Jointly typical blocks for a finite IID multisource

Let $P$ and $Z_t=(X_{1,t},\dots ,X_{k,t})$ be as in Finite-alphabet IID multisource, and let ${\mathcal{X}}_{+}:=\{x\in \mathcal{X}:P(x)>0\}$. For $z^n\in {\mathcal{X}}_{+}^n$, write ${\widehat{P}}_{z^n}$ for its empirical type from Empirical types, n-types, and type classes, and write $I_n(z^n)$ for its normalized block information from Information density and typical sets for a finite source applied to the vector source on $\mathcal{X}$. For each nonempty $S\subseteq [k]$ and supported pair $(x_S^n,x_{S^c}^n)$, define the conditional block information $J_{S,n}(x_S^n\mid x_{S^c}^n):=-\frac{1}{n}\ln P^{\otimes n}(x_S^n\mid x_{S^c}^n),$ where the conditional probability is the finite conditional law of the block under $P^{\otimes n}$. Its positive denominator and coordinatewise factorization on supported tuples are proved in Conditional block law factors across IID coordinates.

For $\delta ,ϵ>0$, the jointly typical set is ${\mathcal{T}}_n(P,\delta ,ϵ):=\{z^n\in {\mathcal{X}}_{+}^n:\mathrm{\parallel }{\widehat{P}}_{z^n}-P{\mathrm{\parallel }}_{\mathrm{\infty }}\le \delta ,\mathrm{\mid }{I}_n(z^n)-H(P)\mathrm{\mid }\le ϵ,\mathrm{\mid }{J}_{S,n}(x_S^n\mid x_{S^c}^n)-H_P(X_S\mid X_{S^c})\mathrm{\mid }\le ϵ\text{ for every nonempty }S\subseteq [k]\}\mathrm{.}$ Here $\mathrm{\parallel }q-P{\mathrm{\parallel }}_{\mathrm{\infty }}:={\max }_{x\in \mathcal{X}}\mathrm{\mid }q(x)-P(x)\mathrm{\mid }$, $H(P)$ is the finite entropy of Finite distributions, entropy, divergence, and stochastic matrices, and each $H_P(X_S\mid X_{S^c})$ is from Finite conditional entropy under the multisource law.