Ergodic-Type Characterizations of Algorithmic Randomness

A useful theorem of Kuˇcera states that given a Martin-L¨of random infinite binary sequence ω and an effectively open set A of measure less than 1, some tail of ω is not in A. We show that this result can be seen as an effective version of Birkhoff’s ergodic theorem (in a special case). We prove several results in the same spirit and generalize them via an effective ergodic theorem for bijective ergodic maps.

• Type: Conference
• Year: 2010
• Where: CIE
• Authors: Laurent Bienvenu, Adam R. Day, Ilya Mezhirov, Alexander Shen
• Keywords: Applied Computing, Birkhoff’s Ergodic Theorem, CIE 2010, Ergodic Theorem, Effective Ergodic Theorem