by Chebyshev’s inequality. Choose K$_{ϵ}$ = [ − a$_{ϵ}$, a$_{ϵ}$ ] , then K$_{ϵ}$ is compact in$_{R}$ and µ ( K$_{ϵ}$ ) ≥ 1 − ϵ for all µ ∈ Λ( γ, q ) , thus Λ( γ, q ) is tight. Let B$_{m}$ = $^{[}$− 1 $_{m}$, 1 m ] for m = 1 , 2 , . . . . Let { µ$_{n}$ } ∞ n =1 be a convergent sequence in Λ( γ, q ) with limit µ$_{0}$ . Since µ$_{n}$ ( B$_{m}$ ) ≥ q for every m, n , we have [15, Section 3 . 1 ] and hence Moreover, let f ( x ) = x 2 which is continuous and bounded below. By weak convergence [15, Section 3 . 1 ], we have Therefore, µ$_{0}$ ∈ Λ( γ, q ) , i.e., Λ( γ, q ) is closed, and the compactness of Λ( γ, q ) then follows. Since the mutual information I ( µ ) is continuous on P [17, Theorem 9 ], it must achieve its maximum on the compact set Λ( γ, q ) . Hence the capacity-achieving distribution µ$_{0}$ exists. According to [17, Corollary 2], the mutual information I ( µ ) is strictly concave. It is easy to see that Λ( γ, q ) is convex. Hence the capacity-achieving distribution µ$_{0}$ must be unique. We denote the finite-power set as Let φ ( · ) defined in (14) be extended to the complex plane. The relative entropy d ( x ; µ ) defined in (16) can be extended to the complex plane$_{C}$ and has the following property: Lemma 2: For any µ ∈ Λ( q ) and z ∈$_{C}$ , 10 d ( z ; µ ) = ∫ ∞ −∞ φ ( y − z ) log φ ( y − z ) p$_{Y}$ ( y ; µ ) d y (23) Λ( q ) = ∪$_{0}$$_{≤}$$_{γ<}$$_{∞}$ Λ( γ, q ) . (22) E$_{µ}$$_{0}$ $^{{}$X 2 } = ∫ f d µ$_{0}$ ≤ lim inf n →∞ ∫ f d µ$_{n}$ ≤ γ. (21) µ$_{0}$ ( { 0 } ) = µ$_{0}$ ( ∞ ⋂ m =1 B$_{m}$$^{)}$ = lim m $_{→∞}$µ$_{0}$ ( B$_{m}$ ) ≥ q. (20) q ≤ lim sup n →∞ µ$_{n}$ ( B$_{m}$ ) ≤ µ$_{0}$ ( B$_{m}$ ) , (19) B. Sufficient and Necessary Conditions