true since Fourier-Motzkin elimination only guarantees the existence of a random variable for $\mathsf{J}_{2}$ (that depends on the information density terms) satisfying the bounds, but $\mathsf{J}_{2}$ must be a constant since it is a parameter of the code construction in [@yassaee2013oneshot].

Theorem \[thm:bc\_cm\] gives the following second order bound. Consider $n$ independent channel uses. Let $\mathsf{L}_{a}=2^{nR_{a}}$ for $a=0,1,2$. By the multi-dimensional Berry-Esseen theorem [@bentkus2003dependence] (using the notation in [@yassaee2013oneshot]), we have $P_{e}\le\epsilon$ if there exists $\bar{R},\hat{R}_{1},\hat{R}_{2}\ge0$ such that $$\left[\begin{array}{c}
\tilde{R}_{1}+\bar{R}\\
\tilde{R}_{2}\\
\tilde{R}_{2}-\bar{R}\\
\tilde{R}_{0}+\tilde{R}_{1}+\bar{R}\\
\tilde{R}_{0}+\tilde{R}_{2}\\
\tilde{R}_{0}+\tilde{R}_{2}-\bar{R}
\end{array}\right]\in\mathbf{E}[I]-\frac{1}{\sqrt{n}}\mathcal{Q}^{-1}\left(\mathrm{Cov}[I],\,\epsilon-\frac{\beta}{\sqrt{n}}\right)-\frac{\beta\log n}{n}$$ if $n>\beta^{2}\epsilon^{-2}$, where $\beta$ is a constant that depends on $P_{U_{0},U_{1},U_{2},Y_{1},Y_{2}}$, and $\tilde{R}_{0}=R_{0}+\hat{R}_{1}+\hat{R}_{2}$, $\tilde{R}_{a}=R_{a}-\hat{R}_{a}$ for $a=1,2$, and $$I=\left[\begin{array}{c}
\iota_{U_{1};Y_{1}|U_{0}}(U_{1};Y_{1}|U_{0})\\
\iota_{U_{2},Y_{2}|U_{0}}(U_{2};Y_{2}|U_{0})\\
\iota_{U_{2},Y_{2}|U_{0}}(U_{2};Y_{2}|U_{0})-\iota_{U_{1};U_{2}|U_{0}}(U_{1};U_{2}|U_{0})\\
\iota_{U_{0},U_{1};Y_{1}}(U_{0},U_{1};Y_{1})\\
\iota_{U_{0},U_{2};Y_{2}}(U_{0},U_{2};Y_{2})\\
\iota_{U_{0},U_{2};Y_{2}}(U_{0},U_{2};Y_{2})-\iota_{U_{1};U_{2}|U_{0}}(U_{1};U_{2}|U_{0})
\end{array}\right].$$

To demonstrate the use of the generalized Poisson matching lemma in place of the mutual covering lemma, we prove a one-shot version of Marton’s inner bound without common message [@marton1979broadcast] (i.e., $\mathsf{L}_{0}=1$). Our bound is stronger than that in [@verdu2012nonasymp] in the sense that our bound implies [@verdu2012nonasymp] (with a slight penalty of having $2^{1-\gamma}+2^{-2\gamma}$ instead of $2^{1-\gamma}+e^{-2^{\gamma}}$), but [@verdu2012nonasymp] does not imply our bound. We also note that a finite-blocklength bound is given in [@yassaee2013binning]. Nevertheless, the analysis in [@yassaee2013binning] only works for discrete auxiliary random variables 