On probability distributions on the field of $p$-adic numbers

It is shown that many naturally defined probability distributions can be realized as distributions on the field of $p$-adic numbers. Following R. Mises and E. Tornier we consider first the stochastic experiment whose elementary outcome is the result of the infinite sequence of tossings of a symmetric coin. It is proved that the distribution corresponding to this experiment is the Haar distribution on the ring of $p$-adic integers. Then we consider the convergence of series of random variables with rational values on the field of $p$-adic numbers. It is shown that the series converge a.s. on the field of $p$-adic numbers but diverge on the field of real numbers.