Weighted Response and Pair-Exponent Quadratic Form on the Synthetic Copy: the Linearised Injection Equation, Comparison with the Profile Injection, and the Observation-Information Bound

Adopt the setting, hypotheses (OC), (X), (W) and notation of Uniform Pair-Exponent Bound on the Synthetic Copy: Removed-Clock Intensities, the Insertion Response on the Clock-Good Event, and Bounds on the Pair Covariances (and hence of Pathwise Reduction of the Symmetrised Move Information on the Synthetic Copy: Removed-Clock Likelihoods, the Shifted Family, and the Averaging Bound, The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record and 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): the probability space $(\Omega,\mathcal{F},P)$, the driving variables $U^{c,j}_i$ and the event $\Omega^{U}_0$, the set $\mathcal{L}$ of transition labels $c=(\sigma,\gamma)$ (with $l(l-1)$ elements), the standard basis vectors $\delta_1,\dots,\delta_l$ of $\mathbb{R}^l$ and the label vectors $v_c=\delta_\gamma-\delta_\sigma$ of Open-Loop Aggregate Solution Driven by Aggregate Transition Clocks, the probability simplex $\Delta^l$, the observation record space $(\mathbf{R},\mathcal{R},\rho)$ with horizon $T$ and $\tilde{l}$ channels, the cells $I_{c,j}=(b^{c}_{j-1},b^{c}_j]$ ($1\le j\le J_c$) of lengths $\mu_{c,j}=\mu_q$ indexed by $q=(c,j)\in\mathsf{L}$ (with $d$ elements), the set $\mathbb{N}_0^{\mathsf{L}}$ of count vectors, the cell-count vector $\mathsf{K}$ with coordinates $\mathsf{K}_q$, the standard basis vectors $e_q=e_{c,j}$ of $\mathbb{R}^{d}$ (coordinates indexed by $\mathsf{L}$), the deterministic-count clocks $\mathsf{P}^{(y)}$ and the copy clocks $\mathsf{P}^{\sharp}$, the regularised paths $\bar\Sigma^{(y),r}(\omega)$ and $\bar\Sigma^{\sharp,r}(\omega)$ with their left limits $\bar\Sigma^{(y),r}_{t-}(\omega)$ and $\bar\Sigma^{\sharp,r}_{t-}(\omega)$, the conflict-free sets $\mathsf{G}^{(y)}$ and $\mathsf{G}^{\sharp}$, the intensities $\lambda^{\sharp,\omega}$ and the effective removed intensities $\lambda^{-q,\omega}$ ($q\in\mathsf{L}$), the pair exponents $E^{\omega}_{qq'}$, the move size $\mathsf{m}$, the data $N$, $l$, $m$, $\tilde{l}$, $B$, $\tilde{B}$, $\tilde{K}$, $K$, $\underline{b}$, $\beta$, $\tilde\beta$, $x_0$, $\mathbb{G}_N$, $T$, $R\ge NBT$, the extensions $(U,W_\beta,\bar\beta)$ and $(\tilde{U},\bar{\tilde\beta})$, the extended aggregate observation drift $\bar{\tilde{b}}$ (which agrees with $\tilde{b}$ on $\Delta^l$), the policy $h$ with record-frozen control paths $a^{r}$, the constants $\Lambda_1$, $\Gamma$, $A_0$ (formed with the discrepancy tolerance $D+\mathsf{m}$), the clock-good event $G_{L,D}$ and the tracked records $\mathsf{T}_\omega$. Adopt also from Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect the label rates $\psi_c$, state gradients $g^{c}$, drift Jacobian $\mathcal{E}$ and the constant $\Lambda_2$, and note that the constant $A_0^{+}$ of Removal Form of the Shared-Clock Insertion Response: Discrepancy on the Enlarged Clock Family, the Base-Clock Insertion Count, and the Linearisation Defect Along Either Path formed with the present $D$ coincides with $A_0$. Adopt from Twice Continuously Differentiable Extension of an Observation-Rate Family the partial derivatives $\partial_\gamma$ on $\tilde{U}$. Write $|\cdot|$ for the Euclidean norm (and the absolute value of real numbers), $x\cdot y$ for the dot product, $\#S$ for the number of elements of a finite set $S$, $\mathbf{1}\{\cdot\}$ for the indicator of a condition, equal to $1$ when it holds and $0$ otherwise, $\int_{[0,t]}\cdot\,ds$ for the Lebesgue integral over the compact interval $[0,t]$ (componentwise for vector maps, and equal to $0$ for $t=0$), and measurable for real functions on $[0,T]$ means measurable with respect to the trace Borel $\sigma$-algebra. Put $\mu_{\max}=\max_{q}\mu_q$ and $\mu_{\min}=\min_q\mu_q$.

Weights. Let $\phi_c:[0,T]\to[0,B]$ ($c\in\mathcal{L}$) be measurable, let $\varpi:[0,T]\to\mathbb{R}^l$, $t\mapsto\varpi_t$, have measurable components with $|\varpi_t|\le\Lambda$ for a real $\Lambda\ge0$ (the profile; it plays the role of the map called $\lambda$ in Injection Weights from a Bounded Profile Along Mean-Field Clocks: Label-Rate Form of the Fluctuation Covariance, Weight Bounds, the Prior Quadratic Form, and the Step-Function Injection, a letter reserved here for intensities), and let $\bar{\mathsf{C}}^{c}$, the time cells, the injection weights $w=(w_q)_{q\in\mathsf{L}}$ and the profile injection $\bar{F}_t$ be those of Injection Weights from a Bounded Profile Along Mean-Field Clocks: Label-Rate Form of the Fluctuation Covariance, Weight Bounds, the Prior Quadratic Form, and the Step-Function Injection formed from $\phi_c$, $\varpi$, $\Lambda$, the present cells, $N$ and $\mathsf{m}$; write $\lVert w\rVert_1=\sum_q|w_q|$.

The response data. Let $G^{\mathsf{m}}$ be the set of $\omega\in G_{L,D}$ with $\mathsf{K}_q(\omega)\ge\mathsf{m}$ for every $q\in\mathsf{L}$. For $\omega\in G^{\mathsf{m}}$, $r\in\mathsf{T}_\omega$, $q=(c,j)\in\mathsf{L}$ and $t\in[0,T]$ put $y^{(q)}=\mathsf{K}(\omega)-\mathsf{m}e_q\in\mathbb{N}_0^{\mathsf{L}}$ and define the response and its left-limit version

$$Y^{(q),r}_t(\omega)=N\bigl(\bar\Sigma^{\sharp,r}_t(\omega)-\bar\Sigma^{(y^{(q)}),r}_t(\omega)\bigr),\qquad Y^{(q),r}_{t-}(\omega)=N\bigl(\bar\Sigma^{\sharp,r}_{t-}(\omega)-\bar\Sigma^{(y^{(q)}),r}_{t-}(\omega)\bigr),$$

the consumed clocks $\mathsf{C}^{\sharp,c',r}_t(\omega)$ and $\mathsf{C}^{(q),c',r}_t(\omega)$ ($c'\in\mathcal{L}$) of the open-loop aggregate solutions $\bar\Sigma^{\sharp,r}(\omega)$ and $\bar\Sigma^{(y^{(q)}),r}(\omega)$ for the data $(\mathsf{P}^{\sharp}(\omega),a^{r},x_0)$ and $(\mathsf{P}^{(y^{(q)})}(\omega),a^{r},x_0)$ (which are the open-loop aggregate solutions by claim 2(c) 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, both data being conflict-free for $r\in\mathsf{T}_\omega$), the base-clock count

$$\iota^{(q),r}_t(\omega)=\#\bigl\{i\in\mathbb{N}:\ \mathsf{K}_q(\omega)-\mathsf{m}<i\le\mathsf{K}_q(\omega),\ U^{q}_i(\omega)\le\mathsf{C}^{(q),c,r}_t(\omega)\bigr\},$$

the weighted response, the realized injection and the weighted defect

$$\hat\psi^{r}_t(\omega)=\frac{1}{N}\sum_{q\in\mathsf{L}}w_q\,Y^{(q),r}_t(\omega),\qquad \hat\psi^{r}_{t-}(\omega)=\frac{1}{N}\sum_{q\in\mathsf{L}}w_q\,Y^{(q),r}_{t-}(\omega),\qquad F^{r}_t(\omega)=\frac{1}{N}\sum_{q=(c,j)\in\mathsf{L}}w_q\,v_c\,\iota^{(q),r}_t(\omega),$$

and $\hat{\mathsf{d}}^{r}_t(\omega)=\hat\psi^{r}_t(\omega)-F^{r}_t(\omega)-\int_{[0,t]}\mathcal{E}\bigl(\bar\Sigma^{\sharp,r}_s(\omega),a^{r}_s\bigr)\hat\psi^{r}_s(\omega)\,ds$ (the integrand being bounded with measurable components by claim 1, so that the integral exists). Finally, for $x\in\Delta^l$ and $\upsilon\in\{1,\dots,\tilde{l}\}$ let $g_\upsilon(x)=(\partial_1\bar{\tilde{b}}^\upsilon(x),\dots,\partial_l\bar{\tilde{b}}^\upsilon(x))\in\mathbb{R}^l$ be the observation gradient and let $\tilde{D}(x)$ be the observation information matrix, the real matrix with $l$ rows and $l$ columns and entries

$$\tilde{D}^{\gamma\delta}(x)=\sum_{\upsilon=1}^{\tilde{l}}\frac{\partial_\gamma\bar{\tilde{b}}^\upsilon(x)\,\partial_\delta\bar{\tilde{b}}^\upsilon(x)}{\tilde{b}^\upsilon(x)}\qquad(\gamma,\delta\in\{1,\dots,l\}),$$

so that $z\cdot\bigl(\tilde{D}(x)z\bigr)=\sum_\upsilon(g_\upsilon(x)\cdot z)^{2}/\tilde{b}^\upsilon(x)$ for $z\in\mathbb{R}^l$, with the matrix-vector product. The superscript $r$ and the argument $\omega$ are dropped when they are fixed.

1. (Weighted response equation.) $G^{\mathsf{m}}\in\mathcal{F}$. Let $\omega\in G^{\mathsf{m}}$ and $r\in\mathsf{T}_\omega$. For every $q=(c,j)\in\mathsf{L}$, the clock family $\mathsf{P}^{\sharp}(\omega)$ is the perturbed clock family of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect obtained from $\mathsf{P}^{(y^{(q)})}(\omega)$ by inserting the $\mathsf{m}$ points $U^{q}_i(\omega)$, $\mathsf{K}_q(\omega)-\mathsf{m}<i\le\mathsf{K}_q(\omega)$, into the clock $c$, and Removal Form of the Shared-Clock Insertion Response: Discrepancy on the Enlarged Clock Family, the Base-Clock Insertion Count, and the Linearisation Defect Along Either Path applies to the data $(\mathsf{P}^{(y^{(q)})}(\omega),a^{r},x_0)$, $(\mathsf{P}^{\sharp}(\omega),a^{r},x_0)$, $R$, $L$, $D$: for every $t\in[0,T]$ and every label $c'$,

$$|Y^{(q)}_t|\le A_0,\qquad |Y^{(q)}_{t-}|\le A_0,\qquad |\mathsf{C}^{\sharp,c'}_t-\mathsf{C}^{(q),c'}_t|\le\Lambda_1TA_0,\qquad Y^{(q)}_t=v_c\,\iota^{(q)}_t+\int_{[0,t]}\mathcal{E}\bigl(\bar\Sigma^{\sharp}_s,a^{r}_s\bigr)Y^{(q)}_s\,ds+\mathsf{d}^{(q)}_t$$

with $|\mathsf{d}^{(q)}_t|\le\sqrt{2}\,l(l-1)\bigl(D+\Lambda_2TA_0^{2}/N\bigr)$. Consequently $s\mapsto\hat\psi_s$, $s\mapsto F_s$ and $s\mapsto\mathcal{E}(\bar\Sigma^{\sharp}_s,a^{r}_s)\hat\psi_s$ are bounded on $[0,T]$ with measurable components, $|\hat\psi_t|\le A_0\lVert w\rVert_1/N$, and

$$\hat\psi_t=F_t+\int_{[0,t]}\mathcal{E}\bigl(\bar\Sigma^{\sharp}_s,a^{r}_s\bigr)\hat\psi_s\,ds+\hat{\mathsf{d}}_t,\qquad |\hat{\mathsf{d}}_t|\le\frac{\sqrt{2}\,l(l-1)\,\lVert w\rVert_1}{N}\Bigl(D+\frac{\Lambda_2TA_0^{2}}{N}\Bigr)\qquad(t\in[0,T]).$$

2. (Realized versus profile injection.) Let $\omega\in G^{\mathsf{m}}$, $r\in\mathsf{T}_\omega$ and $t\in[0,T]$. Then

$$|F_t-\bar{F}_t|\le\frac{2\Lambda}{N}\sum_{c\in\mathcal{L}}\bigl(|\mathsf{C}^{\sharp,c}_t-\bar{\mathsf{C}}^{c}_t|+3\mu_{\max}\bigr)+\frac{4\Lambda\,l(l-1)\,\mu_{\max}\,(\Lambda_1TA_0+\mu_{\max})}{N\,\mu_{\min}} .$$

3. (Pair-exponent quadratic form and the observation-information bound.) For every $\omega\in\Omega$ and $r\in\mathbf{R}$ the pair-exponent quadratic form

$$\mathcal{Q}^{\omega}(r)=\sum_{q\in\mathsf{L}}\sum_{q'\in\mathsf{L}}w_qw_{q'}\,E^{\omega}_{qq'}(r)=\int_{[0,T]}\sum_{\upsilon=1}^{\tilde{l}}\frac{\Bigl(\sum_{q\in\mathsf{L}}w_q\bigl(\lambda^{-q,\omega,\upsilon}_t(r)-\lambda^{\sharp,\omega,\upsilon}_t(r)\bigr)\Bigr)^{2}}{\lambda^{\sharp,\omega,\upsilon}_t(r)}\,dt$$

is a well-defined nonnegative real number. If $\omega\in G^{\mathsf{m}}$ and $r\in\mathsf{T}_\omega$, then the functions $t\mapsto\hat\psi_{t-}\cdot\bigl(\tilde{D}(\bar\Sigma^{\sharp}_{t-})\hat\psi_{t-}\bigr)$ and $t\mapsto\hat\psi_t\cdot\bigl(\tilde{D}(\bar\Sigma^{\sharp}_t)\hat\psi_t\bigr)$ are bounded and measurable on $[0,T]$, differ at only finitely many $t$, and for every real $\zeta>0$

$$\mathcal{Q}^{\omega}(r)\ \le\ (1+\zeta)\,N\int_{[0,T]}\hat\psi_t\cdot\bigl(\tilde{D}(\bar\Sigma^{\sharp}_t)\hat\psi_t\bigr)\,dt\ +\ \Bigl(1+\frac{1}{\zeta}\Bigr)\frac{9\,\tilde{l}\,l^{2}\tilde{K}^{2}\,T\,\lVert w\rVert_1^{2}A_0^{4}}{4\,N^{3}\,\underline{b}} .$$