content_image/308403.jpg content_image/308402.jpg content_image/308401.jpg
According to Proposition 1, for any n , I ( X $^{n}$; Y $^{n}$) is maximized by a stationary distribution. Therefore with n converges to infinity, the capacity in (9) is achieved by a stationary input distribution. 

## B. The Input-output Mutual Information 

Proposition 2: Let the input follows a stationary distribution µ ∈ Λ( γ,q,c ) . The limit of the input- output mutual information per symbol as a function of µ can be expressed as 

<img src='content_image/308403.jpg'>

where I ( X ; Y ) is the mutual information of the AWGN channel between the input X , which follows distribution µ$_{X}$$_{1}$ and the corresponding output Y , h ( Y ) is the differential entropy of Y and h ( Y ) is the differential entropy rate of output process { Y$_{i}$ } . 

Proof: The mutual information between X n and Y n can be expressed using relative entropies 

<img src='content_image/308402.jpg'>

Then 

Proposition 2 is established. 

<img src='content_image/308401.jpg'>

When the input is an i.i.d. random process, the output process is also i.i.d., h ( Y ) = h ( Y ) . This implies the following corollary. 

Corollary 1: Among all i.i.d. distributions, the one that maximizes the mutual information under duty 

