the channel $P_{Y|X}$. The decoder observes $Y$ and recovers $\hat{Z}\in\mathcal{Z}$ with probability of excess distortion $P_{e}=\mathbf{P}\{\mathsf{d}(W,\hat{Z})>\mathsf{D}\}$, where $\mathsf{d}:\mathcal{W}\times\mathcal{Z}\to\mathbb{R}_{\ge0}$ is a distortion measure.

We show a one-shot joint source-channel coding result that achieves the optimal dispersion in [@kostina2013joint].

\[thm:jscc\]Fix any $P_{X}$ and $P_{Z}$. There exists a code for the source distribution $P_{W}$ and channel $P_{Y|X}$, with probability of excess distortion $$P_{e}\le\mathbf{E}\left[\left(1+P_{Z}(\mathcal{B}_{\mathsf{D}}(W))2^{\iota_{X;Y}(X;Y)}\right)^{-1}\right]$$ if $P_{XY}\ll P_{X}\times P_{Y}$, where $(W,X,Y)\sim P_{W}\times P_{X}P_{Y|X}$, and $\mathcal{B}_{\mathsf{D}}(w):=\{z:\,\mathsf{d}(w,z)\le\mathsf{D}\}$.

Let $\{(\bar{X}_{i},\bar{Z}_{i}),T_{i}\}_{i\in\mathbb{N}}$ be the points of a Poisson process with intensity measure $P_{X}\times P_{Z}\times\lambda_{\mathbb{R}_{\ge0}}$ independent of $W$. Let $\rho(w):=P_{Z}(\mathcal{B}_{\mathsf{D}}(w))$. Let $P_{\check{Z}|W}$ be defined as $$P_{\check{Z}|W}(A|w):=\begin{cases}
P_{Z}(A\cap\mathcal{B}_{\mathsf{D}}(w))/\rho(w) & \mathrm{if}\,\rho(w)>0\\
P_{Z}(A) & \mathrm{if}\,\rho(w)=0.
\end{cases}$$ The encoding function is $w\mapsto\tilde{X}_{P_{X}\times P_{\check{Z}|W}(\cdot|w)}$ (i.e., $X=\tilde{X}_{P_{X}\times P_{\check{Z}|W}(\cdot|W)}$). The decoding function is $y\mapsto\tilde{Z}_{P_{X|Y}(\cdot|y)\times P_{Z}}$ (i.e., $\hat{Z}=\tilde{Z}_{P_{X|Y}(\cdot|Y)\times P_{Z}}$). Also define $\check{Z}=\tilde{Z}_{P_{X}\times P_{\check{Z}|W}(\cdot|W)}$. We have $(X,Y,W,\check{Z})\sim P_{X}P_{Y|X}\times P_{W}P_{\check{Z}|W}$. $$\begin{aligned}
 & \mathbf{P}\{\mathsf{d}(W,\hat{Z})>\mathsf{D}\}\\
 & \le\mathbf{P}\{\rho(W)=0\}+\mathbf{P}\{\rho(W)>0\;\mathrm{and}\;\check{Z}\neq\hat{Z}\}\\
 & \le\mathbf{P}\{\rho(W)=0\}+\mathbf{E}\left[\mathbf{1}\{\rho(W)>0\}\mathbf{P}\{(X,\check{Z})\neq(\tilde{X},\tilde{Z})_{P_{X|Y}(\cdot|Y)\times P_{Z}}\,|\,X,Y,W,\check{Z}\}\right]\\
 & \stackrel{(a)}{\le}\mathbf{P}\{\rho(W)=0\}+\mathbf{E}\left[\mathbf{1}\{\rho(W)>0\}\left(1-\left(1+\frac{dP_{X}\times P_{\check{Z}|W}(\cdot|W)}{dP_{X|Y}(\cdot|Y)\times P_{Z}}(X,\check{Z})\right)^{-1}\right)\right]\\
 & =\mathbf{P}\{\rho(W)=0\}+\mathbf{E}\left[\mathbf{1}\{\rho(W)>0\}\left(1-\left(1+(\rho(W))^{-1}2^{-\iota_{X;Y}(X;Y)}\right)^{-1}\right)\right]\\
 & =\mathbf{E}\left[\left(1+\rho(W)2^{\iota_{X;Y}(X;Y)}\right)^{-1}\right],\end{aligned}$$ where (a) is by the conditional Poisson matching lemma on $(W,\,(X,\check{Z}),\,Y,\,P_{X|Y}\times P_{Z})$ (note that $P_{X,\check{Z}|W}=P_{X}\times P_{\check{Z}|W}$). Therefore there exists a fixed $\{(\bar{x}_{i},\bar{z}_{i}),t_{i}\}_{i\in\mathbb{N}}$ attaining the desired bound.

Compare this to Theorem 7 in [@kostina2013joint]: $$P_{e}\le\mathbf{E}\left[\min\left\{ J2^{-\iota_{X;Y}(X;Y)},\,1\right\} \right]+\mathbf{E}\left[(1-P_{Z}(\mathcal{B}_{\mathsf{D}}(W)))^{J}\right]\label{eq:jscc_kostina}$$ for any $P_{J|W}$, $J\in\mathbb{N}$. While neither of the bounds implies the other, our bound is at least within a 