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

Adopt the setting and notation of Pathwise Reduction of the Symmetrised Move Information on the Synthetic Copy: Removed-Clock Likelihoods, the Shifted Family, and the Averaging Bound (and hence of 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 driving variables $U^{c,j}_i$ on $(\Omega,\mathcal{F},P)$, the event $\Omega^{U}_0$, the cells $I_{c,j}$ of lengths $\mu_{c,j}$ indexed by $\mathsf{L}$ ($d$ elements), the cell-count vector $\mathsf{K}$, the copy clocks $\mathsf{P}^{\sharp}$ with $\mathsf{P}^{\sharp}(\omega)=\mathsf{P}^{(\mathsf{K}(\omega))}(\omega)$, the regularised paths $\bar\Sigma^{\sharp,r}(\omega)$ and $\bar\Sigma^{(y),r}(\omega)$ with their left limits, the intensities $\lambda^{\sharp,\omega}=(\lambda^{\sharp,\omega,\upsilon})_\upsilon$ and $\lambda^{(y),\omega}$, the conflict-free sets $\mathsf{G}^{\sharp}$ and $\mathsf{G}^{(y)}$, the likelihoods $\ell^{\sharp,\omega}$ and $\ell^{-(c,j),\omega}$, the move size $\mathsf{m}$, the removal ratios $\varrho_{c,j}$, the observation record space $(\mathbf{R},\mathcal{R},\rho)$ with horizon $T$ and channel set $V=\{1,\dots,\tilde{l}\}$, and the remaining data $N$, $l$, $m$, $B$, $\tilde{B}$, $\beta$, $\tilde\beta$, $x_0$, $\mathbb{G}_N$, the real number $R\ge NBT$, and the policy $h$ with values in $\mathcal{A}$ with its record-frozen control paths $a^r$. Causal intensities and their likelihoods $\ell_\lambda$ are as in those definitions, and conflict-free data are those of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution. Let $\mathcal{L}$ be the set of transition labels, $|\cdot|$ the Euclidean norm on $\mathbb{R}^l$, and $\Delta^l$ the probability simplex. Assume in addition:

(OC) there is a real number $\underline{b}>0$ with $\tilde{b}^\upsilon(\Sigma)\ge\underline{b}$ for all $\Sigma\in\Delta^l$ and all $\upsilon\in V$, where $\tilde{b}$ is the aggregate observation drift of $\tilde\beta$;

(X) $(\tilde{U},\bar{\tilde\beta})$ is a twice continuously differentiable extension of $\tilde\beta$ with derivative bound $\tilde{K}$, and $(U,W_\beta,\bar\beta)$ is a twice continuously differentiable extension of $\beta$ with derivative bound $K$ (its second component is the set called $V$ in that definition, renamed here to avoid the channel set $V$); put $\Lambda_1=\sqrt{l+m}\,(B+K)$ as in Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect and $\Gamma=\sqrt{l}\,(\tilde{B}+\tilde{K})$, the Lipschitz constant of each $\tilde{b}^\upsilon$ on $\Delta^l$ given by claims (i) and (ii) of Regularity and Derivative Bounds of the Extended Aggregate Observation Drift (with $\bar{\tilde{b}}$ the extended aggregate observation drift of $(\tilde{U},\bar{\tilde\beta})$, which agrees with $\tilde{b}$ on $\Delta^l$);

(W) $L\ge0$ and $D\ge0$ are real numbers with $\Lambda_1TA_0<L$, where $A_0=\sqrt{2}\,\bigl(\mathsf{m}+l(l-1)(D+\mathsf{m})\bigr)\exp\bigl(\sqrt{2}\,l(l-1)\Lambda_1T\bigr),$ the constant of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect for the move size $\mathsf{m}$ and the discrepancy tolerance $D+\mathsf{m}$.

The clock-good event. Let $G_{L,D}$ be the set of $\omega\in\Omega^{U}_0$ such that for every label $c\in\mathcal{L}$ and all real $0\le u\le u'\le R$ with $u'-u\le L$, $\bigl|\mathsf{P}^{\sharp,c}_{u'}(\omega)-\mathsf{P}^{\sharp,c}_{u}(\omega)-(u'-u)\bigr|\le D .$

The tracked records. For $\omega\in\Omega$ let $\mathsf{T}_\omega$ be the set of $r\in\mathbf{R}$ with $(r,\omega)\in\mathsf{G}^{\sharp}$ and $(r,\omega)\in\mathsf{G}^{(\mathsf{K}(\omega)-\mathsf{m}e_{c,j})}$ for every $(c,j)\in\mathsf{L}$ with $\mathsf{K}_{c,j}(\omega)\ge\mathsf{m}$.

The effective removed intensities. For $\omega\in\Omega$ and $(c,j)\in\mathsf{L}$ put $\lambda^{-(c,j),\omega}=\lambda^{(\mathsf{K}(\omega)-\mathsf{m}e_{c,j}),\omega}$ if $\mathsf{K}_{c,j}(\omega)\ge\mathsf{m}$ and $\lambda^{-(c,j),\omega}=\lambda^{\sharp,\omega}$ otherwise; and let $E^{\omega}_{(c,j)(c',j')}$, $C^{\omega}_{(c,j)(c',j')}$ and $\mathrm{Var}^{\omega}_{(c,j)}$ be the pair exponents, pair covariances and ratio variances of Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form for the base intensity $\lambda^{\sharp,\omega}$ and the perturbed intensities $(\lambda^{-(c,j),\omega})_{(c,j)\in\mathsf{L}}$ ($n=d$, indexed by $\mathsf{L}$).

1. (Admissibility) For every $\omega$: $\mathsf{T}_\omega\in\mathcal{R}$ and $G_{L,D}\in\mathcal{F}$; $\lambda^{\sharp,\omega}$ is a causal intensity with bound $N\tilde{B}$ and $\lambda^{\sharp,\omega,\upsilon}\ge N\underline{b}$ everywhere; each $\lambda^{-(c,j),\omega}$ is a causal intensity with bound $N\tilde{B}$; and $\varrho_{c,j}(\mathsf{K}_{c,j}(\omega))\,\ell^{-(c,j),\omega}=\varrho_{c,j}(\mathsf{K}_{c,j}(\omega))\,\ell_{\lambda^{-(c,j),\omega}}$ on $\mathbf{R}$. Hence Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form applies to $\lambda^{\sharp,\omega}$ and $(\lambda^{-(c,j),\omega})_{(c,j)}$ with any weights and with the ratio factors $\varrho_{c,j}(\mathsf{K}_{c,j}(\omega))$, and Good-Bad Splitting of the Integrated Symmetrised Score Functional: Chebyshev Bound for the Under-Likelihood Set, Transfer of Mass Between Densities, and the Split Bound applies on $(\mathbf{R},\mathcal{R},\rho)$ to $\ell=\ell^{\sharp,\omega}$, $\ell_{(c,j)}=\ell_{\lambda^{-(c,j),\omega}}$ and the same ratio factors, its integrand $\Phi_w(\ell,(\mathsf{r}_q\ell_q)_q)$ being the pathwise score map $\Psi(\cdot,\omega)$ of Pathwise Reduction of the Symmetrised Move Information on the Synthetic Copy: Removed-Clock Likelihoods, the Shifted Family, and the Averaging Bound.

2. (Response bound on the clock-good event) Let $\omega\in G_{L,D}$, $(c,j)\in\mathsf{L}$ with $\mathsf{K}_{c,j}(\omega)\ge\mathsf{m}$, and $r\in\mathsf{T}_\omega$; put $y=\mathsf{K}(\omega)-\mathsf{m}e_{c,j}$. Then for every $t\in[0,T]$ $\bigl|\bar\Sigma^{\sharp,r}_t(\omega)-\bar\Sigma^{(y),r}_t(\omega)\bigr|\le\frac{A_0}{N},\qquad \bigl|\bar\Sigma^{\sharp,r}_{t-}(\omega)-\bar\Sigma^{(y),r}_{t-}(\omega)\bigr|\le\frac{A_0}{N},$ and consequently $|\lambda^{-(c,j),\omega,\upsilon}_t(r)-\lambda^{\sharp,\omega,\upsilon}_t(r)|\le\Gamma A_0$ for every $\upsilon\in V$ and $t\in[0,T]$. (For $(c,j)$ with $\mathsf{K}_{c,j}(\omega)<\mathsf{m}$ the two intensities coincide.)

3. (Uniform pair-exponent bound) Put $\bar{E}_N=\frac{\tilde{l}\,T\,\Gamma^{2}A_0^{2}}{N\,\underline{b}} .$ For every $\omega\in G_{L,D}$, all $(c,j),(c',j')\in\mathsf{L}$ and every $r\in\mathsf{T}_\omega$, $|E^{\omega}_{(c,j)(c',j')}(r)|\le\bar{E}_N$.

4. (Bounds on the pair covariances) Let $\omega\in G_{L,D}$ be such that $\rho(\mathbf{R}\setminus\mathsf{T}_\omega)=0$. Then for all $(c,j),(c',j')\in\mathsf{L}$, $|C^{\omega}_{(c,j)(c',j')}|\le\exp(\bar{E}_N)-1,\qquad 0\le\mathrm{Var}^{\omega}_{(c,j)}\le\exp(\bar{E}_N)-1,\qquad \Bigl|C^{\omega}_{(c,j)(c',j')}-\int_{\mathbf{R}}\ell_{\tilde\mu^{\omega}_{(c,j)(c',j')}}E^{\omega}_{(c,j)(c',j')}\,d\rho\Bigr|\le\tfrac12\bar{E}_N^{2}\exp(\bar{E}_N),$ where $\tilde\mu^{\omega}_{(c,j)(c',j')}$ is the pair intensity of Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form.