On Stability Property 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. A 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 the Birkhoff’s ergodic theorem is non-uniform. Moreover, a computable non-ergodic measure preserving transformation can be constructed such that ergodic theorem is non-stable. We also show that any universal data compression scheme is also non-stable with respect to the class of all computable ergodic measures. ∗This paper is an extended version of the talk at the Eighth International Conference on Computability, Complexity and Randomness (CCR 2013). September 23-27, 2013, Moscow, Russia; see also the conference paper V’yugin [24]. This work is partially supported by grant RFBR 13-01-12458