Proof 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

Throughout, $\omega$ ranges over $\Omega$, $q,q'$ over $\mathsf{L}$, and we abbreviate $\varrho_q=\varrho_q(\mathsf{K}_q)$ (a random variable by claim 2 of Pathwise Reduction of the Symmetrised Move Information on the Synthetic Copy: Removed-Clock Likelihoods, the Shifted Family, and the Averaging Bound, $\mathcal{V}$-measurable, with $\mathbb{E}\varrho_q=1$ and $\mathbb{E}\varrho_q^{2}=1+\mathsf{j}_q$), $\ell=\ell^{\sharp,\omega}$, $\ell_q=\ell_{\lambda^{-q,\omega}}$ and $L_q=L^{\omega}_q$ when $\omega$ is fixed. By claim 1 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, for every $\omega$ the hypotheses of 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 hold on $(\mathbf{R},\mathcal{R},\rho)$ for $\ell$, $(\ell_q)_q$, the ratio factors $\mathsf{r}_q=\varrho_q(\mathsf{K}_q(\omega))$ and $\delta$, with integrand $\Psi(\cdot,\omega)$ and with $\pi_q=\pi^{\omega}_q$, $\pi=\pi^{\omega}$ and ratio variances $V_q=\mathrm{Var}^{\omega}_q=C^{\omega}_{qq}$; those of Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form hold as well, giving the identity for $Q^{\omega}$ in the statement. Cauchy-Schwarz for random variables is claim 1 of Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm; for nonnegative $X,Y$ with $\mathbb{E}X^{2},\mathbb{E}Y^{2}<\infty$ it reads $\mathbb{E}[XY]\le(\mathbb{E}X^{2})^{1/2}(\mathbb{E}Y^{2})^{1/2}$.

Step 1 (claim 1). Independence. By the preamble of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record and its claim 1, the $\sigma$-algebras $\mathcal{G}_c$ generated by $K^{c}$ and $(V^{c}_i)_i$ are independent over the labels $c$ (Grouping Lemma for Independent Random Variables), each $\mathsf{K}_{c,j}=\sum_{i\le\widetilde{K}^{c}}\mathbf{1}\{V^{c}_i\in I_{c,j}\}$ is $\mathcal{G}_c$-measurable (it is the pointwise limit in $\mathbb{R}$ of the partial sums $\sum_{i\le n}\mathbf{1}\{i\le\widetilde{K}^{c}\}\mathbf{1}\{V^{c}_i\in I_{c,j}\}$, which are $\mathcal{G}_c$-measurable, so claim 5 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions applies), and for each $c$ the counts $\mathsf{K}_{c,1},\dots,\mathsf{K}_{c,J_c}$ are independent (claim 2 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). Hence for Borel sets $B_{c,j}$, $P(\bigcap_{c,j}\{\mathsf{K}_{c,j}\in B_{c,j}\})=\prod_cP(\bigcap_j\{\mathsf{K}_{c,j}\in B_{c,j}\})=\prod_c\prod_jP(\mathsf{K}_{c,j}\in B_{c,j})$, which is the independence of $(\mathsf{K}_q)_q$ in the sense of Independence of Events and of Random Variables (subfamilies are handled by taking $B_{c,j}=\mathbb{R}$). Each $\varrho_q(\mathsf{K}_q)$ is $\sigma(\mathsf{K}_q)$-measurable (claim 2 of Pathwise Reduction of the Symmetrised Move Information on the Synthetic Copy: Removed-Clock Likelihoods, the Shifted Family, and the Averaging Bound shows it is a countable sum of multiples of indicators $\mathbf{1}\{\mathsf{K}_q=k\}$), so the family $(\varrho_q(\mathsf{K}_q))_q$ is independent by the final assertion of Grouping Lemma for Independent Random Variables (index the finite family $(\mathsf{K}_q)_q$ by $\{1,\dots,d\}$, with singleton groups).

The square. Expanding, $(\sum_qw_q(1-\varrho_q))^{2}=\sum_{q,q'}w_qw_{q'}(1-\varrho_q)(1-\varrho_{q'})$, every term having finite expectation (second moments finite). For $q\neq q'$, $1-\varrho_q$ and $1-\varrho_{q'}$ are independent with finite expectations, so $\mathbb{E}[(1-\varrho_q)(1-\varrho_{q'})]=\mathbb{E}[1-\varrho_q]\,\mathbb{E}[1-\varrho_{q'}]=0$ by Expectation of a Product of Independent Random Variables; for $q=q'$, $\mathbb{E}[(1-\varrho_q)^{2}]=1-2+(1+\mathsf{j}_q)=\mathsf{j}_q$. Linearity of the expectation gives claim 1.

Step 2 (measurability). The map $(r,\omega)\mapsto\ell^{\sharp,\omega}(r)$ is $\mathcal{R}\otimes\mathcal{F}$-measurable with $\int\ell^{\sharp,\omega}d\rho=1$ (claim 3 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record); the maps $(r,\omega)\mapsto\varrho_q(\mathsf{K}_q(\omega))\ell^{-q,\omega}(r)=\varrho_q(\mathsf{K}_q(\omega))\ell_{\lambda^{-q,\omega}}(r)$ (claim 1 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 $\Psi(r,\omega)$ are $\mathcal{R}\otimes\mathcal{F}$-measurable and nonnegative (claim 2 of Pathwise Reduction of the Symmetrised Move Information on the Synthetic Copy: Removed-Clock Likelihoods, the Shifted Family, and the Averaging Bound). Moreover $(r,\omega)\mapsto\ell_{\lambda^{-q,\omega}}(r)$ itself is $\mathcal{R}\otimes\mathcal{F}$-measurable: it equals $\ell^{-q,\omega}(r)$ on $\{\mathsf{K}_q\ge\mathsf{m}\}$ (by the definition of the removed-clock likelihood in Pathwise Reduction of the Symmetrised Move Information on the Synthetic Copy: Removed-Clock Likelihoods, the Shifted Family, and the Averaging Bound and of the effective removed intensity) and $\ell^{\sharp,\omega}(r)$ on $\{\mathsf{K}_q<\mathsf{m}\}$, both measurable. Since $\ell^{\sharp,\omega}>0$ (claim 1 of Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form, applicable for every $\omega$ by claim 1 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), $1/\ell$ is jointly measurable by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable with $E=(0,\infty)$ and $g(x)=1/x$, and $(r,\omega)\mapsto\ell(1-L_q)(1-L_{q'})=\ell-\ell_q-\ell_{q'}+\ell_q\ell_{q'}/\ell$ is jointly measurable (claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions); splitting it into positive and negative parts, each of which is integrable in $r$ for every $\omega$ (claim 1 of Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form), the Tonelli theorem on $\mathbf{R}\times\Omega$ ($\rho$ $\sigma$-finite, $P$ finite) shows that $\omega\mapsto C^{\omega}_{qq'}$, the difference of the two section integrals, is $\mathcal{F}$-measurable and finite. Likewise $Q^{\omega}$ is finite for every $\omega$ by claim 2 of Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form. Since $\ell>0$, the set $\{(r,\omega):\varrho_q\ell_q(r)<(1-\delta)\ell(r)\}=\{\varrho_qL_q<1-\delta\}$ is $\mathcal{R}\otimes\mathcal{F}$-measurable (difference of measurable maps, claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions), so $\ell\mathbf{1}\{\varrho_qL_q<1-\delta\}$ is jointly measurable and $\omega\mapsto\pi^{\omega}_q$ is $\mathcal{F}$-measurable by the Tonelli theorem on $\mathbf{R}\times\Omega$ ($\rho$ $\sigma$-finite, $P$ finite); likewise $\omega\mapsto\int\Psi(r,\omega)\rho(dr)$ (claim 2 of Pathwise Reduction of the Symmetrised Move Information on the Synthetic Copy: Removed-Clock Likelihoods, the Shifted Family, and the Averaging Bound) and $\omega\mapsto Q^{\omega}$, whose integrand $\ell(\sum_qw_q(1-\varrho_qL_q))^{2}=(\sum_qw_q(\ell-\varrho_q\ell_q))^{2}/\ell$ is jointly measurable, nonnegative, and $[0,\infty]$-valued. Also $\pi^{\omega}_q\le\int\ell\,d\rho=1$, so $\pi^{\omega}\le d$.

Step 3 (claim 2: under-likelihood mass on $G$). Fix $\omega\in G$ and $q$. If $\mathsf{K}_q(\omega)<\mu_q-x_q$ the bound is trivial ($\pi^{\omega}_q\le1$). Otherwise $\mathsf{K}_q(\omega)\ge\mu_q-x_q$ and $\mathsf{m}(x_q+\mathsf{m})\le\delta\mu_q/2\le\mu_q$, so claim 4 of The Poisson Removal Ratio for Moves of Several Points: Move Score, Mean, Exact Second Moment, Move Information, and Pointwise Bounds gives $\varrho_q\ge1-\mathsf{m}(x_q+\mathsf{m})/\mu_q\ge1-\delta/2>0$; in particular $\mathsf{K}_q(\omega)\ge\mathsf{m}$ (as $\varrho_q(k)=0$ for $k<\mathsf{m}$ by the definition of the removal ratio in that lemma), so $\lambda^{-q,\omega}=\lambda^{(\mathsf{K}(\omega)-\mathsf{m}e_q),\omega}$. On $\{L_q\ge1-\delta/2\}$ we get $\varrho_qL_q\ge(1-\delta/2)^{2}\ge1-\delta$, hence $\{\varrho_qL_q<1-\delta\}\subseteq\{L_q<1-\delta/2\}$ and $\pi^{\omega}_q\le\int\ell\mathbf{1}\{L_q<1-\delta/2\}\,d\rho$.

The clipped intensity. Define $\mu''=(\mu''^\upsilon)_\upsilon$ by $\mu''^\upsilon_s(r)=\min\bigl(\max(\lambda^{-q,\omega,\upsilon}_s(r),(1-\varepsilon_0)\lambda^{\sharp,\omega,\upsilon}_s(r)),(1+\varepsilon_0)\lambda^{\sharp,\omega,\upsilon}_s(r)\bigr)$. It is a causal intensity with bound $(1+\varepsilon_0)N\tilde{B}$: measurability in $(s,r)$ by claim 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and non-anticipation (condition (ii) of that definition, with respect to the strict prefix maps there) because both $\lambda^{-q,\omega}$ and $\lambda^{\sharp,\omega}$ are non-anticipating and $\mu''_s(r)$ is a function of their values at $(s,r)$ only. By construction $|\mu''-\lambda^{\sharp,\omega}|\le\varepsilon_0\lambda^{\sharp,\omega}$ everywhere, and by claim 2 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, for $r\in\mathsf{T}_\omega$ one has $|\lambda^{-q,\omega,\upsilon}_t(r)-\lambda^{\sharp,\omega,\upsilon}_t(r)|\le\Gamma A_0=\varepsilon_0N\underline{b}\le\varepsilon_0\lambda^{\sharp,\omega,\upsilon}_t(r)$ for all $t,\upsilon$, so that $\mu''_t(r)=\lambda^{-q,\omega}_t(r)$ for all $t$ and $\upsilon$ when $r\in\mathsf{T}_\omega$; since the likelihood at $r$ depends only on the intensity values at $r$, $\ell_{\mu''}(r)=\ell_q(r)$ for $r\in\mathsf{T}_\omega$. Apply Chernoff Bound for the Under-Likelihood Set of a Relatively Perturbed Causal Intensity: Elementary Exponential Inequalities, the Tilted Power-Product Exponent, and the Markov Step with base $\lambda^{\sharp,\omega}$ ($\bar\mu=N\tilde{B}$, $\underline\mu=N\underline{b}$, by claim 1 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), perturbation $\mu''$, $\varepsilon=\varepsilon_0\le\tfrac12$, the integer $\theta$ and $\delta'=\delta/2$: its claim 3 gives $\int\ell\mathbf{1}\{\ell_{\mu''}/\ell<1-\delta/2\}\,d\rho\le\exp(-\theta\delta/2+\mathsf{E}^{\mathrm{ch}}_\theta)$ with $\mathsf{E}^{\mathrm{ch}}_\theta=8\tilde{l}TN\tilde{B}\theta^{2}\varepsilon_0^{2}$. The two indicators $\mathbf{1}\{L_q<1-\delta/2\}$ and $\mathbf{1}\{\ell_{\mu''}/\ell<1-\delta/2\}$ agree on $\mathsf{T}_\omega$, whose complement is $\rho$-null by (G); the integrals of $\ell$ times each indicator therefore agree (the integral of a nonnegative measurable function $h$ over the null set $\mathsf{Z}_\omega=\mathbf{R}\setminus\mathsf{T}_\omega$ vanishes: $\int\min(h,k)\mathbf{1}_{\mathsf{Z}_\omega}\,d\rho\le k\rho(\mathsf{Z}_\omega)=0$ for every natural number $k$ by Simple Function and Its Integral and monotonicity, and Monotone Convergence Theorem applies as $k\to\infty$). This proves the pointwise bound of claim 2. Summing over $q$, multiplying by $\mathbf{1}_G$ and taking expectations: $\mathbb{E}[\mathbf{1}\{\mathsf{K}_q<\mu_q-x_q\}]=P(\mathsf{K}_q<\mu_q-x_q)\le P(\mathsf{K}_q\le\mu_q-x_q)\le\exp(-\varpi_{\mu_q}(x_q))$ by claim 3 of Series Formula, Exponential Moments, and Chernoff Tail Bounds for the Poisson Distribution, $\mathsf{K}_q$ being Poisson with parameter $\mu_q$ (claim 2 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); this gives $\mathbb{E}[\mathbf{1}_G\pi^{\omega}]\le\bar\Pi$ by monotonicity and linearity (Linearity and Monotonicity of the Lebesgue Integral).

Step 4 (claim 3). On $G$. Fix $\omega\in G$. Claim 4 of 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 gives $\int_{\mathbf{R}}\Psi(r,\omega)\rho(dr)\le(1+\delta)Q^{\omega}+2d\lVert w\rVert_1^{2}\Bigl[\pi^{\omega}+\sum_q\varrho_q\bigl(\pi^{\omega}+(V_q)^{1/2}(\pi^{\omega})^{1/2}\bigr)\Bigr],$ and claim 4 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 (applicable by (G)) gives $|C^{\omega}_{qq'}|\le\mathsf{c}_N$ and $V_q=\mathrm{Var}^{\omega}_q\le\mathsf{c}_N$. By the identity for $Q^{\omega}$, $Q^{\omega}$ is finite on $G$ and $Q^{\omega}-(\sum_qw_q(1-\varrho_q))^{2}=\sum_{q,q'}w_qw_{q'}\varrho_q\varrho_{q'}C^{\omega}_{qq'}$, whose absolute value is at most $\mathsf{c}_N\sum_{q,q'}|w_qw_{q'}|\varrho_q\varrho_{q'}$, a random variable with finite expectation (Cauchy-Schwarz: $\mathbb{E}[\varrho_q\varrho_{q'}]\le(1+\mathsf{j}_q)^{1/2}(1+\mathsf{j}_{q'})^{1/2}$). Hence $\mathcal{C}$ is a well-defined real number (the integrand $\mathbf{1}_G(Q^{\omega}-(\dots)^{2})$ is measurable by Step 2, dominated by an integrable function), and it equals the stated sum by linearity.

Off $G$. For every $\omega$, claim 2(b) of Weighted Cauchy-Schwarz Inequality on a Measure Space and the Symmetrised Score Functional: Bounds and Averaging gives $\Psi(r,\omega)\le2d\lVert w\rVert_1^{2}(\ell(r)+\sum_q\varrho_q\ell_q(r))$, and integrating ($\int\ell\,d\rho=\int\ell_q\,d\rho=1$) yields $\int\Psi(r,\omega)\rho(dr)\le2d\lVert w\rVert_1^{2}(1+\sum_q\varrho_q)$.

Expectations. By claim 4 of Pathwise Reduction of the Symmetrised Move Information on the Synthetic Copy: Removed-Clock Likelihoods, the Shifted Family, and the Averaging Bound, $\mathsf{J}^{\mathrm{sym}}\le\mathbb{E}[\int\Psi\,d\rho]=\mathbb{E}[\mathbf{1}_G\int\Psi\,d\rho]+\mathbb{E}[\mathbf{1}_{G^{c}}\int\Psi\,d\rho]$ (additivity for nonnegative measurable maps). For the first term, by the bound on $G$ and monotonicity, $\mathbb{E}\Bigl[\mathbf{1}_G\int\Psi\,d\rho\Bigr]\le(1+\delta)\,\mathbb{E}[\mathbf{1}_GQ^{\omega}]+2d\lVert w\rVert_1^{2}\Bigl(\mathbb{E}[\mathbf{1}_G\pi^{\omega}]+\sum_q\mathbb{E}[\mathbf{1}_G\varrho_q\pi^{\omega}]+\sum_q\mathsf{c}_N^{1/2}\,\mathbb{E}[\mathbf{1}_G\varrho_q(\pi^{\omega})^{1/2}]\Bigr).$ Here $\mathbb{E}[\mathbf{1}_GQ^{\omega}]=\mathbb{E}[\mathbf{1}_G(\sum_qw_q(1-\varrho_q))^{2}]+\mathcal{C}\le\sum_qw_q^{2}\mathsf{j}_q+\mathcal{C}$ by Step 1 (dropping $\mathbf{1}_G$ from a nonnegative term). Next, $\mathbb{E}[\mathbf{1}_G\pi^{\omega}]\le\bar\Pi$ by claim 2; $\mathbb{E}[\mathbf{1}_G\varrho_q\pi^{\omega}]\le(\mathbb{E}\varrho_q^{2})^{1/2}(\mathbb{E}[\mathbf{1}_G(\pi^{\omega})^{2}])^{1/2}\le(1+\mathsf{j}_q)^{1/2}(d\,\mathbb{E}[\mathbf{1}_G\pi^{\omega}])^{1/2}\le(1+\mathsf{j}_q)^{1/2}(d\bar\Pi)^{1/2}$ by Cauchy-Schwarz and $(\pi^{\omega})^{2}\le d\,\pi^{\omega}$ (Step 2); and $\mathbb{E}[\mathbf{1}_G\varrho_q(\pi^{\omega})^{1/2}]\le(1+\mathsf{j}_q)^{1/2}(\mathbb{E}[\mathbf{1}_G\pi^{\omega}])^{1/2}\le(1+\mathsf{j}_q)^{1/2}\bar\Pi^{1/2}$ likewise. For the second term, by the bound off $G$, $\mathbb{E}[\mathbf{1}_{G^{c}}\int\Psi\,d\rho]\le2d\lVert w\rVert_1^{2}(\mathsf{g}+\sum_q\mathbb{E}[\mathbf{1}_{G^{c}}\varrho_q])\le2d\lVert w\rVert_1^{2}(\mathsf{g}+\mathsf{g}^{1/2}\sum_q(1+\mathsf{j}_q)^{1/2})$, using Cauchy-Schwarz $\mathbb{E}[\mathbf{1}_{G^{c}}\varrho_q]\le P(G^{c})^{1/2}(\mathbb{E}\varrho_q^{2})^{1/2}$. Collecting the terms gives the assembly bound with the stated $\mathsf{B}$ (the square roots of products being products of square roots, Existence and Uniqueness of the Nonnegative Square Root). $\blacksquare$