\(p\)-adic statistical stabilization of relative frequencies
type: Item
instance of: scholarly article
title: \(p\)-adic statistical stabilization of relative frequencies
zbMATH Open document ID: 0887.60006
published in: Doklady Mathematics
publication date: 1998-05-25T00:00:00Z
review text: When a quantum mechanics with \(p\)-adic-valued wave functions was being constructed [see the author, Russ. Math. Surv. 45, No. 4, 87-125 (1990); translation from Usp. Mat. Nauk 45, No. 4(274), 79-110 (1990; Zbl 0722.46040)], there arose coefficients that had to be probabilistic from the physical standpoint; however, they belonged to the field \(Q_p\) of \(p\)-adic numbers rather than to the interval \([0, 1]\) of the real number field \(\mathbb{R}\) [regarding \(p\)-adic numbers, see, for example, \textit{Z. I. Borevich} and \textit{I. R. Shafarevich}, ``Number theory'' (1985; Zbl 0592.12001) and \textit{V. S. Vladimirov}, Russ. Math. Surv. 43, No. 5, 19-64 (1988); translation from Usp. Mat. Nauk 43, No. 5(263), 17-53 (1988; Zbl 0678.46053)]. Thus, the necessity arose to develop a probabilistic formalism, in which the probabilities could be \(p\)-adic numbers. Such a probabilistic formalism was proposed by the author [Sov. Phys., Dokl. 37, No. 2, 81-83 (1992); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 322, No. 6, 1075-1079 (1992; Zbl 0768.60002)]. In the aforementioned works, the frequency approach to the definition of the probability, similar to the frequency theory put forward by \textit{R. von Mises} [``Wahrscheinlichkeit, Statistik und Wahrheit'' (1936; Zbl 0014.22103) and ``Mathematical theory of probability and statistics'' (1964; Zbl 0132.12303)], was used. In the present note, the formalism of the statistical stabilization in the \(p\)-adic context is discussed axiomatically.
Mathematics Subject Classification ID: 60A99
zbMATH DE Number: 1099403
zbMATH Keywords: statistical stabilization of relative frequencies
zbMATH Keywords: quantum mechanics
zbMATH Keywords: \(p\)-adic numbers
author: Andrei Khrennikov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1099403