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

Adopt the setting and notation of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record: the natural numbers $N\ge1$, $l\ge2$, $m\ge1$, $\tilde{l}\ge1$, the real numbers $B\ge0$, $T>0$ and $R>0$ with $R\ge NBT$, the probability space $(\Omega,\mathcal{F},P)$ carrying the independent family of driving variables $K^{c}$, $V^{c}_i$, $U^{c,j}_i$ (with $c$ ranging over the transition labels), the cells $I_{c,j}=(b^{c}_{j-1},b^{c}_j]\subseteq(0,R]$ (with $0=b^{c}_0<b^{c}_1<\dots<b^{c}_{J_c}=R$ and natural numbers $J_c\ge1$) indexed by the finite set $\mathsf{L}$ of pairs $(c,j)$ with $1\le j\le J_c$, the $\sigma$-algebras $\mathcal{U}$ and $\mathcal{V}$, the event $\Omega^{U}_0$, the set $\mathbb{N}_0^{\mathsf{L}}$, the cell-count vector $\mathsf{K}=(\mathsf{K}_{c,j})_{(c,j)\in\mathsf{L}}$, and, for each transition label $c$, the uniform Poisson path $p^{c}=(p^{c}_u)_{u\in[0,R]}$, the deterministic-count paths $p^{c,(z)}$ ($z\in\mathbb{N}_0^{J_c}$) 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 and the cell-count block $\mathsf{K}_{c,\cdot}=(\mathsf{K}_{c,1},\dots,\mathsf{K}_{c,J_c})$ (a notation coined here; it is the vector of cell counts 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 data of the label $c$, written $C$ there, a symbol otherwise reserved in the present setting for the record cells), all formed from $K^{c}$, $(V^{c}_i)_i$, $(U^{c,j}_i)_{j,i}$ and the cells $I_{c,1},\dots,I_{c,J_c}$, together with the deterministic-count clocks $\mathsf{P}^{(y)}=(\mathsf{P}^{(y),c})_c$ ($y\in\mathbb{N}_0^{\mathsf{L}}$) and the copy clocks $\mathsf{P}^{\sharp}=(\mathsf{P}^{\sharp,c})_c$, $\mathsf{P}^{\sharp,c}_u(\omega)=\mathsf{P}^{(\mathsf{K}(\omega)),c}_u(\omega)$ ($u\ge0$). For a real number $\mu\ge0$ write $P_\mu$ for the Poisson distribution with parameter $\mu$, a probability measure on the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$; independence of random variables and their distributions are as in those definitions.

1. (Law identity for the increments) Let $c$ be a transition label, let $r\ge1$ be a natural number, let $0\le u_0<u_1<\dots<u_r\le R$ be real numbers and let $B'_1,\dots,B'_r\in\mathcal{B}(\mathbb{R})$. Then $P\Bigl(\bigcap_{k=1}^{r}\bigl\{\mathsf{P}^{\sharp,c}_{u_k}-\mathsf{P}^{\sharp,c}_{u_{k-1}}\in B'_k\bigr\}\Bigr)=P\Bigl(\bigcap_{k=1}^{r}\bigl\{p^{c}_{u_k}-p^{c}_{u_{k-1}}\in B'_k\bigr\}\Bigr)=\prod_{k=1}^{r}P_{u_k-u_{k-1}}(B'_k).$

2. (Independent Poisson increments on the clock interval) For every transition label $c$: $\mathsf{P}^{\sharp,c}_0(\omega)=0$ for every $\omega\in\Omega$; for all real $0\le u<u'\le R$ the increment $\mathsf{P}^{\sharp,c}_{u'}-\mathsf{P}^{\sharp,c}_{u}$ has the Poisson distribution with parameter $u'-u$; and for every natural number $r\ge1$ and all real $0\le u_0<u_1<\dots<u_r\le R$ the increments $\mathsf{P}^{\sharp,c}_{u_1}-\mathsf{P}^{\sharp,c}_{u_0},\dots,\mathsf{P}^{\sharp,c}_{u_r}-\mathsf{P}^{\sharp,c}_{u_{r-1}}$ are independent.

3. (Applicability of the window discrepancy bound) For every transition label $c$ and every natural number $n$ with $n\le R$ (such $n$ exist exactly when $R\ge1$), the process $\mathsf{P}^{\sharp,c}=(\mathsf{P}^{\sharp,c}_u)_{u\ge0}$ on $(\Omega,\mathcal{F},P)$ satisfies the hypotheses of Uniform Window Discrepancy Bound for a Counting Process with Poisson Increments on the Integer Grid with this $n$, with any natural number in the role of the window length called $m$ there, and with any real $x>0$: all its paths are counting paths, and for all integers $0\le i<i'\le n$ the increment $\mathsf{P}^{\sharp,c}_{i'}-\mathsf{P}^{\sharp,c}_{i}$ has the Poisson distribution with parameter $i'-i$.