Ergodic Theorems for Densities

Abstract

This chapter is devoted to developing ergodic theorems first for relative entropy densities and then information densities for the general case of AMS processes with standard alphabets. The general results were first developed by Barron [9] using the martingale convergence theorem and a new martingale inequality. The similar results of Algoet and Cover [7] can be proved without direct recourse to martingale theory. They infer the result for the stationary Markov approximation and for the infinite order approximation from the ordinary ergodic theorem. They then demonstrate that the growth rate of the true density is asymptotically sandwiched between that for the kth order Markov approximation and the infinite order approximation and that no gap is left between these asymptotic upper and lower bounds in the limit as k → ∞. They use martingale theory to show that the values between which the limiting density is sandwiched are arbitrarily close to each other, but we shall see that this is not necessary and this property follows from the results of Chapter 6.