  In the online prediction framework, we use generalized entropy of to study
the loss rate of predictors when outcomes are drawn according to stationary
ergodic distributions over the binary alphabet. We show that the notion of
generalized entropy of a regular game \cite{KVV04} is well-defined for
stationary ergodic distributions. In proving this, we obtain new game-theoretic
proofs of some classical information theoretic inequalities. Using Birkhoff's
ergodic theorem and convergence properties of conditional distributions, we
prove that a classical Shannon-McMillan-Breiman theorem holds for a restricted
class of regular games, when no computational constraints are imposed on the
prediction strategies.
  If a game is mixable, then there is an optimal aggregating strategy which
loses at most an additive constant when compared to any other lower
semicomputable strategy. The loss incurred by this algorithm on an infinite
sequence of outcomes is called its predictive complexity. We use our version of
Shannon-McMillan-Breiman theorem to prove that when a restriced regular game
has a predictive complexity, the predictive complexity converges to the
generalized entropy of the game almost everywhere with respect to the
stationary ergodic distribution.
