Conditional information variance and DMC dispersion

Let $P$ be an input probability vector for the finite DMC $W$, and let $P_Y(y)={\sum }_{x}P(x)W(y\mid x)$. On cells with $P(x)W(y\mid x)>0$, define ${\mathrm{ı}}_P(x;y)=\ln [W(y\mid x)\mathrm{/}{P}_Y(y)]$; the denominator is then positive. Put $V(P,W):={\sum }_{x:P(x)>0}P(x){\mathrm{Var}}_{W(\cdot \mid x)}[{\mathrm{ı}}_P(x;Y)]\mathrm{.}$ Let $C(W)={\max }_{P}I(P,W)$ and $\mathrm{\Pi }(W)=\{P:I(P,W)=C(W)\}$, the nonempty set of capacity-achieving input distributions. Define $V_{\min }(W):={\min }_{P\in \mathrm{\Pi }(W)}V(P,W),V_{\max }(W):={\max }_{P\in \mathrm{\Pi }(W)}V(P,W)\mathrm{.}$ The local attainment lemma DMC minimum and maximum dispersion are attained ↗ proves these extrema exist. Dispersion is measured in nats squared. In the normal-approximation theorem on this page only the fixed range $0<ϵ<1\mathrm{/}2$ and premise $V_{\min }(W)>0$ are used.