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 where b$_{i}$ = 1$_{{}$$_{x}$$_{i}$$_{̸}$$_{=0}$$_{}}$ P$_{X}$ = µ$_{X}$ and P$_{X}$$_{2}$$_{|}$$_{X}$$_{1}$ = µ$_{X}$$_{2}$$_{|}$$_{X}$$_{1}$ . Then the maximum of L ( µ ) can be achieved by the follow optimization Since given any q$_{0}$ ≥ q > 0 , I$_{X}$ ( q$_{0}$ ) − I$_{B}$ ( q$_{0}$ ) can be maximized by the maximum of I$_{X}$ ( q$_{0}$ ) and the minimum of I$_{B}$ ( q$_{0}$ ) respectively, the maximization of (119) must be achieved by P$_{X}$ , which maximizes I ( X ; Y ) for given q$_{0}$ . Therefore given q$_{0}$ , the maximization in (120) is similar to the problem in Theorem 1. The difference to Theorem 1 is that in (120) the distribution P$_{X}$ satisfies P$_{X}$ (0) = q$_{0}$ ≥ q , however in Theorem 1 the distribution P$_{X}$ satisfies P$_{X}$ (0) ≥ q . Define where µ is the marginal input distribution of the first-order Markov process. We can establish the following lemma. Lemma 7: Λ$_{0}$( γ, q$_{0}$ ) is compact in the topological space P . Λ$_{0}$( γ, q$_{0}$ ) = $^{{}$µ : µ ( { 0 } ) = q$_{0}$, E$_{µ}$ $^{{}$X 2 $^{}}$≤ γ } (124) subject to I$_{X}$ ( q$_{0}$ ) = maximize P$_{X}$ I ( X ; Y ) (120) I$_{B}$ ( q$_{0}$ ) = minimize P ( B$_{2}$ | B$_{1}$ ) I ( B$_{1}$ ; B$_{2}$ ) (121) P$_{X}$ (0) = P$_{B}$$_{1}$ (0) = P$_{B}$$_{2}$ (0) = q$_{0}$ (122) q$_{0}$ − 2 cq$_{0}$P$_{B}$$_{2}$$_{|}$$_{B}$$_{1}$ (1 | 0) ≥ q. (123) maximize q$_{0}$ I$_{X}$ ( q$_{0}$ ) − I$_{B}$ ( q$_{0}$ ) (119) P$_{X}$$_{2}$$_{|}$$_{X}$$_{1}$ ( x$_{2}$ | x$_{1}$ ) = P$_{B}$$_{2}$$_{|}$$_{B}$$_{1}$ ( b$_{2}$ | b$_{1}$ ) P$_{X}$ ( x$_{2}$ ) P$_{B}$$_{2}$ ( b$_{2}$ ) (118) 24