set. The same is true for $\Upsilon_{P\Vert Q}(j)$.

By the mapping theorem [@kingman1992poisson; @last2017lectures] (also see Appendix A of [@sfrl_trans]), $$\{\bar{U}_{i_{P,j}},\,T_{i_{P,j}}((dP/d\mu)(U_{i_{P,j}}))^{-1}\}_{j\in\mathbb{N}}=\{\tilde{U}_{P}(j),\tilde{T}_{P}(j)\}_{j\in\mathbb{N}}$$ is a Poisson process with intensity measure $P\times\lambda_{\mathbb{R}_{\ge0}}$. Hence $$\tilde{U}_{P}(1),\tilde{U}_{P}(2),\ldots\stackrel{iid}{\sim}P.$$

We present a generalized Poisson matching lemma concerning the indices beyond the first. The proof is given in Appendix \[subsec:pf\_phidiv\].

\[Generalized Poisson matching lemma\]\[lem:phidiv\_gen\]Let $\{\bar{U}_{i},T_{i}\}_{i\in\mathbb{N}}$ be the points of a Poisson process with intensity measure $\mu\times\lambda_{\mathbb{R}_{\ge0}}$ on $\mathcal{U}\times\mathbb{R}_{\ge0}$, and $P,Q$ be probability measures over $\mathcal{U}$ with $P,Q\ll\mu$. Fix any $j\in\mathbb{N}$. Then we have the following almost surely: $$\mathbf{E}\left[\left.\Upsilon_{P\Vert Q}(j)\,\right|\,\tilde{U}_{P}(j)\right]\le j\frac{dP}{dQ}(\tilde{U}_{P}(j))+1,$$ where we write $(dP/dQ)(u)=(dP/d\mu)(u)/((dQ/d\mu)(u))$ as in (we do not require $P\ll Q$). As a result, we have the following almost surely: for all $k\in\mathbb{N}$, $$\begin{aligned}
\mathbf{P}\left\{ \left.\tilde{U}_{P}(j)\notin\{\tilde{U}_{Q}(i)\}_{i\in[1:k]}\,\right|\,\tilde{U}_{P}(j)\right\}  & \le\mathbf{P}\left\{ \left.\Upsilon_{P\Vert Q}(j)>k\,\right|\,\tilde{U}_{P}(j)\right\} \\
 & \le\min\left\{ \frac{j}{k}\frac{dP}{dQ}(\tilde{U}_{P}(j)),\,1\right\} .\end{aligned}$$ For $k=1$, this can be slightly strengthened to $$\mathbf{P}\left\{ \left.\Upsilon_{P\Vert Q}(j)>1\,\right|\,\tilde{U}_{P}(j)\right\} \le1-\left(1-\min\left\{ \frac{dP}{dQ}(\tilde{U}_{P}(j)),\,1\right\} \right)^{j}.$$ For $j=1$, this can be slightly strengthened to: for all $k\in\mathbb{N}$, $$\begin{aligned}
\mathbf{P}\left\{ \left.\Upsilon_{P\Vert Q}(1)>k\,\right|\,\tilde{U}_{P}(1)\right\}  & \le\Bigl(1-\Bigl(1+\frac{dP}{dQ}(\tilde{U}_{P}(1))\Bigr)^{-1}\Bigr)^{k}\\
 & \le1-\Bigl(1+k^{-1}\frac{dP}{dQ}(\tilde{U}_{P}(1))\Bigr)^{-1}.\end{aligned}$$

The exact distribution of $\Upsilon_{P\Vert Q}(j)$ is given in .

Similar to Lemma \[lem:cond\_phidiv\], we can state a conditional version of the generalized Poisson matching lemma. The proof follows the same logic as Lemma \[lem:cond\_phidiv\] and is omitted.

\[Conditional generalized 