V. PROOF OF THEOREM 2 (THE CASE OF REALISTIC DUTY CYCLE CONSTRAINT) A. Stationarity of the Capacity-achieving Input Distribution We first establish the fact that a stationary distribution achieves the capacity of the AWGN channel with the realistic duty cycle constraint and power constraint. Proposition 1: A stationary distribution 1 achieves Proof: Let T$_{k}$ ( · ) as a k -cyclic-shift operator on µ ∈ Λ $^{n}$( γ, q, c ) , defined as where k = 1 , · · · , n − 1 , and specifically T$_{0}$ ( µ ) = µ . For any distribution µ in Λ $^{n}$( γ, q, c ) , a distribution ν on X n can be defined as According the concavity of the mutual information I ( · ) , where I ( T$_{k}$ ( µ )) = I ( µ ) since the AWGN channel (1) is a memoryless and time-invariant. Obviously ν is a stationary distribution and satisfied the duty cycle constraint and power constraint, i.e., ν ∈ Λ $^{n}$( γ, q, c ) , hence Proposition 1 established. $^{1}$The stationarity of distribution ν on X n satisfies for any index s, t, k satisfied 19 ν$_{X}$$_{s}$$_{,}$$_{···}$$_{,X}$$_{t}$ = ν$_{X}$$_{s}$$_{+}$$_{k}$$_{,}$$_{···}$$_{,X}$$_{t}$$_{+}$$_{k}$ 1 ≤ s ≤ t ≤ n 1 ≤ s + k ≤ t + k ≤ n I ( ν ) = I$^{(}$ 1 n n − 1 ∑ k =0 T$_{k}$ ( µ )$^{)}$ (82) ≥ 1 n n − 1 ∑ k =0 I ( T$_{k}$ ( µ )) (83) = I ( µ ) (84) ν = 1 n n − 1 ∑ k =0 T$_{k}$ ( µ ) . (81) T$_{k}$ ( µ ) = µ$_{X}$$_{k}$$_{+1}$$_{,}$$_{···}$$_{,X}$$_{n}$$_{,X}$$_{1}$$_{,}$$_{···}$$_{X}$$_{k}$ (80) max µ ∈ Λ $^{n}$( γ,q,c ) I ( X $^{n}$; Y $^{n}$) . (79)