The lack-of-preference law and the corresponding distributions in frequency probability theory
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.
ordering relations on random variables range
scientific article; zbMATH DE number 5155518