Limit behaviour of sums of independent random variables with respect to the uniform p-adic distribution

Abstract

We investigate (as usual) limit behaviour of sums S n ( ω) of independent equally distributed random variables. However, limits of probabilities are studied with respect to a p-adic metric (where p is a prime number). We found that (despite of rather unusual features of a p-adic metric) limits of classical probabilities exist in a field of p-adic numbers. These probabilities are rational numbers (which can be calculated by using simple combinatorial considerations). Limit theorems are related to divisibility of sums S n ( ω) by p. In fact, limits depend on choices of subsequences { S n k ( ω)}. We obtain two limit theorems which describe all possible limit behaviours. All considerations are based on one special p-adic probability distribution, namely the uniform distribution.