Proof 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

Throughout, "measurable" for a real-valued map means measurable with respect to the relevant $\sigma$-algebra and $\mathcal{B}(\mathbb{R})$; we use claims 1 to 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions for constants, indicators, sums and products of measurable maps, and claims 2 and 4 of Basic Properties of a Measure (monotonicity, countable subadditivity) together with countable additivity of $P$ (Measure, Measure Space, and Probability Measure). Generic Borel sets are written $B'$, the letter $B$ being the rate bound. Fix a transition label $c$ for the whole proof, and write $z=y_{c,\cdot}=(y_{c,1},\dots,y_{c,J_c})\in\mathbb{N}_0^{J_c}$ for the $c$-block of a vector $y\in\mathbb{N}_0^{\mathsf{L}}$.

Step 0 (the clocks depend on $y$ only through the $c$-block). By the definition of the deterministic-count clocks in The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record, $\mathsf{P}^{(y),c}_u=\mathbf{1}_{\Omega^{U}_0}\sum_{j=1}^{J_c}\sum_{i=1}^{y_{c,j}}\mathbf{1}\{U^{c,j}_i\le u\}$ involves $y$ only through $y_{c,\cdot}$. Hence for $z\in\mathbb{N}_0^{J_c}$ we may write $\mathsf{P}^{[z],c}$ for $\mathsf{P}^{(y),c}$ with any $y$ satisfying $y_{c,\cdot}=z$ (a notation coined here), and $\mathsf{P}^{\sharp,c}_u(\omega)=\mathsf{P}^{[\mathsf{K}_{c,\cdot}(\omega)],c}_u(\omega)$ for every $\omega$ and $u\ge0$, the $c$-block of $\mathsf{K}(\omega)$ being $\mathsf{K}_{c,\cdot}(\omega)$. By claim 2 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record, each $\mathsf{P}^{[z],c}_u$ is $\mathcal{U}$-measurable, and $\mathsf{P}^{[z],c}_u(\omega)=p^{c,(z)}_u(\omega)$ for $\omega\in\Omega^{U}_0$ and $u\in[0,R]$. By claim 1 of the same lemma, $\Omega^{U}_0\in\mathcal{U}$ with $P(\Omega^{U}_0)=1$, each $\mathsf{K}_{c,j}$ is $\mathcal{V}$-measurable with values in $\mathbb{N}_0$, and $\mathcal{U}$ and $\mathcal{V}$ are independent, i.e. $P(A\cap A')=P(A)P(A')$ for $A\in\mathcal{V}$ and $A'\in\mathcal{U}$.

Step 1 (claim 1). Fix $r$, $0\le u_0<\dots<u_r\le R$ and $B'_1,\dots,B'_r$ as in claim 1. Write $E^{\sharp}$ for the event on the left of the display in claim 1, and for $z\in\mathbb{N}_0^{J_c}$ put $E_z=\bigcap_{k=1}^{r}\bigl\{\mathsf{P}^{[z],c}_{u_k}-\mathsf{P}^{[z],c}_{u_{k-1}}\in B'_k\bigr\},\qquad E'_z=\bigcap_{k=1}^{r}\bigl\{p^{c,(z)}_{u_k}-p^{c,(z)}_{u_{k-1}}\in B'_k\bigr\},\qquad D_z=\{\mathsf{K}_{c,\cdot}=z\}=\bigcap_{j=1}^{J_c}\{\mathsf{K}_{c,j}=z_j\}.$ Then $E_z\in\mathcal{U}$ (preimages of Borel sets under $\mathcal{U}$-measurable differences), $D_z\in\mathcal{V}$, and $E'_z\in\mathcal{F}$ (each $p^{c,(z)}_u$ is a random variable by claim 2 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). The events $D_z$, $z\in\mathbb{N}_0^{J_c}$, are pairwise disjoint with union $\Omega$, because $\mathsf{K}_{c,\cdot}$ takes values in $\mathbb{N}_0^{J_c}$. The set $\mathbb{N}_0^{J_c}$ is countable: $\mathbb{N}_0=\mathbb{N}\cup\{0\}$ is countable by claims 1 and 6 of Basic Properties of Countable Sets, and its $J_c$-th power is countable by claim 2 of Products and Powers of Countable Sets. Fix a sequence $(z_n)_{n\in\mathbb{N}}$ whose set of terms is $\mathbb{N}_0^{J_c}$. Whenever below we sum a family $(P(A_z))_{z\in\mathbb{N}_0^{J_c}}$ of probabilities of pairwise disjoint events $A_z$, we mean the sum $\sum_{n\in\mathbb{N}}P(A'_n)$ with $A'_n=A_{z_n}$ if $z_n\notin\{z_1,\dots,z_{n-1}\}$ and $A'_n=\emptyset$ otherwise; the events $A'_n$ are pairwise disjoint with $\bigcup_nA'_n=\bigcup_zA_z$, so countable additivity gives $P(\bigcup_zA_z)=\sum_zP(A_z)$ in this sense, and the same convention (dropping repeated terms) is used for sums of products $P(D_z)P(E_z)$ and $P(D_z)P(E'_z)$, which are then compared term by term.

(a) $E^{\sharp}=\bigcup_{z}(D_z\cap E_z)$. Indeed, for $\omega\in D_z$ we have $\mathsf{K}_{c,\cdot}(\omega)=z$, so by Step 0 $\mathsf{P}^{\sharp,c}_u(\omega)=\mathsf{P}^{[z],c}_u(\omega)$ for every $u$, and therefore $\omega\in E^{\sharp}$ if and only if $\omega\in E_z$. By countable additivity and the independence of $\mathcal{U}$ and $\mathcal{V}$, $P(E^{\sharp})=\sum_{z}P(D_z\cap E_z)=\sum_{z}P(D_z)\,P(E_z).$

(b) $P(E_z)=P(E'_z)$ for every $z$. By Step 0, $E_z\cap\Omega^{U}_0=E'_z\cap\Omega^{U}_0$ (all $u_k$ lie in $[0,R]$), so each of $E_z$, $E'_z$ is contained in the union of the other with $\Omega\setminus\Omega^{U}_0$, and monotonicity with subadditivity give $P(E_z)\le P(E'_z)+P(\Omega\setminus\Omega^{U}_0)=P(E'_z)$ and symmetrically $P(E'_z)\le P(E_z)$, using $P(\Omega\setminus\Omega^{U}_0)=1-P(\Omega^{U}_0)=0$ (claim 3 of Basic Properties of a Measure).

(c) The conditional-law identity. Let $\mathbb{R}^{[0,R]}$ carry the cylinder $\sigma$-algebra $\mathcal{C}$ 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, generated by the coordinate maps $\mathrm{ev}_u$, and put $F=\mathbf{1}_{A},\qquad A=\bigcap_{k=1}^{r}\bigl\{x\in\mathbb{R}^{[0,R]}:\ \mathrm{ev}_{u_k}(x)-\mathrm{ev}_{u_{k-1}}(x)\in B'_k\bigr\}.$ Each $\mathrm{ev}_u$ is measurable from $\mathcal{C}$ to $\mathcal{B}(\mathbb{R})$ (the sets $\mathrm{ev}_u^{-1}(B')$ generate $\mathcal{C}$), so each difference $\mathrm{ev}_{u_k}-\mathrm{ev}_{u_{k-1}}$ is measurable, $A$ is a finite intersection of preimages of Borel sets and lies in $\mathcal{C}$, and $F:\mathbb{R}^{[0,R]}\to[0,\infty]$ is $\mathcal{C}$-measurable. For the maps $p^{c},p^{c,(z)}:\Omega\to\mathbb{R}^{[0,R]}$ (measurable with respect to $\mathcal{F}$ and $\mathcal{C}$ by claim 3 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) we have $F(p^{c,(z)})=\mathbf{1}_{E'_z}$ and $F(p^{c})=\mathbf{1}_{E'}$, where $E'=\bigcap_{k=1}^{r}\bigl\{p^{c}_{u_k}-p^{c}_{u_{k-1}}\in B'_k\bigr\},$ the event on the right of the first equality in claim 1. Claim 3 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, applied for the label $c$ (whose data $K^{c}$, $(V^{c}_i)_i$, $(U^{c,j}_i)_{j,i}$ and cells $I_{c,1},\dots,I_{c,J_c}$ are those of that lemma, as recorded in The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record), with this $F$ and $y=z$ gives, the expectation of the indicator of an event being the probability of that event (Simple Function and Its Integral), $P(E'\cap D_z)=\mathbb{E}\bigl[F(p^{c})\mathbf{1}\{\mathsf{K}_{c,\cdot}=z\}\bigr]=P(\mathsf{K}_{c,\cdot}=z)\,\mathbb{E}\bigl[F(p^{c,(z)})\bigr]=P(D_z)\,P(E'_z).$

(d) Assembly. By (a), (b), (c) and countable additivity over the partition $(D_z)_z$ of $\Omega$, $P(E^{\sharp})=\sum_{z}P(D_z)P(E_z)=\sum_{z}P(D_z)P(E'_z)=\sum_{z}P(E'\cap D_z)=P(E'),$ which is the first equality of claim 1. For the second, claim 2 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 states that the increments $p^{c}_{u_1}-p^{c}_{u_0},\dots,p^{c}_{u_r}-p^{c}_{u_{r-1}}$ are independent with $p^{c}_{u_k}-p^{c}_{u_{k-1}}$ Poisson with parameter $u_k-u_{k-1}$; by Independence of Events and of Random Variables (with the full index set) and Distribution and Cumulative Distribution Function of a Random Variable, $P(E')=\prod_{k=1}^{r}P\bigl(p^{c}_{u_k}-p^{c}_{u_{k-1}}\in B'_k\bigr)=\prod_{k=1}^{r}P_{u_k-u_{k-1}}(B'_k).$

Step 2 (claim 2). Every path of $\mathsf{P}^{\sharp,c}$ is a counting path by claim 2 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record, and a counting path vanishes at $0$ by condition 1 of Counting Path and Its Jump Times; hence $\mathsf{P}^{\sharp,c}_0=0$ everywhere. Next let $0\le u<u'\le R$. Claim 1 with $r=1$, $u_0=u$, $u_1=u'$ gives $P(\mathsf{P}^{\sharp,c}_{u'}-\mathsf{P}^{\sharp,c}_{u}\in B')=P_{u'-u}(B')$ for every Borel set $B'$; the increment is a random variable (a difference of the $\mathcal{F}$-measurable variables of claim 2 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record), so its distribution is $P_{u'-u}$, i.e. it is Poisson with parameter $u'-u$. Finally let $0\le u_0<\dots<u_r\le R$ and write $X_k=\mathsf{P}^{\sharp,c}_{u_k}-\mathsf{P}^{\sharp,c}_{u_{k-1}}$. For Borel sets $B'_1,\dots,B'_r$ and a nonempty $S\subseteq\{1,\dots,r\}$, apply claim 1 with $B'_k$ replaced by $\mathbb{R}$ for $k\notin S$ (so that $\{X_k\in\mathbb{R}\}=\Omega$ and $P_{u_k-u_{k-1}}(\mathbb{R})=1$): $P(\bigcap_{k\in S}\{X_k\in B'_k\})=\prod_{k\in S}P_{u_k-u_{k-1}}(B'_k)=\prod_{k\in S}P(X_k\in B'_k)$, the last step by the single-increment case just proved. This is the independence of $X_1,\dots,X_r$ in the sense of Independence of Events and of Random Variables.

Step 3 (claim 3). Let $n\le R$ be a natural number. Every path of $\mathsf{P}^{\sharp,c}$ is a counting path and every $\mathsf{P}^{\sharp,c}_u$ is $\mathcal{F}$-measurable by claim 2 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record, so $\mathsf{P}^{\sharp,c}$ is a stochastic process on $(\Omega,\mathcal{F},P)$ with counting paths; and for integers $0\le i<i'\le n\le R$ the increment $\mathsf{P}^{\sharp,c}_{i'}-\mathsf{P}^{\sharp,c}_{i}$ is Poisson with parameter $i'-i$ by claim 2. These are exactly the hypotheses of Uniform Window Discrepancy Bound for a Counting Process with Poisson Increments on the Integer Grid for this $n$ and an arbitrary natural number in the role of the window length called $m$ there (the $m$ of the present setting is the control dimension and plays no role here).