content_image/302476.jpg content_image/302471.jpg content_image/302470.jpg content_image/302472.jpg
| log p$_{Y}$ ( y ; µ$_{1}$ ) | ≤ 1 $_{2}$y 2 + ay + b . Therefore, 

<img src='content_image/302476.jpg'>

due to the assumption that µ$_{2}$ ∈ Λ( q ) . 

Therefore, the dominated convergence theorem provides that 

<img src='content_image/302471.jpg'>

Lemma 3 is thus proved. 

We establish the following sufficient and necessary condition for the optimal input distribution. 

Lemma 5: Let 

<img src='content_image/302470.jpg'>

Then µ$_{0}$ ∈ Λ( γ, q ) achieves the capacity if and only if there exists λ ≥ 0 such that λ E$_{µ}$$_{0}$ $^{{}$X 2 − γ } = 0 and E$_{µ}$ { f$_{λ}$ ( X ; µ$_{0}$ ) } ≤ 0 for all µ ∈ Λ( q ) . 

Proof: Define the Lagrangian 

<img src='content_image/302472.jpg'>

where λ is the Lagrange multiplier. Since Λ( q ) is a convex set and I ( µ ) < ∞ on Λ( q ) , µ$_{0}$ is capacity- achieving if and only if there exists λ ≥ 0 such that the following conditions hold [19]: 

(i) λ E$_{µ}$$_{0}$ $^{{}$X 2 − γ } = 0 ; 

(ii) for all µ ∈ Λ( q ) , J ( µ$_{0}$ ) ≥ J ( µ ) . 

Due to concavity of I ( µ ) , J ( µ ) is also concave. Condition (ii) is then equivalent to that the weak derivative J ′ $_{µ$_{0}$}$( µ ) ≤ 0 for all µ ∈ Λ( q ) . 

