Mean-Square Assembly of the Estimand Linearisation on the Synthetic Copy: Approximation of the Recentred Copy Endpoint by an Affine Function of the Parameter

Adopt the setting, hypotheses (OC), (X), (W), (G), (G$'$), (CL) and notation of Information Bound on the Synthetic Copy with Mean-Field Data: Prior Energy of the Profile, Observation Information Along the Profile Response, and the Non-Close Record Mass (and hence of Assembly of the Symmetrised Move Information on the Synthetic Copy: Prior Part, Record Part, and the Bad Part with Exponentially Small Under-Likelihood Mass, Closeness to a Mean-Field Pair on the Synthetic Copy: Clock Discrepancy, Gronwall Comparison of the Weighted Response with the Profile Response, and the Observation-Information Bound with Mean-Field Data, 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, 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 $(\Omega,\mathcal{F},P)$ with expectation $\mathbb{E}$; the natural numbers $N\ge1$, $l\ge2$, $m\ge1$, $\tilde{l}\ge1$; the real numbers $B\ge0$, $\tilde{B}\ge0$, $K\ge0$, $\tilde{K}\ge0$, $\underline{b}>0$, $T>0$ and $D\ge0$; the control set $\mathcal{A}\subseteq\mathbb{R}^m$; the transition-rate family $\beta$ on $l$ states with rate bound $B$, its twice continuously differentiable extension $(U,W_\beta,\bar\beta)$ with derivative bound $K$ of (X), the set $\mathcal{L}$ of transition labels $c=(\sigma,\gamma)$ with vectors $v_c=\delta_\gamma-\delta_\sigma$, the label rates $\psi_c$, state gradients $g^{c}$ and drift Jacobian $\mathcal{E}$, and the constants $\Lambda_1=\sqrt{l+m}\,(B+K)$, $\Lambda_2=\tfrac32(l+m)K$, $\Lambda_3=3K\sqrt{l(l+m)}$ and $\Lambda_{\mathcal{E}}=\sqrt{2}\,l(l-1)\sqrt{l}\,(B+K)$; the probability simplex $\Delta^l$, the aggregate lattice $\mathbb{G}_N\subseteq\Delta^l$ and the point $x_0\in\mathbb{G}_N$; the observation-driven control policy $h$ with values in $\mathcal{A}$ and its record-frozen control paths $a^{r}$; the observation record space $(\mathbf{R},\mathcal{R},\rho)$ with horizon $T$ and $\tilde{l}$ channels; the clock horizon $R\ge NBT$, the cell boundaries $0=\mathsf{b}^{c}_0<\mathsf{b}^{c}_1<\dots<\mathsf{b}^{c}_{J_c}=R$ (written $b^{c}_j$ in The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record and $\mathsf{b}^{c}_j$ here, to keep them apart from the drift $b$), the cells $I_{c,j}$ with lengths $\mu_{c,j}$, indexed by the finite set $\mathsf{L}$ of pairs $q=(c,j)$ with $d$ elements and identified with $\{1,\dots,d\}$ by the fixed bijection, so that $\mathbb{R}^{d}$ has coordinates indexed by $\mathsf{L}$; the maximal cell length $\mu_{\max}$; the cell-count vector $\mathsf{K}=(\mathsf{K}_q)_{q\in\mathsf{L}}$; the event $\Omega^{U}_0$; the real number $L\ge0$ of (W) and the clock-good event $G_{L,D}\subseteq\Omega^{U}_0$; the move size $\mathsf{m}$ and the event $G^{\mathsf{m}}$; the copy clocks $\mathsf{P}^{\sharp}=(\mathsf{P}^{\sharp,c})_{c\in\mathcal{L}}$, all of whose paths are counting paths; the regularised paths $\bar\Sigma^{\sharp,r}(\omega)$ with consumed clocks $\mathsf{C}^{\sharp,c,r}_t(\omega)$ and the conflict-free set $\mathsf{G}^{\sharp}$; the tracked records $\mathsf{T}_\omega$; the likelihoods $\ell^{\sharp,\omega}$; the smoothing parameter $\eta\in(0,1]$ with the Gaussian smoothing weight $\varphi_\eta$ on $\mathbb{R}^d$; the synthetic copy $(\Omega^{\sharp},\mathcal{F}^{\sharp},\mu^{\sharp})$ with $\Omega^{\sharp}=\Omega\times\mathbb{R}^d\times\mathbf{R}$, its density $\mathsf{q}^{\sharp}(\omega,\theta,r)=\varphi_\eta(\theta-\mathsf{K}(\omega)/\sqrt{N})\,\ell^{\sharp,\omega}(r)$ with respect to $(P\otimes\lambda_d)\otimes\rho$, and its coordinate maps, the parameter $\Theta(\omega,\theta,r)=\theta$ and the record $\mathsf{D}(\omega,\theta,r)=r$; the event $G$ with $\mathsf{g}=P(\Omega\setminus G)$; the comparison pair $(S,\mathsf{A})$ with the mean-field label rates $\phi_c(t)=\psi_c(S_t,\mathsf{A}_t)$ and mean-field clocks $\bar{\mathsf{C}}^{c}_t=N\int_{[0,t]}\phi_c(u)\,du$; the real numbers $\varepsilon_S\ge0$ and $\varepsilon_{\mathrm{ctl}}\ge0$, the close records $\mathsf{R}^{\mathrm{cl}}_\omega\subseteq\mathsf{T}_\omega$, the non-close masses $\pi^{\mathrm{nc}}_\omega$ and $\bar\pi^{\mathrm{nc}}=\mathbb{E}[\mathbf{1}_G\pi^{\mathrm{nc}}]$ of (CL); and the control discrepancy $\mathsf{D}^{r}_{\mathrm{ctl}}$.

Notational reservations. In the present lemma $\Theta$ denotes the parameter coordinate of the copy, never the aggregate fluctuation covariance (which is not used here, nor are the objects $\mathcal{P}$, $\bar\psi$, $\mathsf{M}$, $\mathsf{e}_F$, $\epsilon_\psi$, $\kappa$ and $\mathsf{Q}^{\mathrm{cl}}$ of the adopted setting defined through it); hypothesis (P) of Assembly of the Symmetrised Move Information on the Synthetic Copy: Prior Part, Record Part, and the Bad Part with Exponentially Small Under-Likelihood Mass is not assumed, and the objects $\delta$, $\theta$, $x_q$, $\bar\Pi$ and $\mathsf{B}$ defined through it are not used, so that $\theta$ below is free to denote the second coordinate of a point of $\Omega^{\sharp}$; $\mu=(\mu_q)_{q\in\mathsf{L}}\in\mathbb{R}^d$ denotes the vector of cell lengths, distinct from the copy measure $\mu^{\sharp}$; $\rho$ is the reference measure of the record space, and the residual written $\rho_u$ in Linearisation of a Perturbed Controlled Aggregate Flow Along a Comparison Pair: Exact Variation-of-Constants Identity and Residual Bound is unrelated to it; $R$ is the clock horizon; $K$ is the derivative bound of (X), unrelated to the recursion index written $K$ in Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution; $D$ is the discrepancy tolerance of (W) and $\mathsf{D}$ the record coordinate; the bold letter $\mathbf{c}$ below is an element of $\mathbb{R}^l$ unrelated to the labels $c$; the letter $G$ denotes the event of (G); the set of times $\mathsf{Q}_T$ below is unrelated to the tracked records $\mathsf{T}_\omega$; the constant map $\varsigma$ below is unrelated to the real number $\zeta$ of the adopted setting; the injection weights $w=(w_q)_{q\in\mathsf{L}}$, their norm $\lVert w\rVert_1$ and the profile $\varpi$ of the adopted setting are not used here, and $\mathsf{w}^{\mathrm{clk}}$ below is a clock tolerance (written $w_1$ in Pathwise Estimand Linearisation of an Open-Loop Aggregate Solution Along a Comparison Pair: Cell-Count Coefficients from the Fundamental Solution and the Three Error Terms); $\alpha$ with a subscript in $\mathsf{L}$, and the vector $\alpha\in\mathbb{R}^d$, are cell coefficients, never control values (control values are written $\alpha'$ where needed); and the recursion times written $\theta_k$ in 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 are written $\vartheta_k$ where they occur. Write $|\cdot|$ for the Euclidean norm, $x\cdot y$ for the dot product, $t^{1/2}=\sqrt{t}$ for the nonnegative square root of a real $t\ge0$ and $t^{1/4}=(t^{1/2})^{1/2}$, $\mathbb{Q}$ for the rational numbers, $\sup$ for the least upper bound, $\int_{[0,t]}\cdot\,du$ ($0\le t\le T$) for the Lebesgue integral over the compact interval $[0,t]$, and $\otimes$ for the product $\sigma$-algebra; a real-valued map on a measurable space is measurable when it is measurable with respect to the named $\sigma$-algebra and the Borel $\sigma$-algebra of the real line. Since $\mu^{\sharp}$ is a probability measure (claim 4 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record), random variables on $(\Omega^{\sharp},\mathcal{F}^{\sharp},\mu^{\sharp})$ have the notions of square-integrability and mean-square norm $\lVert\cdot\rVert_2$ of that definition; write $\int_{\Omega^{\sharp}}\cdot\,d\mu^{\sharp}$ for the integral of a nonnegative measurable function on the copy. A function of $(\omega,r)$ or of $\omega$ alone is regarded as a function on $\Omega^{\sharp}$ through the coordinate maps, and measurability with respect to $\mathcal{R}\otimes\mathcal{F}$ on $\mathbf{R}\times\Omega$ is transported to $\mathcal{F}\otimes\mathcal{R}$ on $\Omega\times\mathbf{R}$ by the exchange of coordinates.

Assume in addition:

(AF) (affine data) $(\beta_0,\beta_1)$ is an affine-controlled transition-rate family on $l$ states with control set $\mathcal{A}$, which is nonempty, convex and compact (this strengthens the adopted hypothesis on $\mathcal{A}$), and with Lipschitz constant $\Lambda^{\mathrm{aff}}$ (written $\Lambda$ in The Transition-Rate Family and Aggregate Drift of Affine-Controlled Data, Stability of the Mean-Field Flow under Perturbation of the Initial State and Weak Convergence of Controls and The Record-Frozen Control as a Measurable Map into the Weakly Metrized Control Set, and the Record-Frozen Mean-Field Flow: Joint Measurability, Flow Equation, and Measurability in the Record, a letter reserved here for the profile bound of the adopted setting); $\beta$ is the transition-rate family of claim 2 of The Transition-Rate Family and Aggregate Drift of Affine-Controlled Data formed from $(\beta_0,\beta_1)$, and the rate bound $B$ of the adopted setting is equal to the number $B=\sup\beta(\sigma,\gamma,\Sigma,\alpha')$ (supremum over all labels $(\sigma,\gamma)$, all $\Sigma\in\Delta^l$ and all $\alpha'\in\mathcal{A}$) defined in that lemma (so that $B$ is this supremum and not merely an upper bound, and the constants of Stability of the Mean-Field Flow under Perturbation of the Initial State and Weak Convergence of Controls are formed with this same $B$); the control bound $\sup_{\alpha'\in\mathcal{A}}|\alpha'|$, written $R$ in those three lemmas, is written $R^{\mathcal{A}}$ here and is not used, the letter $R$ being the clock horizon; and $h$ is $\mathcal{A}$-valued. Fix a point $z_0\in\Delta^l$ (the base point of the flow, in general different from $x_0$) and a sequence $(\mathsf{v}_n)_{n\in\mathbb{N}}$ in $L^{2}([0,T];\mathbb{R}^{m})$ whose set of terms is dense there (written $(w_n)$ in The Record-Frozen Control as a Measurable Map into the Weakly Metrized Control Set, and the Record-Frozen Mean-Field Flow: Joint Measurability, Flow Equation, and Measurability in the Record; its weak metric, written $\rho$ there, is not used here). Write $\mathsf{S}(x,\xi)$, with values $\mathsf{S}_t(x,\xi)$, for the mean-field flow of claim 2 of Stability of the Mean-Field Flow under Perturbation of the Initial State and Weak Convergence of Controls (written $S(x,\xi)$ there; the letter $S$ without arguments is reserved for the comparison path). With The Record-Frozen Control as a Measurable Map into the Weakly Metrized Control Set, and the Record-Frozen Mean-Field Flow: Joint Measurability, Flow Equation, and Measurability in the Record applied as stated in claim 1, let

be its record-frozen flow (claim 4 of that lemma).

(CP) (comparison pair with continuous Jacobian) every entry of $u\mapsto\mathcal{E}^{\star}_u=\mathcal{E}(S_u,\mathsf{A}_u)$ is continuous on $[0,T]$; let $\Phi^{\mathcal{E}}(t,u)$ ($t,u\in[0,T]$) be the two-parameter fundamental solution of $u\mapsto\mathcal{E}^{\star}_u$ and let $\bar\Phi\ge0$ be a real number with $|\Phi^{\mathcal{E}}(t,u)y|\le\bar\Phi^{2}|y|$ for all $t,u\in[0,T]$ and $y\in\mathbb{R}^l$, as in Linearisation of a Perturbed Controlled Aggregate Flow Along a Comparison Pair: Exact Variation-of-Constants Identity and Residual Bound. With Pathwise Estimand Linearisation of an Open-Loop Aggregate Solution Along a Comparison Pair: Cell-Count Coefficients from the Fundamental Solution and the Three Error Terms applied as stated in claim 1, adopt from that lemma, for an estimand direction $\mathbf{c}\in\mathbb{R}^l$ fixed from now on, the maps $H^{c}_u=\Phi^{\mathcal{E}}(T,u)\,\mathcal{E}^{\star}_u\,v_c$ with $\lVert H^{c}\rVert_1=\int_{[0,T]}|H^{c}_u|\,du$, the entry times $\bar\tau_q$ and the cell coefficients $\alpha_q$ ($q\in\mathsf{L}$; $\alpha_q=\mathbf{c}\cdot(\Phi^{\mathcal{E}}(T,\bar\tau_q)v_c)$ when $\bar{\mathsf{C}}^{c}_T\ge \mathsf{b}^{c}_j$ for $q=(c,j)$, and $\alpha_q=0$ otherwise), collected in the vector $\alpha=(\alpha_q)_{q\in\mathsf{L}}\in\mathbb{R}^d$, and the window discrepancies $\mathrm{Disc}_w(p)$ of Cell-Count Form of a Compensated Counting-Path Functional Along a Time Change: Window-Discrepancy Bounds for the Time-Change and Partial-Cell Errors, formed with the clock horizon $R$, of a counting path $p$. Put

The deviation and the recentred endpoint. For $\omega\in\Omega$ and $r\in\mathbf{R}$ put

(FM) (fourth moment of the deviation) $\mathsf{c}_4\ge0$ is a real number with $\int_{\Omega^{\sharp}}\bar{\mathsf{e}}^{4}\,d\mu^{\sharp}\le\mathsf{c}_4$ (the integrand being measurable by claim 2).

(DM) (discrepancy majorants) for every label $c\in\mathcal{L}$, $\Xi^{c}:\Omega\to[0,\infty)$ is an $\mathcal{F}$-measurable map with $\mathbb{E}[(\Xi^{c})^{2}]<\infty$ such that, for every $\omega\in G$,

where $\mathsf{P}^{\sharp,c}(\omega)$ is the counting path $u\mapsto\mathsf{P}^{\sharp,c}_u(\omega)$.

Finally put $\mathsf{k}_4=\bigl(\mathbb{E}\bigl[(\alpha\cdot(\mathsf{K}-\mu))^{4}\bigr]\bigr)^{1/4}$ (finite by claim 2) and let $\varsigma:\mathbf{R}\to\mathbb{R}$ be the constant map $\varsigma(r)=-\alpha\cdot\mu/\sqrt{N}$.

1. (Instantiation.) The affine data of (AF), the horizon $T$, the channel number $\tilde{l}$, the policy $h$, the point $z_0$ in the role of $x_0$ and the sequence $(\mathsf{v}_n)$ satisfy the hypotheses of The Record-Frozen Control as a Measurable Map into the Weakly Metrized Control Set, and the Record-Frozen Mean-Field Flow: Joint Measurability, Flow Equation, and Measurability in the Record; hence, for every $r\in\mathbf{R}$, the record-frozen flow $\Phi^{r}$ is a measurable map from $[0,T]$ to $\Delta^l$ with continuous components, $|\Phi^{r}_t-\Phi^{r}_u|\le K_b|t-u|$ for all $t,u\in[0,T]$ with $K_b=2\sqrt{l}\,(l-1)B$, and $\Phi^{r}_t=z_0+\int_{[0,t]}b(\Phi^{r}_u,a^{r}_u)\,du$ for every $t\in[0,T]$, where $b$ is the aggregate state drift of $\beta$; the map $(t,r)\mapsto\Phi^{r,\gamma}_t$ is measurable with respect to $\mathcal{B}_{[0,T]}\otimes\mathcal{R}$ for every coordinate $\gamma$, $\mathcal{B}_{[0,T]}$ being the trace Borel $\sigma$-algebra of $[0,T]$. Moreover the numbers $l$, $m$, $N$, $B$, $K$, $T$ and $R$, the control set $\mathcal{A}$, the transition-rate family $\beta$ with its extension $(U,W_\beta,\bar\beta)$ (its second component $W_\beta$ being the set written $V$ in Linearisation of a Perturbed Controlled Aggregate Flow Along a Comparison Pair: Exact Variation-of-Constants Identity and Residual Bound, the letter $V$ being the channel set of the adopted setting), the lattice $\mathbb{G}_N$, the comparison pair $(S,\mathsf{A})$ with the fundamental solution $\Phi^{\mathcal{E}}$ and its bound $\bar\Phi$ of (CP), the estimand direction $\mathbf{c}$, the cell boundaries $\mathsf{b}^{c}_j$ and the mean-field label rates $\phi_c$ are data of the kind required by the setting of Pathwise Estimand Linearisation of an Open-Loop Aggregate Solution Along a Comparison Pair: Cell-Count Coefficients from the Fundamental Solution and the Three Error Terms (the drift $b$ of that lemma being the aggregate state drift of $\beta$), and the objects $H^{c}$, $\bar\tau_q$ and $\alpha_q$ of that lemma are defined from these data alone: for every $q=(c,j)$ with $\bar{\mathsf{C}}^{c}_T\ge\mathsf{b}^{c}_j$ the set $\{u\in[0,T]:\bar{\mathsf{C}}^{c}_u\ge\mathsf{b}^{c}_j\}$ has a least element $\bar\tau_q$, so that $\alpha_q$ is defined, and whenever that lemma is applied to an instance with these data (as in claim 3) its cell coefficients are these $\alpha_q$. Moreover, for every $t\in[0,T]$ and every $\gamma$, the map $r\mapsto\Phi^{r,\gamma}_t$ is measurable with respect to $\mathcal{R}$.

2. (Measurability and moments.) The map $(\omega,r)\mapsto\bar{\mathsf{e}}(\omega,r)$ equals $\sup_{t\in\mathsf{Q}_T}\sqrt{N}|\bar\Sigma^{\sharp,r}_t(\omega)-\Phi^{r}_t|$ with the countable set $\mathsf{Q}_T=(\mathbb{Q}\cap[0,T])\cup\{T\}$, is measurable with respect to $\mathcal{F}\otimes\mathcal{R}$ and takes values in $[0,\sqrt{2N}]$; each component of $(\omega,r)\mapsto X''(\omega,r)$ is measurable with respect to $\mathcal{F}\otimes\mathcal{R}$, and $|\mathbf{c}\cdot X''|\le|\mathbf{c}|\,\bar{\mathsf{e}}$ pointwise. Consequently $\bar{\mathsf{e}}$ and $\mathbf{c}\cdot X''$ are bounded random variables on the copy, and for every $\mathcal{F}\otimes\mathcal{R}$-measurable $F:\Omega\times\mathbf{R}\to[0,\infty)$,

the inner integral being an $\mathcal{F}$-measurable function of $\omega$; in particular $\int_{\Omega^{\sharp}}F\,d\mu^{\sharp}=\mathbb{E}[F]$ when $F$ depends on $\omega$ alone. Each $\mathsf{K}_q$ has the Poisson distribution with parameter $\mu_q$, the random variable $(\alpha\cdot(\mathsf{K}-\mu))^{4}$ is integrable, so that $\mathsf{k}_4$ is a finite nonnegative real number, and the coordinates $\Theta_q$ of the parameter are square-integrable on the copy with

3. (Pathwise linearisation on the close records.) Let $\omega\in G$ and $r\in\mathsf{R}^{\mathrm{cl}}_\omega$. Then $\bar\Sigma^{\sharp,r}(\omega)$ is the unique open-loop aggregate solution on $[0,T]$ for the data $(\mathsf{P}^{\sharp}(\omega),a^{r},x_0)$, and its consumed clock times are the $\mathsf{C}^{\sharp,c,r}_t(\omega)$ of the adopted setting; the cell counts of Pathwise Estimand Linearisation of an Open-Loop Aggregate Solution Along a Comparison Pair: Cell-Count Coefficients from the Fundamental Solution and the Three Error Terms formed from the clock family $\mathsf{P}^{\sharp}(\omega)$ and the boundaries $\mathsf{b}^{c}_j$ are the $\mathsf{K}_q(\omega)$; the flow $\Phi^{r}$, which by claim 1 satisfies the flow equation for the control path $a^{r}$, is admissible as the flow $y$ of that lemma, with $y_0=z_0$; the clock-discrepancy hypothesis (CD) of that lemma, $|\mathsf{C}^{\sharp,c,r}_u(\omega)-\bar{\mathsf{C}}^{c}_u|\le \mathsf{w}^{\mathrm{clk}}$ for all $c$ and $u$, holds; and, with $\bar{\mathsf{e}}=\bar{\mathsf{e}}(\omega,r)$,

where

a formula defining $\tilde{Z}$ as an $\mathcal{F}\otimes\mathcal{R}$-measurable function on all of $\Omega\times\mathbf{R}$.

4. (Mean-square assembly.) The random variables $\alpha\cdot\Theta=\sum_{q\in\mathsf{L}}\alpha_q\Theta_q$, $\varsigma(\mathsf{D})=\varsigma\circ\mathsf{D}$ and $\mathbf{c}\cdot X''$ on the copy are square-integrable, $\varsigma$ is measurable with respect to $\mathcal{R}$, and (beyond claim 3, using of (CL) only that each section $\mathsf{R}^{\mathrm{cl}}_\omega$ belongs to $\mathcal{R}$, the defining formula of $\pi^{\mathrm{nc}}_\omega$ with the $\mathcal{F}$-measurability of $\omega\mapsto\pi^{\mathrm{nc}}_\omega$, and $\bar\pi^{\mathrm{nc}}=\mathbb{E}[\mathbf{1}_G\pi^{\mathrm{nc}}]$; the set $\{(\omega,r):r\in\mathsf{R}^{\mathrm{cl}}_\omega\}$ is not assumed $\mathcal{F}\otimes\mathcal{R}$-measurable)

where

Remark (not part of the claims). The five terms are, in order: the initial offset between the lattice start $x_0$ and the base point $z_0$; the quadratic and control-gradient residuals of the linearisation, controlled through (FM), $\varepsilon_S$ and $\varepsilon_{\mathrm{ctl}}$; the cell-count and time-change errors of the compensated clocks, controlled through (DM); the Gaussian smoothing of the parameter; and the contribution of the pairs $(\omega,r)$ with $\omega\notin G$ or $r\notin\mathsf{R}^{\mathrm{cl}}_\omega$, controlled through $\mathsf{g}$ and $\bar\pi^{\mathrm{nc}}$.