Pathwise Reduction of the Symmetrised Move Information on the Synthetic Copy: Removed-Clock Likelihoods, the Shifted Family, and the Averaging 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, in particular $N$, $\tilde{B}$, $\eta$, the Poisson mass functions $\mathrm{poi}_\mu$, and the $\sigma$-finite reference measure $\rho$: the probability space $(\Omega,\mathcal{F},P)$ with the driving variables $U^{c,j}_i$, the $\sigma$-algebras $\mathcal{U}$ and $\mathcal{V}$, the event $\Omega^{U}_0$, the cells $I_{c,j}$ with lengths $\mu_{c,j}=|I_{c,j}|>0$ indexed by the finite set $\mathsf{L}$ with $d$ elements, the cell-count vector $\mathsf{K}$, the deterministic-count clocks $\mathsf{P}^{(y)}$ and the copy clocks $\mathsf{P}^{\sharp}$ (with $\mathsf{P}^{\sharp}(\omega)=\mathsf{P}^{(\mathsf{K}(\omega))}(\omega)$), the likelihoods $\ell^{(y),\omega}$ and $\ell^{\sharp,\omega}=\ell^{(\mathsf{K}(\omega)),\omega}$, the observation record space $(\mathbf{R},\mathcal{R},\rho)$ with $\mathbf{R}^{(k)}$ the records with $k$ events, the parameter lattice $\mathsf{S}=\{y/\sqrt{N}:y\in\mathbb{N}_0^{\mathsf{L}}\}$, the count mass function $\mathsf{p}$, the record kernel $f$, and the standard basis vectors $e_{c,j}$ of $\mathbb{R}^d$; $\mathbb{E}$ is the expectation on $(\Omega,\mathcal{F},P)$, extended to $[0,\infty]$-valued measurable maps, $\mathbf{1}\{\cdot\}$ the indicator of an event, and $\mathbb{N}_0=\{0\}\cup\mathbb{N}$ with $\mathbb{N}$ the natural numbers. Fix a natural number $\mathsf{m}\ge1$ (the move size) and weights $w=(w_{c,j})_{(c,j)\in\mathsf{L}}\in\mathbb{R}^d$, and consider the $d$ moves $a_{c,j}=\mathsf{m}e_{c,j}/\sqrt{N}$ ($(c,j)\in\mathsf{L}$), indexed by $\mathsf{L}$ through its fixed bijection with $\{1,\dots,d\}$; by claim 5 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record, $x+a_{c,j}\in\mathsf{S}$ for all $x\in\mathsf{S}$, and Gaussian Smoothing of a Kernel-Weighted Discrete Family with Countable Support: Joint Density, van Trees Regularity, and the Symmetrised Directional Score Bound applies to $(d,\eta,\mathsf{p},f)$ on $(\mathbf{R},\mathcal{R},\rho)$ with $n=d$ moves $a_1,\dots,a_d$ and weights $w_1,\dots,w_d$, the move $a_q$ being $a_{c,j}$ and the weight $w_q$ being $w_{c,j}$ for $q$ the image of $(c,j)$ under the fixed bijection; let $\mathsf{p}_{c,j}$, $f_{c,j}$ be its shifted family and $\mathsf{J}^{\mathrm{sym}}\in[0,\infty]$ its symmetrised kernel-weighted move information, and let $\Phi_w:[0,\infty)^{1+d}\to[0,\infty)$ be the symmetrised score functional for the weights $w$ (its last $d$ arguments indexed by $\mathsf{L}$).

For $(c,j)\in\mathsf{L}$ let $\varrho_{c,j}=\varrho^{\mathsf{m}}_{\mu_{c,j}}:\mathbb{N}_0\to[0,\infty)$ be the removal ratio with parameter $\mu_{c,j}$ and move size $\mathsf{m}$, so that $\varrho_{c,j}(k)=\mathrm{poi}_{\mu_{c,j}}(k-\mathsf{m})/\mathrm{poi}_{\mu_{c,j}}(k)$ for $k\ge\mathsf{m}$ and $\varrho_{c,j}(k)=0$ for $k<\mathsf{m}$ (the symbol $\varrho$ is distinct from the reference measure $\rho$, and the cell lengths $\mu_{c,j}$ from the copy measure $\mu^{\sharp}$). The removed-clock likelihood for the cell $(c,j)$ is the map $(r,\omega)\mapsto\ell^{-(c,j),\omega}(r)$ on $\mathbf{R}\times\Omega$ given by $\ell^{-(c,j),\omega}=\ell^{(\mathsf{K}(\omega)-\mathsf{m}e_{c,j}),\omega}\ \text{ if }\mathsf{K}_{c,j}(\omega)\ge\mathsf{m},\qquad \ell^{-(c,j),\omega}=0\ \text{ otherwise}.$

1. (Removed clocks) Let $(c,j)\in\mathsf{L}$ and $\omega\in\Omega$ with $\mathsf{K}_{c,j}(\omega)\ge\mathsf{m}$, and put $y=\mathsf{K}(\omega)-\mathsf{m}e_{c,j}\in\mathbb{N}_0^{\mathsf{L}}$. Then $\mathsf{P}^{\sharp,c'}_u(\omega)=\mathsf{P}^{(y),c'}_u(\omega)$ for every label $c'\neq c$ and every $u\ge0$, and $\mathsf{P}^{\sharp,c}_u(\omega)=\mathsf{P}^{(y),c}_u(\omega)+\mathbf{1}_{\Omega^{U}_0}(\omega)\sum_{i=\mathsf{K}_{c,j}(\omega)-\mathsf{m}+1}^{\mathsf{K}_{c,j}(\omega)}\mathbf{1}\{U^{c,j}_i(\omega)\le u\}\qquad(u\ge0):$ the copy clocks are the clocks $\mathsf{P}^{(y)}(\omega)$ with the $\mathsf{m}$ points $U^{c,j}_i(\omega)$, $\mathsf{K}_{c,j}(\omega)-\mathsf{m}<i\le\mathsf{K}_{c,j}(\omega)$, inserted into the cell $I_{c,j}$ of the clock $c$; on $\Omega^{U}_0$ these points lie in $I_{c,j}$, are pairwise distinct, and differ from every point $U^{c,j'}_{i'}(\omega)$ with $1\le j'\le J_c$ and $1\le i'\le y_{c,j'}$. The likelihood $\ell^{-(c,j),\omega}=\ell^{(y),\omega}$ is the record-driven likelihood 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 for the clock family $\mathsf{P}^{(y)}(\omega)$.

2. (Measurability and moments) For every $(c,j)\in\mathsf{L}$: the map $(r,\omega)\mapsto\ell^{-(c,j),\omega}(r)$ is measurable with respect to the product $\sigma$-algebra $\mathcal{R}\otimes\mathcal{F}$ and takes values in $[0,(N\tilde{B})^{k}]$ on $\mathbf{R}^{(k)}\times\Omega$ (with $(N\tilde{B})^{0}=1$); $\varrho_{c,j}(\mathsf{K}_{c,j})$ is a $\mathcal{V}$-measurable random variable with $\mathbb{E}\bigl[\varrho_{c,j}(\mathsf{K}_{c,j})\bigr]=1,\qquad \mathbb{E}\bigl[\varrho_{c,j}(\mathsf{K}_{c,j})^{2}\bigr]=\sum_{i=0}^{\mathsf{m}}\binom{\mathsf{m}}{i}^{2}\,i!\,\mu_{c,j}^{-i};$ and the pathwise score map $\Psi(r,\omega)=\Phi_w\Bigl(\ell^{\sharp,\omega}(r),\ \bigl(\varrho_{c,j}(\mathsf{K}_{c,j}(\omega))\,\ell^{-(c,j),\omega}(r)\bigr)_{(c,j)\in\mathsf{L}}\Bigr)\qquad((r,\omega)\in\mathbf{R}\times\Omega)$ is $\mathcal{R}\otimes\mathcal{F}$-measurable with values in $[0,\infty)$; consequently $r\mapsto\mathbb{E}[\Psi(r,\cdot)]$ is $\mathcal{R}$-measurable and $\omega\mapsto\int_{\mathbf{R}}\Psi(r,\omega)\,\rho(dr)$ is $\mathcal{F}$-measurable, both with values in $[0,\infty]$ (by the Tonelli theorem, $\rho$ being $\sigma$-finite and $P$ finite).

3. (The shifted family on the copy) For every $x=y/\sqrt{N}\in\mathsf{S}$, every $r\in\mathbf{R}$ and every $(c,j)\in\mathsf{L}$, $\mathsf{p}(x)f(x,r)=\mathbb{E}\bigl[\mathbf{1}\{\mathsf{K}=y\}\,\ell^{\sharp,\cdot}(r)\bigr],\qquad \mathsf{p}_{c,j}(x)f_{c,j}(x,r)=\mathbb{E}\bigl[\mathbf{1}\{\mathsf{K}=y\}\,\varrho_{c,j}(\mathsf{K}_{c,j})\,\ell^{-(c,j),\cdot}(r)\bigr].$

4. (Pathwise reduction) In $[0,\infty]$, $\mathsf{J}^{\mathrm{sym}}\ \le\ \mathbb{E}\Bigl[\int_{\mathbf{R}}\Psi(r,\cdot)\,\rho(dr)\Bigr]=\int_{\mathbf{R}}\mathbb{E}\bigl[\Psi(r,\cdot)\bigr]\,\rho(dr).$