The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record

Adopt the setting and notation of The Record-Driven Causal Intensity of the Open-Loop Aggregate Solution: Joint Measurability of the Record-Frozen Control, the Regularised Recursion Path, Non-Anticipation, and Measurability of the Likelihood: natural numbers $N\ge1$, $l\ge2$, $m\ge1$, $\tilde{l}\ge1$, the control set $\mathcal{A}\subseteq\mathbb{R}^m$, real numbers $B\ge0$, $\tilde{B}\ge0$ and $T>0$, the transition-rate family $\beta$, the observation-rate family $\tilde\beta$ with aggregate observation drift $\tilde{b}$, the aggregate lattice $\mathbb{G}_N$ and the point $x_0\in\mathbb{G}_N$, the transition labels $c$ (the ordered pairs $(\sigma,\gamma)$ of distinct elements of $\{1,\dots,l\}$, of which there are $l(l-1)$), the observation record space $(\mathbf{R},\mathcal{R},\rho)$ with horizon $T$ and $\tilde{l}$ channels, with cells $C_\emptyset$ and $C_{k,v}$ and with $\mathbf{R}^{(k)}$ the set of records with exactly $k$ events (the measure $\rho$ is $\sigma$-finite: there are countably many cells, $\rho(C_\emptyset)=1$, and $\rho(C_{k,v})$ is the volume of the ordered time simplex $D_k(T)$, which is finite), and the observation-driven control policy $h$ with values in $\mathcal{A}$, with record-frozen control paths $a^r$. Let $\mathbb{N}$ be the set of natural numbers and $\mathbb{N}_0=\mathbb{N}\cup\{0\}$. Let $R>0$ be a real number with $R\ge NBT$, and for every transition label $c$ let $J_c\ge1$ be a natural number and $0=b^{c}_0<b^{c}_1<\dots<b^{c}_{J_c}=R$ real numbers; the cells of the clock $c$ are the intervals $I_{c,j}=(b^{c}_{j-1},b^{c}_j]$, $1\le j\le J_c$, of lengths $|I_{c,j}|=b^{c}_j-b^{c}_{j-1}$. Let $\mathsf{L}$ be the set of all pairs $(c,j)$ with $c$ a transition label and $1\le j\le J_c$, a finite set with $d=\sum_{c}J_c\ge1$ elements; fix a bijection of $\mathsf{L}$ with $\{1,\dots,d\}$ and index the coordinates of points of Euclidean space $\mathbb{R}^d$ by $\mathsf{L}$, writing $e_{c,j}$ for the standard basis vector of the coordinate $(c,j)$ and $\mathbb{N}_0^{\mathsf{L}}\subseteq\mathbb{R}^d$ for the set of points with all coordinates in $\mathbb{N}_0$. Let $\eta$ be a real number with $0<\eta\le1$ and let $\varphi_\eta$ be the Gaussian smoothing weight on $\mathbb{R}^d$. Write $\mathrm{poi}_\mu(k)=\exp(-\mu)\mu^{k}/k!$ ($k\in\mathbb{N}_0$, $\mu\ge0$) for the mass function of the Poisson distribution with parameter $\mu$, with the exponential function $\exp$; $\mathcal{B}(\mathbb{R}^d)$ for the Borel $\sigma$-algebra and $\lambda_d$ for Lebesgue measure on $\mathbb{R}^d$; $\otimes$ for product $\sigma$-algebras and product measures, triple products always being written with explicit parentheses; $\mathbb{E}$ for the expectation on $(\Omega,\mathcal{F},P)$, extended to $[0,\infty]$-valued measurable maps as their integrals with respect to $P$; $\mathbf{1}\{\cdot\}$ for the indicator of an event; and $\lVert\cdot\rVert$ for the Euclidean norm.

The driving variables. Let $(\Omega,\mathcal{F},P)$ be a probability space carrying an independent family of random variables $K^{c}\ (c\text{ a label}),\qquad V^{c}_i\ (c\text{ a label},\ i\in\mathbb{N}),\qquad U^{c,j}_i\ ((c,j)\in\mathsf{L},\ i\in\mathbb{N}),$ where $K^{c}$ has the Poisson distribution with parameter $R$, $V^{c}_i$ has the uniform law on $(0,R]$ and $U^{c,j}_i$ has the uniform law on $I_{c,j}$, in the sense 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; such a space exists by Existence of Independent Sequences with Prescribed Distributions. For each label $c$, the variables $K^{c}$, $(V^{c}_i)_i$, $(U^{c,j}_i)_{j,i}$ (an independent family, as a subfamily of an independent family) with the cells $I_{c,1},\dots,I_{c,J_c}$ form the data 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; let $\widetilde{K}^{c}$, the uniform Poisson path $p^{c}$, the deterministic-count paths $p^{c,(y)}$ ($y\in\mathbb{N}_0^{J_c}$), the cell counts (written $\mathsf{K}_{c,j}$ here, $1\le j\le J_c$; they are the $C_j$ of that lemma, a symbol reserved here for the cells $C_{k,v}$ of the record space) and the event $\Omega^{c}_0$ be the corresponding objects of that lemma. Let $\mathcal{V}$ be the $\sigma$-algebra generated by all $K^{c}$ and $V^{c}_i$, let $\mathcal{U}$ be the $\sigma$-algebra generated by all $U^{c,j}_i$, let $\Omega^{U}_0$ be 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 with $U^{c,j}_i\in I_{c,j}$, and let $\mathsf{K}:\Omega\to\mathbb{N}_0^{\mathsf{L}}$ be the cell-count vector with coordinates $\mathsf{K}_{c,j}$.

The clocks. For $y\in\mathbb{N}_0^{\mathsf{L}}$, a label $c$ and $u\ge0$ put $\mathsf{P}^{(y),c}_u=\mathbf{1}_{\Omega^{U}_0}\sum_{j=1}^{J_c}\sum_{i=1}^{y_{c,j}}\mathbf{1}\{U^{c,j}_i\le u\},\qquad \mathsf{P}^{\sharp,c}_u(\omega)=\mathsf{P}^{(\mathsf{K}(\omega)),c}_u(\omega),$ the deterministic-count clocks $\mathsf{P}^{(y)}=(\mathsf{P}^{(y),c})_c$ and the copy clocks $\mathsf{P}^{\sharp}=(\mathsf{P}^{\sharp,c})_c$.

1. (Independent cell structure) $\mathcal{U}$ and $\mathcal{V}$ are independent; $\Omega^{U}_0\in\mathcal{U}$ and $P(\Omega^{U}_0)=1$; each $\mathsf{K}_{c,j}$ is a $\mathcal{V}$-measurable random variable with values in $\mathbb{N}_0$ and $\mathbb{E}[\mathsf{K}_{c,j}]=|I_{c,j}|$; $P(\mathsf{K}=y)=\prod_{(c,j)\in\mathsf{L}}\mathrm{poi}_{|I_{c,j}|}(y_{c,j})>0$ for every $y\in\mathbb{N}_0^{\mathsf{L}}$, and $\sum_{y\in\mathbb{N}_0^{\mathsf{L}}}P(\mathsf{K}=y)=1$; and for every $y\in\mathbb{N}_0^{\mathsf{L}}$ and every $\mathcal{U}$-measurable $Z:\Omega\to[0,\infty]$, $\mathbb{E}\bigl[\mathbf{1}\{\mathsf{K}=y\}\,Z\bigr]=P(\mathsf{K}=y)\,\mathbb{E}[Z]\qquad\text{in }[0,\infty].$

2. (Clock structure) For every $y\in\mathbb{N}_0^{\mathsf{L}}$ and every label $c$: every path $u\mapsto\mathsf{P}^{(y),c}_u(\omega)$ is a counting path, each $\mathsf{P}^{(y),c}_u$ is $\mathcal{U}$-measurable, and $\mathsf{P}^{(y),c}_u(\omega)=p^{c,(y_{c,\cdot})}_u(\omega)$ for $\omega\in\Omega^{U}_0$ and $u\in[0,R]$, where $y_{c,\cdot}=(y_{c,1},\dots,y_{c,J_c})$. Every path of $\mathsf{P}^{\sharp,c}$ is a counting path and each $\mathsf{P}^{\sharp,c}_u$ is $\mathcal{F}$-measurable. For $\mathsf{m}\in\mathbb{N}$ and $(c_0,j_0)\in\mathsf{L}$, one has $\mathsf{P}^{(y+\mathsf{m}e_{c_0,j_0}),c}=\mathsf{P}^{(y),c}$ for $c\neq c_0$, and $\mathsf{P}^{(y+\mathsf{m}e_{c_0,j_0}),c_0}_u=\mathsf{P}^{(y),c_0}_u+\mathbf{1}_{\Omega^{U}_0}\sum_{i=y_{c_0,j_0}+1}^{y_{c_0,j_0}+\mathsf{m}}\mathbf{1}\{U^{c_0,j_0}_i\le u\}\qquad(u\ge0),$ where on $\Omega^{U}_0$ the $\mathsf{m}$ inserted points $U^{c_0,j_0}_i$ ($y_{c_0,j_0}<i\le y_{c_0,j_0}+\mathsf{m}$) lie in $I_{c_0,j_0}$, are pairwise distinct, and differ from every point $U^{c_0,j}_{i'}$ with $1\le j\le J_{c_0}$ and $1\le i'\le y_{c_0,j}$.

3. (Record-driven likelihoods on the copy) The Record-Driven Causal Intensity of the Open-Loop Aggregate Solution: Joint Measurability of the Record-Frozen Control, the Regularised Recursion Path, Non-Anticipation, and Measurability of the Likelihood applies to the probability space $(\Omega,\mathcal{U},P|_{\mathcal{U}})$ with the clocks $\mathsf{P}^{(y)}$, for each $y\in\mathbb{N}_0^{\mathsf{L}}$, and to $(\Omega,\mathcal{F},P)$ with the clocks $\mathsf{P}^{\sharp}$; write $\bar\Sigma^{(y),r}(\omega)$, $\lambda^{(y),\omega}$, $\ell^{(y),\omega}$, $\mathsf{G}^{(y)}$ and $\bar\Sigma^{\sharp,r}(\omega)$, $\lambda^{\sharp,\omega}$, $\ell^{\sharp,\omega}$, $\mathsf{G}^{\sharp}$ for the regularised paths, intensities, likelihoods and conflict-free sets so obtained. Then $(r,\omega)\mapsto\ell^{(y),\omega}(r)$ is $\mathcal{R}\otimes\mathcal{U}$-measurable, $(r,\omega)\mapsto\ell^{\sharp,\omega}(r)$ is $\mathcal{R}\otimes\mathcal{F}$-measurable, both take values in $[0,(N\tilde{B})^{k}]$ on $\mathbf{R}^{(k)}$ and integrate to $1$ against $\rho$ for every $\omega$, and for every $\omega\in\Omega$, with $y=\mathsf{K}(\omega)$: $\bar\Sigma^{\sharp,r}_t(\omega)=\bar\Sigma^{(y),r}_t(\omega)$ for all $r$ and $t$, $\ell^{\sharp,\omega}=\ell^{(y),\omega}$, and $(r,\omega)\in\mathsf{G}^{\sharp}$ if and only if $(r,\omega)\in\mathsf{G}^{(y)}$.

The copy. With the likelihoods of claim 3 and $\sqrt{N}$ the nonnegative square root, the synthetic copy is the triple $(\Omega^{\sharp},\mathcal{F}^{\sharp},\mu^{\sharp})$ with $\Omega^{\sharp}=\Omega\times\mathbb{R}^d\times\mathbf{R}$, $\mathcal{F}^{\sharp}=(\mathcal{F}\otimes\mathcal{B}(\mathbb{R}^d))\otimes\mathcal{R}$, and $\mu^{\sharp}$ the measure with density $\mathsf{q}^{\sharp}(\omega,\theta,r)=\varphi_\eta\bigl(\theta-\mathsf{K}(\omega)/\sqrt{N}\bigr)\,\ell^{\sharp,\omega}(r)$ ($\mathcal{F}^{\sharp}$-measurable by claim 4 below) with respect to the reference measure $(P\otimes\lambda_d)\otimes\rho$ (which exists, $P$ being finite and $\lambda_d$ and $\rho$ being $\sigma$-finite). Its coordinate maps are the parameter $\Theta(\omega,\theta,r)=\theta$ and the record $\mathsf{D}(\omega,\theta,r)=r$. The parameter lattice is $\mathsf{S}=\{y/\sqrt{N}:y\in\mathbb{N}_0^{\mathsf{L}}\}\subseteq\mathbb{R}^d$, the count mass function is $\mathsf{p}:\mathbb{R}^d\to[0,1]$, $\mathsf{p}(x)=P(\mathsf{K}=\sqrt{N}x)$ for $x\in\mathsf{S}$ and $\mathsf{p}(x)=0$ otherwise, and the record kernel is $f:\mathsf{S}\times\mathbf{R}\to[0,\infty)$, $f(x,r)=\mathbb{E}\bigl[\ell^{(\sqrt{N}x),\cdot}(r)\bigr]$.

4. (The copy is a probability space) $\mathsf{q}^{\sharp}$ is $\mathcal{F}^{\sharp}$-measurable with values in $[0,\infty)$, and $\mu^{\sharp}$ is a probability measure on $(\Omega^{\sharp},\mathcal{F}^{\sharp})$. The maps $\Theta$ and $\mathsf{D}$ are measurable from $\mathcal{F}^{\sharp}$ to $\mathcal{B}(\mathbb{R}^d)$ and to $\mathcal{R}$ respectively.

5. (Smoothed joint density of parameter and record) $\mathsf{S}$ is countable in the sense of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions; $\mathsf{p}$ is a discrete probability mass function on $\mathbb{R}^d$ with support $\{x:\mathsf{p}(x)>0\}=\mathsf{S}$ and $\sum_{x\in\mathsf{S}}\mathsf{p}(x)\lVert x\rVert\le l(l-1)R/\sqrt{N}<\infty$; and the kernel $f$ satisfies conditions (K1) and (K2) of Gaussian Smoothing of a Kernel-Weighted Discrete Family with Countable Support: Joint Density, van Trees Regularity, and the Symmetrised Directional Score Bound, the latter with the $\mathcal{R}$-measurable majorant $M(r)=(N\tilde{B})^{k}$ for $r\in\mathbf{R}^{(k)}$ (with $(N\tilde{B})^{0}=1$). Consequently all hypotheses of Gaussian Smoothing of a Kernel-Weighted Discrete Family with Countable Support: Joint Density, van Trees Regularity, and the Symmetrised Directional Score Bound on $(d,\eta,\mathsf{p},f)$ hold, and its claims 1 to 3 are available for every natural number $n\ge1$, every choice of moves $a_1,\dots,a_n\in\mathbb{R}^d$ with $x+a_q\in\mathsf{S}$ for all $x\in\mathsf{S}$ and all $q$ (for instance $a_q=\mathsf{m}_qe_{c_q,j_q}/\sqrt{N}$ with $\mathsf{m}_q\in\mathbb{N}$ and $(c_q,j_q)\in\mathsf{L}$), and every choice of weights $w\in\mathbb{R}^n$. Moreover the smoothed joint density $g$ of that lemma, $g(\theta,r)=\sum_{x\in\mathsf{S}}\mathsf{p}(x)\varphi_\eta(\theta-x)f(x,r)$, satisfies $g(\theta,r)=\mathbb{E}\bigl[\varphi_\eta(\theta-\mathsf{K}/\sqrt{N})\,\ell^{\sharp,\cdot}(r)\bigr]\qquad\text{for all }(\theta,r)\in\mathbb{R}^d\times\mathbf{R},$ and is the joint density of $(\Theta,\mathsf{D})$ under $\mu^{\sharp}$: for every $F:\mathbb{R}^d\times\mathbf{R}\to[0,\infty]$ measurable with respect to $\mathcal{B}(\mathbb{R}^d)\otimes\mathcal{R}$, $\int_{\Omega^{\sharp}}F(\Theta,\mathsf{D})\,d\mu^{\sharp}=\int_{\mathbb{R}^d\times\mathbf{R}}F\,g\,d(\lambda_d\otimes\rho)\qquad\text{in }[0,\infty].$