The lack-of-preference law and the corresponding distributions in frequency probability theory
type: Item
instance of: scholarly article
title: The lack-of-preference law and the corresponding distributions in frequency probability theory
zbMATH Open document ID: 1122.62013
author: V. P. Maslov
published in: Mathematical Notes
publication date: 2007-05-21T00:00:00Z
review text: The author considers the frequency probability theory and imposes the additional condition to the notion of random variable that its range, i.e., the random variable values, to be ordered. Because of the ordering relation within the range values is the \(s\)-tuple which the random variable is defined on it can also be viewed as a finite time series. Moreover, the frequency probability theory developed here differs also from the classical probability theory in that not only the number of trials but also the number of \(s\)-tuples can tend to infinity. The notions of cumulative probabilities are also introduced as the partial sums of the first (usual) probabilities in the sequence whose values are ordered. This is the distribution considered by the present paper in the framework of frequency probability theory. Actually, the intensive normalized pair of dual distributions of this type are studied. Two theorems synthesize the main asymptotic properties of the considered distributions in the frequency probability theory. They are illustrated on various examples from physics, mathematical finance, but especially on word frequencies in dictionaries and the ``tails'' of the graphs in frequency dictionaries.
Mathematics Subject Classification ID: 62E99
Mathematics Subject Classification ID: 62A01
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 60C05
zbMATH DE Number: 5155518
zbMATH Keywords: frequency probability theory
zbMATH Keywords: ordering relations on random variables range
zbMATH Keywords: cumulative probabilities
zbMATH Keywords: asymptotic properties
zbMATH Keywords: Zipf-Mandelbrot law
zbMATH Keywords: distribution tail
zbMATH Keywords: Shannon entropy
zbMATH Keywords: Wiener process
zbMATH Keywords: Kolmogorov complexity
zbMATH Keywords: mathematical finance
zbMATH Keywords: frequency dictionary
reviewed by: Neculai Curteanu
MaRDI profile type: Publication
full work available at URL: s11006-006-0130-7
OpenAlex ID: W2053317258
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DOI: 10.1007/S11006-006-0130-7
description: scientific article; zbMATH DE number 5155518