Strictly feasible Gaussian inputs yield cost-feasible Gaussian codes

Use [F1] to work in orthogonal coordinates, and draw each vector use independently with branch inputs $X_j\sim N(0,p_j)$, independent across branches. Put $\mu ={\sum }_j{p}_j<P$. By [F2], the normalized sum of information densities converges in probability to $I(p)$.

If $R=0$, one-message codes meet the assertion. Suppose $R>0$, choose $\delta >0$ with $R+2\delta <I(p)$, and put $M_n=\lceil e^{nR}\rceil$. The threshold $\ln (M_n-1)+n\delta$ is eventually below $n(R+2\delta )$. The weak law in [F2] makes the miss probability tend to zero; [F3] then shows the ensemble expected average error $a_n$ tends to zero.

For one vector use put $Q={\sum }_j{X}_j^2$. By [F4], $\mathbb{E}{X}_j^2=p_j$ and $\mathbb{E}{X}_j^4=3p_j^2$; [F7] then gives $\mathrm{Var}(X_j^2)=2p_j^2$, and independence factors the cross moments to yield $\mathrm{Var}(Q)=2{\sum }_j{p}_j^2<\mathrm{\infty }$. Across $n$ uses the codeword average cost ${\bar{Q}}_n$ has mean $\mu$ and variance $\mathrm{Var}(Q)\mathrm{/}n$. With $\mathrm{\Delta }=P-\mu >0$, the event ${\bar{Q}}_n>P$ is contained in $\{\mathrm{\mid }{\bar{Q}}_n-\mu \mathrm{\mid }\ge \mathrm{\Delta }\}$, whose probability is at most $\mathrm{Var}(Q)\mathrm{/}(n{\mathrm{\Delta }}^2)$ by [F5] applied to the squared deviation. This probability is the expected fraction $q_n$ of over-budget codewords, so $q_n\to 0$.

The codewords are drawn IID, hence are exchangeable, and the random code ensemble has $q_n+a_n\to 0$ by steps 2.1 and 2.2. Apply [F6] to prune every over-budget word; [A2] identifies the retained deterministic codes as feasible no-feedback codes, with average power at most $P$, error tending to zero, and rate loss tending to zero. Orthogonal invariance in [F1] carries these codes back to the original channel.