Understanding ergodic capacity: how to prove : two channels c1, c2 - capacities, but we do not know what channel has c1, what c2 nevertheless capacity is c1+c2

Question

As far as I understand if we have channel which has random capacity for each discrete time moment (for example Rayleigh fading ), then

the capacity is just average capacity.

I am trying to understand it.

It seems the following simpliefied situation should be understand first: consider the two channels assume we know that capacity of one of them is c1 , capacity of another is c2, but we do not know what channel has c1, what channel has c2. Nevetheless it seems that capacity is c1+c2, the same as as if we know what channel has c1, what channel has c2...

For example if we consider AWGN channels - can one suggest some simple proof of this ?

(Similar to proof of capacity for AWGN with balls filling )

Answer 1

I recommend O. Sarig, Lecture Notes on Ergodic Theory, Dec., 2009, available here. Chapter 2 is about Ergodic Theorems.

Another interesting reference is P. Billingsley, Ergodic Theory and Information, John Wiley & Sons, 1965. For example, Chapter 5, Section 16 (The noisy channel) have a subsection about channel capacity and ergodicity of the input-output process.