An average-cost DMC block input has mutual information at most n C_I(Gamma)

Given: A finite DMC with finite letter cost, a feasible budget, and a possibly dependent block input satisfying the stated expected average cost constraint.

Finite entropy and mutual-information chain rules: finite joint entropy and mutual information satisfy the chain rules.

Finite mutual information and Finite conditional entropy: $I(X;Y)=H(Y)-H(Y\mid X)$ for finite variables.

Finite DMC with a nonnegative input cost and feasible budget, Average-cost fixed-length codes and operational DMC capacity, Single-letter information capacity of a finite DMC, and Finite probability vectors: $C_I(\gamma )={\sup }_{P_X:\mathbb{E}c(X)\le \gamma }I(P_X;W)$ for every feasible $\gamma$.

Constrained DMC information capacity is attained, nondecreasing, and concave: $C_I$ is nondecreasing and concave on $[c_{\min },\mathrm{\infty })$.

Finite discrete memoryless channels: conditioned on $X^n$, the output coordinates have product law ${\prod }_{t}W(y_t\mid X_t)$.

Jensen's inequality for an integrable cost random variable: an integrable cost variable in $[c_{\min },\mathrm{\infty })$ can be passed through a finite concave function by Jensen's inequality.

Finite KL divergence is nonnegative with equality exactly at equal laws: finite mutual information is nonnegative because it is a KL divergence.