A limit theorem for $p$-adic-valued probability distributions

Probability models in which probabilities, defined in the sense of von Mises as limits of relative frequencies, can belong to $p$-adic number fields appeared in connection with the problem of the probabilistic interpretation of wave functions in $p$-adic-valued quantum mechanics and field theory. Here we present a variant of axiomatic $p$-adic probability theory in the framework of the theory of analytic distributions on $p$-adic spaces. We prove a theorem on the existence of $p$-adic-valued probability distributions on $p$-adic sequences and obtain a limit theorem for sums of independent random variables (an analogue of the law of large numbers).