Let \(\varOmega _p\) be the group of p-adic numbers, and \( \xi _1\), \(\xi _2\), \(\xi _3\) be independent random variables with values in \(\varOmega _p\) and distributions \(\mu _1\), \(\mu _2\), \(\mu _3\). Let \(\alpha _j, \beta _j, \gamma _j\) be topological automorphisms of \(\varOmega _p\). We consider linear forms \(L_1 = \alpha _1\xi _1 + \alpha _2 \xi _2+ \alpha _3 \xi _3\), \(L_2=\beta _1\xi _1 + \beta _2 \xi _2+ \beta _3 \xi _3\) and \(L_3=\gamma _1\xi _1 + \gamma _2 \xi _2+ \gamma _3 \xi _3\). We describe all coefficients of the linear forms for which the independence of \(L_1\), \(L_2\) and \(L_3\) implies that distributions \(\mu _1\), \(\mu _2\), \(\mu _3\) are idempotent. This theorem is an analogue of the well-known Skitovich–Darmois theorem, where a Gaussian distribution on the real line is characterized by the independence of two linear forms.
Group of p-adic numbers Characterization theorem Skitovich–Darmois theorem
Mathematics Subject Classification (2010)
60B15 62E10 43A35
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The author is grateful to a reviewer for careful reading of the article and for valuable remarks.
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