Proof of The Copy Clocks Are Independent Poisson Clocks with Horizon R, and Every Record Is Almost Surely Conflict-Free for Them

Throughout, $c$ ranges over the transition labels, a finite nonempty set. Recall that $\Omega^{U}_0$ is the event that, for every label $c$, the points $U^{c,j}_i$ ($1\le j\le J_c$, $i\in\mathbb{N}$) are pairwise distinct and $U^{c,j}_i\in I_{c,j}$; it is an event with $P(\Omega^{U}_0)=1$ by claim 1 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record (it contains the intersection over $c$ of the probability-one events $\Omega^{c}_0$ of claim 1 of Uniform Representation of the Rate-One Poisson Counting Path on a Bounded Interval: Distinct Points, Poisson Increments, Conditional Law Given the Cell Counts, and Point Insertion attached to the $c$-th block of driving variables). Each $\mathsf{P}^{\sharp,c}_u$ is a random variable: $\mathsf{P}^{\sharp,c}_u=\mathbf{1}_{\Omega^{U}_0}\sum_{j=1}^{J_c}\sum_{i\in\mathbb{N}}\mathbf{1}\{i\le\mathsf{K}_{c,j}\}\mathbf{1}\{U^{c,j}_i\le u\}$ is a pointwise limit of finite sums of products of indicators of events (Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions).

Claim 1. Fix $c$ and $\omega$. Off $\Omega^{U}_0$ the path is identically $0$, a counting path. On $\Omega^{U}_0$ the path is $u\mapsto\sum_{j=1}^{J_c}\sum_{i=1}^{\mathsf{K}_{c,j}(\omega)}\mathbf{1}\{U^{c,j}_i(\omega)\le u\}$, a finite sum of indicators of half-lines $[U^{c,j}_i(\omega),\infty)$ with pairwise distinct left endpoints in $(0,R]$: it vanishes at $0$, takes values in $\mathbb{N}_0$, is nondecreasing and right-continuous (each indicator is), and has unit jumps because the endpoints are distinct; so it is a counting path (Counting Path and Its Jump Times). For $u\ge R$ every indicator equals $1$, so the path is constant on $[R,\infty)$.

Claim 2. $\mathsf{P}^{\sharp,c}$ is a stochastic process by the measurability just noted, and property 1 of Poisson Clock with a Horizon is claim 1. For the remaining properties, let $0\le u_0<u_1<\dots<u_r$ be real numbers and write $\Delta_q=\mathsf{P}^{\sharp,c}_{u_q}-\mathsf{P}^{\sharp,c}_{u_{q-1}}$ ($1\le q\le r$). By claim 1, $\Delta_q=0$ identically if $u_{q-1}\ge R$, and $\Delta_q=\mathsf{P}^{\sharp,c}_{R}-\mathsf{P}^{\sharp,c}_{u_{q-1}}$ if $u_{q-1}<R\le u_q$; in all cases $\Delta_q=\mathsf{P}^{\sharp,c}_{u_q\wedge R}-\mathsf{P}^{\sharp,c}_{u_{q-1}\wedge R}$. Let $q^*$ be the number of indices $q$ with $u_{q-1}<R$. The increments $\Delta_1,\dots,\Delta_{q^*}$ are the increments of $\mathsf{P}^{\sharp,c}$ over the partition $u_0<u_1<\dots<u_{q^*-1}<u_{q^*}\wedge R$ of a subinterval of $[0,R]$ (the group being empty when $q^*=0$), hence are independent, $\Delta_q$ having the Poisson distribution with parameter $(u_q\wedge R)-(u_{q-1}\wedge R)$, by claim 2 of The Copy Clocks Have Independent Poisson Increments on the Clock Interval: Law Identity with the Uniform Poisson Path and Applicability of the Window Discrepancy Bound; and $\Delta_{q^*+1},\dots,\Delta_r$ are identically $0$, which has the Poisson distribution with parameter $0=(u_q\wedge R)-(u_{q-1}\wedge R)$ (the unit mass at $0$, Poisson Distribution). A random variable that is identically $0$ is independent of every family of random variables: for a Borel set $B'$ the event $\{\Delta_q\in B'\}$ is $\Omega$ or $\emptyset$, so every joint probability factorizes (Independence of Events and of Random Variables). Hence $\Delta_1,\dots,\Delta_r$ are independent: this is property 2 of Poisson Clock with a Horizon, and the increment laws are property 3.

Claim 3. For each label $c$ let $\mathcal{D}^{c}$ be the $\sigma$-algebra generated by the $c$-th block of driving variables, $K^{c}$, $V^{c}_i$ ($i\in\mathbb{N}$) and $U^{c,j}_i$ ($1\le j\le J_c$, $i\in\mathbb{N}$). Since the whole family of driving variables is independent and the blocks are disjoint, the family $(\mathcal{D}^{c})_c$ is independent by the grouping lemma (applied after enumerating the countable index set of the driving variables, independence being unaffected by reindexing). Put $\tilde{\mathsf{P}}^{c}_u=\sum_{j=1}^{J_c}\sum_{i=1}^{\mathsf{K}_{c,j}}\mathbf{1}\{U^{c,j}_i\le u\}$ ($u\ge0$); as $\mathsf{K}_{c,j}=\sum_{i=1}^{\widetilde{K}^{c}}\mathbf{1}\{V^{c}_i\in I_{c,j}\}$ is the cell count of Uniform Representation of the Rate-One Poisson Counting Path on a Bounded Interval: Distinct Points, Poisson Increments, Conditional Law Given the Cell Counts, and Point Insertion for the $c$-th block, hence measurable with respect to the $\sigma$-algebra generated by $K^{c}$ and the $V^{c}_i$ and a fortiori $\mathcal{D}^{c}$-measurable, each $\tilde{\mathsf{P}}^{c}_u$ is $\mathcal{D}^{c}$-measurable (Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions: the sum equals $\sum_j\sum_{i\in\mathbb{N}}\mathbf{1}\{i\le\mathsf{K}_{c,j}\}\mathbf{1}\{U^{c,j}_i\le u\}$, a pointwise limit of finite sums of products of $\mathcal{D}^{c}$-measurable indicators), and $\mathsf{P}^{\sharp,c}_u=\tilde{\mathsf{P}}^{c}_u$ on $\Omega^{U}_0$. Let $\mathcal{E}^{c}$ be the class of events $A\in\sigma(\mathsf{P}^{\sharp,c}_u:u\ge0)$ for which there is an event $\tilde{A}\in\mathcal{D}^{c}$ with $A\cap\Omega^{U}_0=\tilde{A}\cap\Omega^{U}_0$. This class is a $\sigma$-algebra (it contains $\Omega$, and complements and countable unions of the $\tilde{A}$'s serve for complements and countable unions of the $A$'s), and it contains the generators $\{\mathsf{P}^{\sharp,c}_u\in B'\}$ ($u\ge0$, $B'$ a Borel set), with $\tilde{A}=\{\tilde{\mathsf{P}}^{c}_u\in B'\}$; hence every $A\in\sigma(\mathsf{P}^{\sharp,c}_u:u\ge0)$ admits such an $\tilde{A}$, and $P(A)=P(\tilde{A})$ since $P(\Omega^{U}_0)=1$. Now let $A_c\in\sigma(\mathsf{P}^{\sharp,c}_u:u\ge0)$ be given for the labels $c$ in a finite set of labels, with $\tilde{A}_c\in\mathcal{D}^{c}$ as above. Then $\bigcap_cA_c$ and $\bigcap_c\tilde{A}_c$ agree on $\Omega^{U}_0$, so

$$P\Bigl(\bigcap_cA_c\Bigr)=P\Bigl(\bigcap_c\tilde{A}_c\Bigr)=\prod_cP(\tilde{A}_c)=\prod_cP(A_c),$$

the middle equality by the independence of $(\mathcal{D}^{c})_c$. This is the independence of the family $\sigma(\mathsf{P}^{\sharp,c}_u:u\ge0)$, $c$ ranging over the labels (Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras).

Claim 4. Fix $r\in\mathbf{R}$. For $y\in\mathbb{N}_0^{\mathsf{L}}$, on the event $\{\mathsf{K}=y\}$ one has $\mathsf{P}^{\sharp,c}_u(\omega)=\mathsf{P}^{(y),c}_u(\omega)$ for all $c$ and $u\ge0$, by the definition $\mathsf{P}^{\sharp,c}_u(\omega)=\mathsf{P}^{(\mathsf{K}(\omega)),c}_u(\omega)$; hence the data $(\mathsf{P}^{\sharp}(\omega),a^{r},x_0)$ and $(\mathsf{P}^{(y)}(\omega),a^{r},x_0)$ coincide, and $(r,\omega)\in\mathsf{G}^{\sharp}$ if and only if $(r,\omega)\in\mathsf{G}^{(y)}$. Therefore

$$\Omega^{r,\sharp}=\bigcup_{y\in\mathbb{N}_0^{\mathsf{L}}}\bigl(\{\mathsf{K}=y\}\cap\Omega^{r,y}\bigr),\qquad \Omega^{r,y}=\{\omega:(r,\omega)\notin\mathsf{G}^{(y)}\},$$

a countable union ($\mathbb{N}_0^{\mathsf{L}}$ is countable, Products and Powers of Countable Sets) of events ($\{\mathsf{K}=y\}$ is an event since $\mathsf{K}$ is a random vector with values in $\mathbb{N}_0^{\mathsf{L}}$, and $\Omega^{r,y}$ is an event by claim 2 of Almost Sure Tracking on the Synthetic Copy: Jump Times of the Deterministic-Count Clocks, Almost Sure Conflict-Freeness of Every Record, Almost Sure Null Mass of the Untracked Records, and Trimming an Event to the Tracked Set), so $\Omega^{r,\sharp}$ is an event; each member of the union has probability zero, since $P(\Omega^{r,y})=0$ by the same claim 2. By countable subadditivity (claim 4 of Basic Properties of a Measure), $P(\Omega^{r,\sharp})=0$.