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