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\(p\)-adic and group valued probabilities. (English) Zbl 1146.60005
Chuong, N.M. (ed.) et al., Harmonic, wavelet and \(p\)-adic analysis. Based on the summer school, Quy Nhon, Vietnam, June 10–15, 2005. Hackensack, NJ: World Scientific (ISBN 978-981-270-549-5/hbk). 291-309 (2007).
Motivated by problems of Mathematical Physics, the author investigated probabilities taking values in a group \(\mathbb Q_p\) of \(p\)-adic numbers. The paper under review is a survey of a theory of such probabilities. There exists an impressing number of publications on this subject, a considerably large part of which was published by the author. Considering the original frequency model for probabilities, i.e. considering the relative frequencies \(\left(\nu_N/N\right)_{N\in\mathbb N}\) of repetitions of an experiment, then, according to the law of large numbers these quantities converge (in measure resp. a.e.) towards the probability of the event. The quantities \(\nu_N/N\) are rational numbers, and may therefore be interpreted as \(p\)-adic numbers, for some fixed prime \(p\). Then a limit point of the relative frequencies within the \(p\)-adic numbers \(\mathbb Q_p\) (convergence along a fixed subsequence) defines a \(p\)-adic probability. The paper contains a survey of \(p\)-adic measures and corresponding integration as well as (probabilistic) \(p\)-adic limit theorems, in particular, \(p\)-adic versions of the law of large numbers etc. Furthermore, various open problems in \(p\)-adic, and more generally, in group-valued probability theory are discussed.
For the entire collection see [Zbl 1117.42001].
60A05 Axioms; other general questions in probability
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
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