Identifying randomness given by high descriptive complexity

We discuss methods for identifying random phenomena which are logical consequences of the fact that almost all real numbers are of a high descriptive complexity. These methods enable us to find complexity-theoretic versions of classical results in the theory of random series. In addition, we show how the outcomes which are of high descriptive complexity of random processes bring into realisation the highly reflective (self-similar) nature of various combinatorial configurations. The paper uses methods of recursion theory.