Information of the Smoothed Record Family

Data. Let $l\ge2$, $\tilde{l}\ge1$, $m\ge1$ be natural numbers and $T>0$ a real number. Let $\beta$ be a transition-rate family on $l$ states with control dimension $m$ and rate bound $B$, Lipschitz in the state argument with constant $K_\beta$ in the sense of Shared-Clock Coupling of One-Agent-Moved Reconstructions, and let $\tilde{\beta}$ be an observation-rate family with $\tilde{l}$ channels and rate bound $\tilde{B}$, Lipschitz in the state argument with constant $K_{\tilde{\beta}}$ in the same sense, satisfying the positivity bound $\tilde{\beta}(\sigma,\upsilon,\Sigma)\ge\tilde{\beta}_{\min}>0$ for all $\sigma$, $\upsilon$, and all $\Sigma$ in the probability simplex $\Delta^l$. Let $(\tilde{U},\bar{\tilde{\beta}})$ be a twice continuously differentiable extension of $\tilde{\beta}$ with derivative bound $\tilde{K}$, with extended aggregate observation drift $\bar{\tilde{b}}$, whose regularity is given by Regularity and Derivative Bounds of the Extended Aggregate Observation Drift. Real matrices, matrix products, transposes, and identity matrices are used throughout. For $\Sigma\in\tilde{U}$ let $\tilde{\mathcal{E}}(\Sigma)$ be the $\tilde{l}\times l$ matrix with entries $\tilde{\mathcal{E}}(\Sigma)_{\upsilon\delta}=\partial_\delta\bar{\tilde{b}}^\upsilon(\Sigma)$ (partial derivatives), and let $\tilde{\Theta}^\star(\Sigma)$ be the $\tilde{l}\times\tilde{l}$ diagonal matrix with entries $\tilde{\Theta}^\star(\Sigma)_{\upsilon\upsilon'}=\mathbf{1}_{\{\upsilon=\upsilon'\}}\bar{\tilde{b}}^\upsilon(\Sigma)$; along a path $S$ these are the observation matrix and observation noise covariance evaluated at $S_t$. For $\Sigma\in\Delta^l$ one has $\bar{\tilde{b}}^\upsilon(\Sigma)=\tilde{b}^\upsilon(\Sigma)\ge\tilde{\beta}_{\min}$, by part (i) of Regularity and Derivative Bounds of the Extended Aggregate Observation Drift, the positivity bound, and the coordinates of $\Sigma$ summing to $1$, so the inverse $\tilde{\Theta}^\star(\Sigma)^{-1}$ exists. Let $S^*:[0,T]\to\Delta^l$ be continuous with respect to the Euclidean distance.

Setup for a fixed $N\ge1$. Let $h$ be an observation-driven control policy with horizon $T$, control dimension $m$, and $\tilde{l}$ channels (the policy may be chosen separately for each $N$ in claim 4). Let $\mathbb{G}=\{x\in\Delta^l:\ Nx^\gamma\ \text{is an integer for every}\ \gamma\}$, a finite set, fix an anchor $\bar{S}\in\mathbb{G}$, and let $H_0=\{z\in\mathbb{R}^l:\ z^1+\dots+z^l=0\}$ and $\mathbb{L}=\{\sqrt{N}(x-\bar{S}):\ x\in\mathbb{G}\}\subseteq H_0$, with $\sqrt{N}$ the square root. Let the configuration map $c:\mathbb{G}\to\{1,\dots,l\}^N$ assign to $x$ the nondecreasing configuration with exactly $Nx^\gamma$ agents in state $\gamma$. Fix a matrix $\iota\in\mathbb{R}^{l\times(l-1)}$ whose columns span $H_0$ and which satisfies $\iota^\top\iota=I_{l-1}$; such matrices exist, by orthonormalization applied to a basis of $H_0$. Write $\mathcal{B}_{l-1}$ and $\lambda_{l-1}$ for the Borel $\sigma$-algebra and product Lebesgue measure on $\mathbb{R}^{l-1}$.

Let $(\Omega,\mathcal{F},P)$ be a probability space carrying: an $\mathbb{L}$-valued random variable $M$; an $\mathbb{R}^{l-1}$-valued random variable $\xi$; and transition clocks $Y^{i,\sigma\gamma}$ and observation clocks $\tilde{Y}^{i,\upsilon}$ as in conditions 2 and 3 of N-Agent Driving System; such that the following family of $\sigma$-algebras is independent: $\sigma(M)$; $\sigma(\xi)$; and the $\sigma$-algebra of each clock. Let $\eta=\eta_N\in(0,1]$ and assume $\xi$ has the density $\varphi_\eta$ with respect to $\lambda_{l-1}$ (measure with density), where $\varphi_\eta(z)=c_\eta\exp(-(z\cdot z)/(2\eta))$ with $\exp$ the real exponential function, $z\cdot z$ the sum of the squared coordinates, and $c_\eta>0$ the unique real for which $\varphi_\eta$ integrates to $1$. Define the smoothed parameter $\Theta=\iota^\top M+\xi$.

For each $x\in\mathbb{G}$: the space $(\Omega,\mathcal{F},P)$ with the deterministic initial states given by $c(x)$ and the clocks above is an $N$-agent driving system; a solution for $h$ on it exists by Existence, Uniqueness, and Regularity for the Controlled N-Agent Dynamics, with empirical state measure written $\Sigma^{(x)}_u$; fix reconstruction data for it and let $f^{(x)}$ be its record density kernel on the observation record space $(\mathbf{R},\mathcal{R},\rho)$ with horizon $T$. The kernel $f^{(x)}$ is measurable with respect to $\mathcal{R}\otimes\mathcal{T}_c$, where $\mathcal{T}_c$ is the $\sigma$-algebra generated by the transition-clock variables: the $\sigma$-algebra called $\mathcal{T}$ in Conditional Density of the Observation Record Given the Initial States and Transition Clocks is here generated by the deterministic initial states, which generate the trivial $\sigma$-algebra, together with the transition clocks. Likewise $(\Omega,\mathcal{F},P)$ with the random initial states $c(\bar{S}+N^{-1/2}M)$ and the same clocks is an $N$-agent driving system; fix a solution for $h$ on it, with observation record $W$. Define $g:\mathbb{R}^{l-1}\times\mathbf{R}\times\Omega\to[0,\infty)$ by $g(\theta,r,\omega)=\sum_{y\in\mathbb{L}}P(M=y)\,\varphi_\eta\bigl(\theta-\iota^\top y\bigr)\,f^{(\bar{S}+N^{-1/2}y)}(r,\omega),$ and the prior density $q:\mathbb{R}^{l-1}\to(0,\infty)$ by $q(\theta)=\sum_{y\in\mathbb{L}}P(M=y)\varphi_\eta(\theta-\iota^\top y)$.

Moment and propagator hypotheses. There are reals $C_M\ge1$ and $C_*\ge1$ with $\mathbb{E}[(M\cdot M)^2]\le C_M$ and, for every $x\in\mathbb{G}$, $\mathbb{E}\Bigl[\sup_{u\in[0,T]}\Bigl(\sum_{\delta=1}^{l}\bigl|\Sigma^{(x),\delta}_u-S^{*,\delta}_u\bigr|\Bigr)^{4}\Bigr]\le C_*\bigl(1+(N(x-\bar{S})\cdot(x-\bar{S}))^{2}\bigr)N^{-2},$ the suprema being measurable since the paths are right-continuous. Moreover (used in claim 4 only) there are a map $\Phi:[0,T]\to\mathbb{R}^{l\times l}$ with continuous entries and $\Phi_0=I_l$, fixed independently of $N$, and reals $C_\Phi\ge1$ and $\epsilon_\Phi\in(0,1/2]$, such that for every $y\in\mathbb{L}$, writing $x=\bar{S}+N^{-1/2}y$, $\mathbb{E}\Bigl[\sup_{u\in[0,T]}\Bigl(\sum_{\delta=1}^{l}\bigl|\sqrt{N}\bigl(\Sigma^{(x),\delta}_u-\Sigma^{(\bar{S}),\delta}_u\bigr)-(\Phi_u\,y)^\delta\bigr|\Bigr)^{2}\Bigr]\le C_\Phi\bigl(1+(y\cdot y)^{2}\bigr)N^{-2\epsilon_\Phi}.$

Claims.

1. (Conditional joint density) The map $g$ is measurable with respect to $\mathcal{B}_{l-1}\otimes\mathcal{R}\otimes\mathcal{T}^P_c$, where $\mathcal{T}^P_c$ is $\mathcal{T}_c$ enlarged by the events of probability zero, and for every $\mathcal{T}_c$-measurable $Z:\Omega\to[0,\infty]$, every $\mathcal{B}_{l-1}$-measurable $\phi:\mathbb{R}^{l-1}\to[0,\infty]$, and every $\mathcal{R}$-measurable $g_0:\mathbf{R}\to[0,\infty]$, $\mathbb{E}\bigl[Z\,\phi(\Theta)\,g_0(W)\bigr]=\mathbb{E}\Bigl[Z\int_{\mathbb{R}^{l-1}}\int_{\mathbf{R}}\phi(\theta)\,g_0(r)\,g(\theta,r,\cdot)\,\rho(dr)\,\lambda_{l-1}(d\theta)\Bigr]\qquad\text{in }[0,\infty].$

2. (van Trees hypotheses) For $P$-almost every $\omega$: $g(\theta,r,\omega)>0$ for every $(\theta,r)$; each section $\theta\mapsto g(\theta,r,\omega)$ is a $C^1$ map on $\mathbb{R}^{l-1}$; and the density $(\theta,r)\mapsto g(\theta,r,\omega)$ together with any pair of random elements whose joint law is the measure with density $g(\cdot,\cdot,\omega)$ with respect to $\lambda_{l-1}\otimes\rho$ satisfies hypotheses (i)--(iv) of The Multivariate van Trees Inequality with $l$ there replaced by $l-1$ and with data space $(\mathbf{R},\mathcal{R},\rho)$. Write $J(\omega)$ for the resulting van Trees information matrix, $I(\theta,\omega)$ for the $(l-1)\times(l-1)$ matrix with entries $I_{ab}(\theta,\omega)=\int_{\mathbf{R}}(\partial_a p)(\partial_b p)/p\,(\theta,r,\omega)\,\rho(dr)$, where $p(\theta,r,\omega)=g(\theta,r,\omega)/q(\theta)$ and $\partial_a$ is the partial derivative in the coordinate $\theta_a$, and $J(q)$ for the $(l-1)\times(l-1)$ matrix with entries $\int_{\mathbb{R}^{l-1}}(\partial_a q)(\partial_b q)/q\,d\lambda_{l-1}$, which is finite and positive definite.

3. (Information decomposition) For $P$-almost every $\omega$, every entry of $I(\cdot,\omega)$ is measurable in $\theta$ and $J(\omega)=J(q)+\int_{\mathbb{R}^{l-1}}I(\theta,\omega)\,q(\theta)\,d\lambda_{l-1}(\theta),$ all entries being finite, the matrix integral taken entrywise.

4. (Information limit) Let data as above be given for every $N$ in an unbounded set $\mathcal{N}$ of natural numbers, with $(\beta,\tilde{\beta},\tilde{U},\bar{\tilde{\beta}},T,S^*,\Phi)$ and the constants $B,\tilde{B},K_\beta,K_{\tilde{\beta}},\tilde{K},\tilde{\beta}_{\min},C_M,C_*,C_\Phi,\epsilon_\Phi$ fixed, and with $\eta_N\to0$ and $N\eta_N\to\infty$ as $N\to\infty$ in $\mathcal{N}$. Then the entrywise expectation $\bar{I}_N=\mathbb{E}\bigl[\int_{\mathbb{R}^{l-1}}I(\theta,\cdot)\,q(\theta)\,d\lambda_{l-1}(\theta)\bigr]$ (the inner integral being the one of claim 3, defined for $P$-almost every $\omega$) is finite for every $N\in\mathcal{N}$, and there are reals $C\ge0$ and $N_0$, depending only on the fixed data and constants, such that for every $N\in\mathcal{N}$ with $N\ge N_0$ every entry of $\bar{I}_N-\iota^\top\Bigl(\int_{[0,T]}\bigl(\tilde{\mathcal{E}}(S^*_u)\,\Phi_u\bigr)^\top\,\tilde{\Theta}^\star(S^*_u)^{-1}\,\bigl(\tilde{\mathcal{E}}(S^*_u)\,\Phi_u\bigr)\,du\Bigr)\iota$ has absolute value at most $C\bigl(N^{-1/2}\eta_N^{-1/2}+\eta_N^{1/2}+N^{-1}\eta_N^{-1}+N^{-\epsilon_\Phi}\bigr)$, the matrix integral taken entrywise, its integrands being bounded and continuous by Regularity and Derivative Bounds of the Extended Aggregate Observation Drift and the continuity of $S^*$ and $\Phi$.