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. 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 where I ( X ; Y ) is the mutual information of the AWGN channel between the input X , which follows distribution µ$_{X}$$_{1}$ and the corresponding output Y , h ( Y ) is the differential entropy of Y and h ( Y ) is the differential entropy rate of output process { Y$_{i}$ } . Proof: The mutual information between X n and Y n can be expressed using relative entropies Then Proposition 2 is established. When the input is an i.i.d. random process, the output process is also i.i.d., h ( Y ) = h ( Y ) . This implies the following corollary. Corollary 1: Among all i.i.d. distributions, the one that maximizes the mutual information under duty 20 I ( µ ) = lim n →∞ 1 n I ( X $^{n}$; Y $^{n}$) (90) = I ( X ; Y ) − h ( Y ) + lim n →∞ 1 n h ( Y $^{n}$) (91) = I ( X ; Y ) − h ( Y ) + h ( Y ) . (92) I ( X $^{n}$; Y $^{n}$) = D ( P$_{Y}$ n | X n ‖ P$_{Y}$ n | P$_{X}$ n ) (86) = D ( P$_{Y}$ n | X n ‖ P$_{Y}$$_{1}$ × · · · × P$_{Y}$$_{n}$ | P$_{X}$ n ) − D ( P$_{Y}$ n ‖ P$_{Y}$$_{1}$$_{×···×}$$_{P}$$_{Y}$$_{n}$ ) (87) = n ∑ k =1 D ( P$_{Y}$$_{k}$$_{|}$$_{X}$$_{k}$ | P$_{Y}$$_{k}$ | P$_{X}$ n ) − E$^{{}$ log P$_{Y}$ n ( Y $^{n}$) − n ∑ i =1 log P$_{Y}$$_{i}$ ( Y$_{i}$ )$^{}}$ (88) = nI ( X ; Y ) − nh ( Y ) + h ( Y $^{n}$) . (89) I ( µ ) = I ( X ; Y ) − h ( Y ) + h ( Y ) (85) B. The Input-output Mutual Information