Approximation of probability measures by convex combinations of measures of spherically invariant processes

1981 ◽  
Vol 29 (4) ◽  
pp. 320-324
Author(s):  
I. V. Kozin
Keyword(s):  
Probability Measures ◽  
Approximation Of Probability Measures

scholarly journals ITERATED FUNCTION SYSTEMS WITH PLACE-DEPENDENT PROBABILITIES AND THE INVERSE PROBLEM OF MEASURE APPROXIMATION USING MOMENTS

Fractals ◽  
2018 ◽  
Vol 26 (05) ◽  
pp. 1850076 ◽  
Author(s):  
D. LA TORRE ◽  
E. MAKI ◽  
F. MENDIVIL ◽  
E. R. VRSCAY
Keyword(s):  
Invariant Measures ◽  
Iterated Function Systems ◽  
Probability Measures ◽  
Moment Matching ◽  
Computational Results ◽  
Compact Metric Space ◽  
Moment Vector ◽  
Iterated Function ◽  
Approximation Of Probability Measures ◽  
Vector Formula

We are concerned with the approximation of probability measures on a compact metric space [Formula: see text] by invariant measures of iterated function systems with place-dependent probabilities (IFSPDPs). The approximation is performed by moment matching. Associated with an IFSPDP is a linear operator [Formula: see text], where [Formula: see text] denotes the set of all infinite moment vectors of probability measures on [Formula: see text]. Let [Formula: see text] be a probability measure that we desire to approximate, with moment vector [Formula: see text]. We then look for an IFSPDP which maps [Formula: see text] as close to itself as possible in terms of an appropriate metric on [Formula: see text]. Some computational results are presented.

LIFTING THE FUNCTOR ITO THE CATEGORY OF UNIFORM SPACES

2020 ◽  
Vol 4 (1) ◽  
pp. 29-39
Author(s):  
Dilrabo Eshkobilova ◽  
Keyword(s):  
Compact Support ◽  
Probability Measures ◽  
Uniform Spaces ◽  
Continuous Maps ◽  
Uniformly Continuous

Uniform properties of the functor Iof idempotent probability measures with compact support are studied. It is proved that this functor can be lifted to the category Unif of uniform spaces and uniformly continuous maps

Generic rotation sets

2020 ◽  
pp. 1-13
Author(s):  
SEBASTIÁN PAVEZ-MOLINA
Keyword(s):  
Dynamical System ◽  
Convex Body ◽  
Probability Measures ◽  
Topological Dynamical System ◽  
Continuous Vector ◽  
Rotation Set ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Uniquely Ergodic ◽  
Dense Set

Abstract Let $(X,T)$ be a topological dynamical system. Given a continuous vector-valued function $F \in C(X, \mathbb {R}^{d})$ called a potential, we define its rotation set $R(F)$ as the set of integrals of F with respect to all T-invariant probability measures, which is a convex body of $\mathbb {R}^{d}$ . In this paper we study the geometry of rotation sets. We prove that if T is a non-uniquely ergodic topological dynamical system with a dense set of periodic measures, then the map $R(\cdot )$ is open with respect to the uniform topologies. As a consequence, we obtain that the rotation set of a generic potential is strictly convex and has $C^{1}$ boundary. Furthermore, we prove that the map $R(\cdot )$ is surjective, extending a result of Kucherenko and Wolf.

scholarly journals On a Variational Definition for the Jensen-Shannon Symmetrization of Distances Based on the Information Radius

Entropy ◽  
2021 ◽  
Vol 23 (4) ◽  
pp. 464
Author(s):  
Frank Nielsen
Keyword(s):  
Diversity Index ◽  
Diversity Indices ◽  
Shannon Diversity Index ◽  
Probability Measures ◽  
Variational Optimization ◽  
Shannon Diversity ◽  
Concept Of Information ◽  
Jensen Shannon Divergence

We generalize the Jensen-Shannon divergence and the Jensen-Shannon diversity index by considering a variational definition with respect to a generic mean, thereby extending the notion of Sibson’s information radius. The variational definition applies to any arbitrary distance and yields a new way to define a Jensen-Shannon symmetrization of distances. When the variational optimization is further constrained to belong to prescribed families of probability measures, we get relative Jensen-Shannon divergences and their equivalent Jensen-Shannon symmetrizations of distances that generalize the concept of information projections. Finally, we touch upon applications of these variational Jensen-Shannon divergences and diversity indices to clustering and quantization tasks of probability measures, including statistical mixtures.

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