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The Bernoulli theorem for probabilities that take \(p\)-adic values. (English. Russian original) Zbl 0971.60034
Dokl. Math. 55, No. 3, 402-405 (1997); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 354, No. 4, 461-464 (1997).
The author considers the sequence of normalized sums \[ T_n=\tfrac{1}n(\xi_1+\cdots +\xi_n),\quad n=1,2,\ldots , \] where \(\xi_1,\ldots ,\xi_n,\ldots\) are independent random variables taking values 0 and 1 with probability 1/2. The probability is understood as a \(\mathbb Q_p\)-valued bounded measure. It is shown that if a sequence \(\{n_k\}\) converges in a \(p\)-adic metric, then the subsequence \(\{T_{n_k}\}\) of the normalized sums converges in a certain weak sense. For a more detailed exposition see the author’s book “Non-Archimedean analysis: Quantum paradoxes, dynamical systems and biological models” (1997; Zbl 0920.11087).

60F99 Limit theorems in probability theory
11K41 Continuous, \(p\)-adic and abstract analogues
60A99 Foundations of probability theory