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The discreteness of the optimized input distribution is proved in the following. According to Properties (b) and (c), lower bound L ( · ) is maximized by a first-order Markov process, the transition probability distribution of which P$_{X}$$_{2}$$_{|}$$_{X}$$_{1}$ can be expressed as 

<img src='content_image/308632.jpg'>

where b$_{i}$ = 1$_{{}$$_{x}$$_{i}$$_{̸}$$_{=0}$$_{}}$ P$_{X}$ = µ$_{X}$ and P$_{X}$$_{2}$$_{|}$$_{X}$$_{1}$ = µ$_{X}$$_{2}$$_{|}$$_{X}$$_{1}$ . Then the maximum of L ( µ ) can be achieved by the follow optimization 

<img src='content_image/308631.jpg'>

<img src='content_image/308627.jpg'>

<img src='content_image/308628.jpg'>

<img src='content_image/308629.jpg'>

<img src='content_image/308630.jpg'>

Since given any q$_{0}$ ≥ q > 0 , I$_{X}$ ( q$_{0}$ ) − I$_{B}$ ( q$_{0}$ ) can be maximized by the maximum of I$_{X}$ ( q$_{0}$ ) and the minimum of I$_{B}$ ( q$_{0}$ ) respectively, the maximization of (119) must be achieved by P$_{X}$ , which maximizes I ( X ; Y ) for given q$_{0}$ . Therefore given q$_{0}$ , the maximization in (120) is similar to the problem in Theorem 1. The difference to Theorem 1 is that in (120) the distribution P$_{X}$ satisfies P$_{X}$ (0) = q$_{0}$ ≥ q , however in Theorem 1 the distribution P$_{X}$ satisfies P$_{X}$ (0) ≥ q . Define 

<img src='content_image/308626.jpg'>

where µ is the marginal input distribution of the first-order Markov process. We can establish the following lemma. 

Lemma 7: Λ$_{0}$( γ, q$_{0}$ ) is compact in the topological space P . 

