On the unimodality of discrete probability measures

The notion of utility maximising entropy (u-entropy) of a probability density, which was introduced and studied in [37], is extended in two directions. First, the relative u-entropy of two probability measures in arbitrary probability spaces is defined. Then, specialising to discrete probability spaces, we also introduce the absolute u-entropy of a probability measure. Both notions are based on the idea, borrowed from mathematical finance, of maximising the expected utility of the terminal wealth of an investor. Moreover, u-entropy is also relevant in thermodynamics, as it can replace the standard Boltzmann-Shannon entropy in the Second Law. If the utility function is logarithmic or isoelastic (a power function), then the well-known notions of Boltzmann-Shannon and Rényi relative entropy are recovered. We establish the principal properties of relative and discrete u-entropy and discuss the links with several related approaches in the literature.

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Optimal Quantization for Piecewise Uniform Distributions

Uniform distribution theory ◽

10.2478/udt-2018-0009 ◽

2018 ◽

Vol 13 (2) ◽

pp. 23-55 ◽

Cited By ~ 3

Author(s):

Joseph Rosenblatt ◽

Mrinal Kanti Roychowdhury

Keyword(s):

Finite Number ◽

Uniform Distribution ◽

Infinite Number ◽

Independent Random Variables ◽

Probability Measures ◽

Discrete Probability ◽

Uniform Distributions ◽

Optimal Quantization ◽

Positive Integers ◽

Optimal Set

Abstract Quantization for a probability distribution refers to the idea of estimating a given probability by a discrete probability supported by a finite number of points. In this paper, firstly a general approach to this process is outlined using independent random variables and ergodic maps; these give asymptotically the optimal sets of n-means and the nth quantization errors for all positive integers n. Secondly two piecewise uniform distributions are considered on R: one with infinite number of pieces and one with finite number of pieces. For these two probability measures, we describe the optimal sets of n-means and the nth quantization errors for all n ∈ N. It is seen that for a uniform distribution with infinite number of pieces to determine the optimal sets of n-means for n ≥ 2 one needs to know an optimal set of (n − 1)-means, but for a uniform distribution with finite number of pieces one can directly determine the optimal sets of n-means and the nth quantization errors for all n ∈ N.

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LIFTING THE FUNCTOR ITO THE CATEGORY OF UNIFORM SPACES

PHYSICAL AND MATHEMATICAL SCIENCES ◽

10.26739/2181-0656-2020-4-4 ◽

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

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Something for Nothing: Probability Measures, Expected Returns and a Pricing Puzzle

SSRN Electronic Journal ◽

10.2139/ssrn.2695812 ◽

2015 ◽

Author(s):

Jamil Baz ◽

Ludovic Feuillet ◽

Nicolas Le Roux

Keyword(s):

Expected Returns ◽

Probability Measures

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Isoperimetry for spherically symmetric log-concave probability measures

Revista Matemática Iberoamericana ◽

10.4171/rmi/631 ◽

2011 ◽

pp. 93-122 ◽

Cited By ~ 2

Author(s):

Nolwen Huet

Keyword(s):

Probability Measures ◽

Spherically Symmetric

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Generic rotation sets

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.129 ◽

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.

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