On Stability of Probability Laws with Respect to Small Violations of Algorithmic Randomness

Abstract

We study a stability property of probability laws with respect to small violations of algorithmic randomness. Some sufficient condition of stability is presented in terms of Schnorr tests of algorithmic randomness. Most probability laws, like the strong law of large numbers, the law of iterated logarithm, and even Birkhoff's pointwise ergodic theorem for ergodic transformations, are stable in this sense. Nevertheless, the phenomenon of instability occurs in ergodic theory. Firstly, the stability property of Birkhoff's ergodic theorem is non-uniform. Moreover, a computable non-ergodic measure-preserving transformation can be constructed such that the ergodic theorem is non-stable.