Some results of multidimensional discrete probability measures represented by Euler products

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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On the magnitude of asymptotic probability measures of Dedekind zeta-functions and other Euler products

Acta Arithmetica ◽

10.4064/aa-60-2-125-147 ◽

1991 ◽

Vol 60 (2) ◽

pp. 125-147 ◽

Cited By ~ 7

Author(s):

Kohji Matsumoto

Keyword(s):

Zeta Functions ◽

Probability Measures ◽

Euler Products ◽

Asymptotic Probability

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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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