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

TL;DR

This paper investigates the stability of probability laws under small violations of algorithmic randomness, identifying conditions for stability and demonstrating both stable and unstable cases across various laws and transformations.

Contribution

It provides a sufficient condition for stability using Schnorr tests and shows that many probability laws are stable, while also constructing examples of instability in ergodic theory and data compression.

Findings

Most probability laws are stable under small violations of randomness.

Stability of Birkhoff's ergodic theorem is non-uniform.

Universal data compression schemes are non-stable with respect to ergodic measures.

Abstract

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.