measures ◽  
Borel Measures ◽  
Underlying Space

AbstractFor certain problems in the weak convergence theory of probability measures on non-separable metric spaces it is convenient to consider measures defined on a σ-field B0 smaller than the Borel σ-field. A suitable theory has been developed by Dudley and Wichura. In this paper it is shown that some of the key results in that theory can be deduced directly from the better known weak convergence theory for Borel measures. This is achieved by a remetrization of the underlying space to make it separable. The σ-field B0 contains the new Borel σ-field and weak convergence in Dudley's sense becomes equivalent to convergence of restrictions of the measures to the latter σ-field.1980 Mathematics Subject Classification (Amer. Math. Soc.): primary 60 B 10, 60 B 05.

Beyond the heuristic approach to Kolmogorov-Smirnov theorems

1982 ◽  
Vol 19 (A) ◽  
pp. 359-365 ◽  
Author(s):  
David Pollard
Keyword(s):  
Weak Convergence ◽  
Goodness Of Fit ◽  
Empirical Distribution ◽  
Distribution Functions ◽  
Heuristic Approach ◽  
Convergence Theory ◽  
Simple Method ◽  
Starting Point ◽  
Kolmogorov Smirnov ◽  
The Subject

The theory of weak convergence has developed into an extensive and useful, but technical, subject. One of its most important applications is in the study of empirical distribution functions: the explication of the asymptotic behavior of the Kolmogorov goodness-of-fit statistic is one of its greatest successes. In this article a simple method for understanding this aspect of the subject is sketched. The starting point is Doob's heuristic approach to the Kolmogorov-Smirnov theorems, and the rigorous justification of that approach offered by Donsker. The ideas can be carried over to other applications of weak convergence theory.

scholarly journals Value-distribution of twisted L-functions of normalized cusp forms

2010 ◽  
Vol 51 ◽  
Author(s):  
Alesia Kolupayeva
Keyword(s):  
Limit Theorem ◽  
Weak Convergence ◽  
Complex Plane ◽  
Dirichlet Character ◽  
Value Distribution ◽  
Cusp Forms ◽  
Probability Measures ◽  
Convergence Of Probability Measures

A limit theorem in the sense of weak convergence of probability measures on the complex plane for twisted with Dirichlet character L-functions of holomorphic normalized Hecke eigen cusp forms with an increasing modulus of the character is proved.

scholarly journals Geometrical properties of the space of idempotent probability measures

2021 ◽  
Vol 22 (2) ◽  
pp. 399
Author(s):  
Kholsaid Fayzullayevich Kholturayev
Keyword(s):  
Metric Space ◽  
Category Theory ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Probability Measures ◽  
Compact Metric Space ◽  
Geometrical Properties ◽  
Idempotent Mathematics

Although traditional and idempotent mathematics are "parallel'', by an application of the category theory we show that objects obtained the similar rules over traditional and idempotent mathematics must not be "parallel''. At first we establish for a compact metric space X the spaces P(X) of probability measures and I(X) idempotent probability measures are homeomorphic ("parallelism''). Then we construct an example which shows that the constructions P and I form distinguished functors from each other ("parallelism'' negation). Further for a compact Hausdorff space X we establish that the hereditary normality of I<sub>3</sub>(X)\ X implies the metrizability of X.

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