fixed $\{(\bar{u}_{i},\bar{m}_{i}),t_{i}\}_{i\in\mathbb{N}}$ attaining the desired bound.

Compared to Theorem 3 in [@verdu2012nonasymp]: $$P_{e}\le\mathbf{P}\{\iota_{U;S}(U;S)>\log\mathsf{J}-\gamma\}+\mathbf{P}\{\iota_{U;Y}(U;Y)\le\log\mathsf{L}\mathsf{J}+\gamma\}+2^{-\gamma}+e^{-2^{\gamma}}$$ for any $\gamma>0$, $\mathsf{J}\in\mathbb{N}$, our result is strictly stronger since $$\begin{aligned}
 & \mathbf{E}\left[1-\left(1+\mathsf{L}2^{\iota_{U;S}(U;S)-\iota_{U;Y}(U;Y)}\right)^{-1}\right]\\
 & \le\mathbf{P}\{\iota_{U;S}(U;S)>\log\mathsf{J}-\gamma\}+\mathbf{P}\{\iota_{U;Y}(U;Y)\le\log\mathsf{L}\mathsf{J}+\gamma\}\\
 & \;\;\;\;+\mathbf{E}\left[1-\left(1+\mathsf{L}2^{\iota_{U;S}(U;S)-\iota_{U;Y}(U;Y)}\right)^{-1}\,|\,\iota_{U;S}(U;S)\le\log\mathsf{J}-\gamma,\,\iota_{U;Y}(U;Y)>\log\mathsf{L}\mathsf{J}+\gamma\right]\\
 & \le\mathbf{P}\{\iota_{U;S}(U;S)>\log\mathsf{J}-\gamma\}+\mathbf{P}\{\iota_{U;Y}(U;Y)\le\log\mathsf{L}\mathsf{J}+\gamma\}+2^{-2\gamma}\\
 & <\mathbf{P}\{\iota_{U;S}(U;S)>\log\mathsf{J}-\gamma\}+\mathbf{P}\{\iota_{U;Y}(U;Y)\le\log\mathsf{L}\mathsf{J}+\gamma\}+2^{-\gamma}+e^{-2^{\gamma}}.\end{aligned}$$ This is due to the fact that the Poisson matching lemma simultaneously replaces both the covering and the packing lemma, resulting in only one error event.

Next, we prove a second-order result. Fix $\epsilon>0$. Let $C:=I(U;Y)-I(U;S)$, $V:=\mathrm{Var}[\iota_{U;S}(U;S)-\iota_{U;Y}(U;Y)]$. We apply Theorem \[thm:channel\_state\] on $n$ uses of the memoryless channel with i.i.d. state sequence $S^{n}=(S_{1},\ldots,S_{n})$, and $$\mathsf{L}:=\left\lfloor \mathrm{exp}_{2}\left(nC-\sqrt{nV}\mathcal{Q}^{-1}\left(\epsilon-\frac{\alpha}{\sqrt{n}}\right)-\frac{1}{2}\log n\right)\right\rfloor ,$$ where $\alpha$ is a constant that depends on $P_{S,U,Y}$. For $n>\alpha^{2}\epsilon^{-2}$, by the Berry-Esseen theorem [@berry1941accuracy; @esseen1942liapunov; @feller1971introduction], we have $$\begin{aligned}
P_{e} & \le\mathbf{E}\left[\min\left\{ 2^{\log\mathsf{L}+\iota_{U^{n};S^{n}}(U^{n};S^{n})-\iota_{U^{n};Y^{n}}(U^{n};Y^{n})},\,1\right\} \right]\\
 & \le\frac{1}{\sqrt{n}}+\mathbf{P}\left\{ 2^{\log\mathsf{L}+\iota_{U^{n};S^{n}}(U^{n};S^{n})-\iota_{U^{n};Y^{n}}(U^{n};Y^{n})}>\frac{1}{\sqrt{n}}\right\} \\
 & \le\frac{1}{\sqrt{n}}+\mathbf{P}\left\{ \frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left(\iota_{U;Y}(U_{i};Y_{i})-\iota_{U;S}(U_{i};S_{i})-C\right)<-\sqrt{V}\mathcal{Q}^{-1}\left(\epsilon-\frac{\alpha}{\sqrt{n}}\right)\right\} \\
 & \le\frac{1}{\sqrt{n}}+\epsilon-\frac{\alpha}{\sqrt{n}}+\frac{\alpha-1}{\sqrt{n}}\\
 & \le\epsilon\end{aligned}$$ if we let $\alpha-1$ be the constant given by the Berry-Esseen theorem. This coincides with the best known second order result in [@scarlett2015dispersions], which is stronger than the second order results implied by [@verdu2012nonasymp; @yassaee2013oneshot; @watanabe2015nonasymp]. We bound $\iota_{U;S}(U;S)-\iota_{U;Y}(U;Y)$ as a single quantity, instead of bounding the two terms separately as in [@verdu2012nonasymp; @yassaee2013oneshot; @watanabe2015nonasymp], resulting in a sharper second order bound.

One-shot Lossy Source Coding with Side Information at 