On Instability of the Ergodic Limit Theorems with Respect to Small   Violations of Algorithmic Randomness
  An instability property of the Birkhoff's ergodic theorem and related asymptotic laws with respect to small violations of algorithmic randomness is studied. The Shannon--McMillan--Breiman theorem and all universal compression schemes are also among them. 
Vladimir V'yugin
cs.IT math.IT