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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).

##### MSC:
 60F99 Limit theorems in probability theory 11K41 Continuous, $$p$$-adic and abstract analogues 60A99 Foundations of probability theory