On the concept of random sequence with respect to $p$-adic valued probabilities

This paper continues investigations on generalized probability models in which probabilities belong to fields of $p$-adic numbers. We study a $p$-adic generalization of Martin–Löf's theory based on tests for randomness. Such generalization appears to be the most natural approach to $p$-adic randomness. Each test for randomness induces a series of limit theorems. We proved that it is possible to enumerate all $p$-adic tests for randomness. However, in contrast to Martin–Löf's theory for real probabilities we proved that a universal test for randomness does not exist.