Proof of The Fluctuation LQG Value Is the Asymptotically Optimal Value of the Recentred N-Agent Cost over Admissible Families of Observation-Driven Policies, and Is Attained by the Approximate Kalman Policies, under Deterministic Initial States

Throughout, the certificate lemma means Injection Certificates on the Trimmed Synthetic Copy for the Family of N-Agent Solutions: the Law-Transported Van Trees Certificate Hypothesis Holds under Deterministic Initial States, Uniform Observation Positivity and a C^2 Observation Extension, the LQG lower bound theorem means Asymptotic Lower Bound for the Recentred N-Agent Cost by the Fluctuation LQG Value, under Law-Transported Injection Certificates, the policy lemma means The Approximate Kalman Filter and Policy for the Controlled N-Agent Dynamics, the attainment theorem means Existence of Asymptotically Optimal-Value Observation-Driven Policies, and the measurability lemma means Joint Measurability of the Tracked Events and of the Tracked Energy Density over the Block Cascade, and Measurability in Time of Their Expectations. The item Asymptotic Lower Bound for the Recentred N-Agent Cost by the Fluctuation LQG Value, under Law-Transported Injection Certificates is the earlier published version of the LQG lower bound theorem, the one cited by the certificate lemma; its hypothesis (OC), its Observation data paragraph, its displayed definitions of $\psi_{\lambda}$ and $\mathcal{A}_{s}(\lambda)$ and its displayed definition of $X'_{t}$ are given there by the same words and formulas as in the LQG lower bound theorem; the two versions differ elsewhere, in particular in the hypotheses (I0) and (VT$'$) and in the naming of the driving system and the record prefixes. This textual identity of the five items just listed is all that is used below to transfer what the certificate lemma asserts about the earlier version to the LQG lower bound theorem. The fundamental solution written $\Phi$ and its inverse $\Psi$ in conclusion 3 of the policy lemma play no role below; $\Phi$ denotes the realized mean-field flow throughout.

Claim 1.

The two inverses. Under (H1) — which holds with $r=c_{J}$ by claim 5 of Localized Joint Coercivity of the Recentred N-Agent Cost Integrand under a Positive-Definite Fluctuation Hessian under (JC), as Ledger Decomposition of the Recentred N-Agent Cost over the Block Cascade and Its Near-Field Filtering Lower Bound records — conclusion (a) of the completion-of-squares theorem gives that every $R_{t}$ is symmetric positive definite, hence invertible by Invertibility of Symmetric Positive Definite Matrices. Fix $t$. By item 7 of Fluctuation LQG Data of a Stationary Mean-Field Triple, $\tilde{\Theta}^{\star}_{t}$ is the diagonal matrix with diagonal entries $\tilde{b}^{\upsilon}(S_{t})$, which are at least $\underline{b}>0$ by (OC) because $S_{t}\in\Delta^{l}$; the diagonal matrix with diagonal entries $1/\tilde{b}^{\upsilon}(S_{t})$ is its inverse, as the index formula for the matrix product shows. So $\Xi_{t}$ and $\tilde{D}_{t}$ are defined.

The setting of the policy lemma. The policy lemma adopts the setting of Fluctuation LQG Data of a Stationary Mean-Field Triple: the natural numbers $l\ge2$, $m\ge1$, $\tilde{l}\ge1$, a nonempty control set $\mathcal{A}\subseteq\mathbb{R}^{m}$, a transition-rate family $\beta$ with control set $\mathcal{A}$ and rate bound $B$ and its extension $(U,V,\bar\beta)$ with derivative bound $K$, the cost extension $(U_{c},\bar{L},\bar{G})$ of $(L,G)$ with its second-derivative bound, the horizon $T$, the triple $(S,A,P)$, the observation-rate family $\tilde\beta$ with $\tilde{l}$ channels and rate bound $\tilde{B}$ and its extension $(\tilde{U},\bar{\tilde\beta})$ with derivative bound $\tilde{K}$. All of these are part of the common data: $\beta$ is the transition-rate family with rate bound $B$ furnished by claim 2 of the affine rate family lemma, $(U,V,\bar\beta)$ with $K$ and $(U_{c},\bar{L},\bar{G})$ with its bound are the extensions of the common data, $\tilde{B}$ is the rate bound of $\tilde\beta$, and $(\tilde{U},\bar{\tilde\beta})$ with $\tilde{K}$ is the extension of (X$'$). The matrices $\mathcal{E}_{t}$, $\mathcal{B}_{t}$, $\tilde{\mathcal{E}}_{t}$, $\Theta^{\star}_{t}$, $\tilde{\Theta}^{\star}_{t}$ of the policy lemma are the fluctuation LQG data of these data, that is, the matrices so named in the statement.

The coefficient matrices. Write $H_{ij}(t)$ ($i,j\in\{1,\dots,l+m\}$) and $F_{\gamma\delta}$ for the fluctuation Hessian coefficients of $(U,V,\bar\beta)$, $(U_{c},\bar{L},\bar{G})$ and $(S,A,P)$. The policy lemma defines $Q_{t}=\tfrac14(H^{SS}_{t}+(H^{SS}_{t})^{\top})$, $V_{t}=\tfrac12(H^{SA}_{t}+(H^{AS}_{t})^{\top})$, $R_{t}=\tfrac14(H^{AA}_{t}+(H^{AA}_{t})^{\top})$ and $\hat{F}=\tfrac14(F^{\star}+(F^{\star})^{\top})$ from the Hessian blocks and the terminal matrix of Fluctuation LQG Data of a Stationary Mean-Field Triple, whose entries are $(H^{SS}_{t})_{\gamma\delta}=H_{\gamma\delta}(t)$, $(H^{SA}_{t})_{\gamma j}=H_{\gamma,l+j}(t)$, $(H^{AS}_{t})_{j\gamma}=H_{l+j,\gamma}(t)$, $(H^{AA}_{t})_{jk}=H_{l+j,l+k}(t)$ and $(F^{\star})_{\gamma\delta}=F_{\gamma\delta}$. By the index formula for the transpose, the entries of these four matrices are $Q^{\gamma\delta}_{t}=\tfrac14(H_{\gamma\delta}(t)+H_{\delta\gamma}(t))$, $V^{\gamma j}_{t}=\tfrac12(H_{\gamma,l+j}(t)+H_{l+j,\gamma}(t))$, $R^{ij}_{t}=\tfrac14(H_{l+i,l+j}(t)+H_{l+j,l+i}(t))$ and $\hat{F}^{\gamma\delta}=\tfrac14(F_{\gamma\delta}+F_{\delta\gamma})$, which are the defining formulas of the matrices $Q_{t}$, $V_{t}$, $R_{t}$, $\hat{F}$ of the completion-of-squares theorem in terms of the same fluctuation Hessian coefficients (that theorem adopts the fluctuation linear-quadratic cost of The Fluctuation Linear-Quadratic Cost Functional for the same extensions and the same triple). So the four coefficient matrices of the policy lemma are those of the common data. Likewise the state matrix has entries $(\mathcal{E}_{t})_{\gamma\delta}=\partial_{\delta}\bar{b}^{\gamma}(S_{t},A_{t})$ and the control matrix has entries $(\mathcal{B}_{t})_{\gamma j}=\partial_{l+j}\bar{b}^{\gamma}(S_{t},A_{t})$, which are the defining formulas $E^{\gamma\delta}_{t}=\partial_{\delta}\bar{b}^{\gamma}(S_{t},A_{t})$ and $\mathsf{B}^{\gamma j}_{t}=\partial_{l+j}\bar{b}^{\gamma}(S_{t},A_{t})$ of the completion-of-squares theorem; so $\mathcal{E}_{t}=E_{t}$ and $\mathcal{B}_{t}=\mathsf{B}_{t}$, and the matrix $W_{t}=Z_{t}\mathcal{B}_{t}+\tfrac12V_{t}$ of the policy lemma, formed with the family $Z$, is the matrix $W_{t}=Z_{t}\mathsf{B}_{t}+\tfrac12V_{t}$ of the statement.

Hypotheses (H1)--(H4) and (C). Hypothesis (H1) of the policy lemma asks for a real $r>0$ with $\sum_{i,j}R^{ij}_{t}a^{i}a^{j}\ge r|a|^{2}$ for all $t$ and $a\in\mathbb{R}^{m}$; the left-hand side is the entry pairing $a\cdot R_{t}a$ of the completion-of-squares theorem, so this is hypothesis (H1), which holds with $r=c_{J}$. Hypothesis (H2) of the policy lemma asks for a family of symmetric matrices, continuously differentiable in integral form as in Weighted Second-Moment Evolution of the State Fluctuation Process, with terminal value $\hat{F}$ and densities $-(\mathcal{E}_{t}^{\top}Z_{t}+Z_{t}\mathcal{E}_{t}-W_{t}R_{t}^{-1}W_{t}^{\top}+Q_{t})$ with $W_{t}=Z_{t}\mathcal{B}_{t}+\tfrac12V_{t}$; with the identifications just made this is hypothesis (H2) of the completion-of-squares theorem, which the family $Z$ of the common data satisfies. So $Z$ can be taken as the Riccati family of (H2) of the policy lemma. Hypothesis (H3) asks for a real $\tilde{r}>0$ with $\tilde{b}^{\upsilon}(S_{t})\ge\tilde{r}$ for all $\upsilon$ and $t$; since $S_{t}\in\Delta^{l}$ for every $t$, hypothesis (OC) gives this with $\tilde{r}=\underline{b}$. Hypothesis (H4) asks that $\Pi_{0}$ be symmetric positive semidefinite, which the zero matrix is, $x\cdot(0x)=0$ for every $x$. Hypothesis (C) asks that $\mathbb{R}^{m}\setminus\mathcal{A}$ be an open subset of $\mathbb{R}^{m}$. The set $\mathcal{A}$ is compact in $(\mathbb{R}^{m},\mathcal{T}_{d_{E}})$, so by Compact Subset of $\mathbb{R}^n$ is Closed (with $n=m$) it is closed there, that is, $\mathbb{R}^{m}\setminus\mathcal{A}\in\mathcal{T}_{d_{E}}$ is open in the metric space $(\mathbb{R}^{m},d_{E})$; by Euclidean Openness Agrees with Metric Openness on $\mathbb{R}^n$ it is then open in the Euclidean sense. So (C) holds. Hence the policy lemma applies to the data named in the statement.

The filter covariance. Conclusion 2 of the policy lemma states that $\tilde{D}_{r}=\tilde{\mathcal{E}}_{r}^{\top}(\tilde{\Theta}^{\star}_{r})^{-1}\tilde{\mathcal{E}}_{r}$ is symmetric positive semidefinite with continuous entries, that every $\Theta^{\star}_{r}$ is positive semidefinite, and that consequently Global Existence and Uniqueness for the Kalman Covariance Riccati Equation furnishes exactly one assignment $\Pi'$ of a matrix with $l$ rows and $l$ columns to each $t\in[0,T]$, with continuous entries, satisfying $\Pi'_{t}=\Pi_{0}+\int_{0}^{t}(\mathcal{E}_{r}\Pi'_{r}+\Pi'_{r}\mathcal{E}_{r}^{\top}-\Pi'_{r}\tilde{D}_{r}\Pi'_{r}+\Theta^{\star}_{r})\,dr$ with $\Pi_{0}$ the zero matrix — its filter covariance — and that every $\Pi'_{t}$ is symmetric positive semidefinite. This is the Riccati theorem applied on $[0,T]$ with $k=l$ and the data $A=\mathcal{E}$, $C=\Theta^{\star}$, $D=\tilde{D}$, $P_{0}=0$; the assignment $\Pi$ of the statement is by definition the solution furnished by that theorem for these data, which is unique. Hence the Riccati theorem applies, $\Pi$ exists and is unique with continuous entries, every $\Pi_{t}$ is symmetric positive semidefinite, and $\Pi$ is the filter covariance of the policy lemma.

The attainment theorem applies; continuity of the integrand; the value. The attainment theorem adopts the setting, hypotheses (H1)--(H4) and (C) and notation of the policy lemma, with the Riccati family $Z$ fixed and the matrix $\Pi_{0}$ of (H4) — here the zero matrix; and for its conclusions 2 and 3 it assumes in addition that $\mathcal{A}$ is convex, which is part of the common data, and hypothesis (H5) of Mean-Square Error Covariance of the Approximate Kalman Filter as assumed in Cost Limit Along the Approximate Kalman Policy, which is the hypothesis (H5) assumed here with $\varrho=\varrho_{\mathcal{A}}$. So the attainment theorem applies. Its setting records that the map $t\mapsto Z_{t}\cdot\Theta^{\star}_{t}+(W_{t}R_{t}^{-1}W_{t}^{\top})\cdot\Pi_{t}$, with the matrix pairing $A'\cdot B'=\sum_{\gamma,\delta}A'^{\gamma\delta}B'^{\gamma\delta}$, is continuous on $[0,T]$; its $\Pi$ being the filter covariance, that is, the present $\Pi$, and its $W_{t}R_{t}^{-1}W_{t}^{\top}$ being the present $\Xi_{t}$, this map is $t\mapsto\sum_{\gamma,\delta}Z^{\gamma\delta}_{t}\Theta^{\star\gamma\delta}_{t}+\sum_{\gamma,\delta}\Xi^{\gamma\delta}_{t}\Pi^{\gamma\delta}_{t}$, the integrand defining $V^{*}$, which is therefore continuous on $[0,T]$. The number $V^{*}$ of the attainment theorem is $Z_{0}\cdot\Pi_{0}+\int_{[0,T]}(Z_{t}\cdot\Theta^{\star}_{t}+(W_{t}R_{t}^{-1}W_{t}^{\top})\cdot\Pi_{t})\,dt$; since $\Pi_{0}$ is the zero matrix, $Z_{0}\cdot\Pi_{0}=0$, so it is the $V^{*}$ of the statement. This proves claim 1.

Claim 2. Let a family of solutions be admissible, with the constants $\kappa^{\sharp}$, $\mathcal{J}^{\sharp}$ and the points $\mathsf{x}^{N}_{0}$ of (I$'$), (CB), (D0).

Hypothesis (I) from (D0). Fix $N\ge1$ and write $\Omega_{*}=\{\Sigma_{0}=\mathsf{x}^{N}_{0}\}$, an event of probability one by (D0) (an event because each component of $\Sigma_{0}$ is a random variable by the definition of a solution). The empirical state measure takes its values in $\Delta^{l}$ by that definition, and any two points $x,y\in\Delta^{l}$ satisfy $|x-y|\le2$: each $|x^{\gamma}-y^{\gamma}|\le1$, so $(x^{\gamma}-y^{\gamma})^{2}\le|x^{\gamma}-y^{\gamma}|\le x^{\gamma}+y^{\gamma}$, whence $|x-y|^{2}\le\sum_{\gamma}(x^{\gamma}+y^{\gamma})=2\le4$. Hence every component of $\mathfrak{s}_{0}=\sqrt{N}(\Sigma_{0}-S_{0})$ is bounded in absolute value by $2\sqrt{N}$ at every point of $\Omega^{\mathrm{ag}}$, so each product $\mathfrak{s}^{\gamma}_{0}\mathfrak{s}^{\delta}_{0}$ is bounded in absolute value by $4N$ everywhere, and on $\Omega_{*}$ it equals the constant $N(\mathsf{x}^{N,\gamma}_{0}-S^{\gamma}_{0})(\mathsf{x}^{N,\delta}_{0}-S^{\delta}_{0})$, itself bounded by $4N$ since $\mathsf{x}^{N}_{0},S_{0}\in\Delta^{l}$. By claim 2 of Almost Sure Inequalities Between Bounded Random Variables Pass to Expectations (with $\Omega_{*}$, the common bound $4N$ and the constant random variable), and since the expectation of a constant random variable on a probability space is that constant by the definition of the expectation, $\mathbb{E}[\mathfrak{s}^{\gamma}_{0}\mathfrak{s}^{\delta}_{0}]=N(\mathsf{x}^{N,\gamma}_{0}-S^{\gamma}_{0})(\mathsf{x}^{N,\delta}_{0}-S^{\delta}_{0})$, whose absolute value is at most $N|\mathsf{x}^{N}_{0}-S_{0}|^{2}=(\sqrt{N}|\mathsf{x}^{N}_{0}-S_{0}|)^{2}$ (as $|u^{\gamma}u^{\delta}|\le|u|^{2}$ for $u\in\mathbb{R}^{l}$). The last sequence has limit $0$ by (D0) and claim 2 of the arithmetic of limits, so by claim 3 of the order properties of limits the sequence $(\mathbb{E}[\mathfrak{s}^{\gamma}_{0}\mathfrak{s}^{\delta}_{0}])_{N\ge1}$ has limit $0$ for all $\gamma,\delta$. This is hypothesis (I) of Asymptotic Lower Bound for the Recentred N-Agent Cost without Uniform Control-Moment Hypotheses with the zero matrix as $\Pi_{0}$; the limit of a real sequence being unique (claim 1 of the order properties of limits, applied to a sequence and itself in both directions, gives that two limits of the same sequence are equal), the matrix $\Pi_{0}$ of hypothesis (I) is the zero matrix, which is hypothesis (I0) of the LQG lower bound theorem. Thus the family satisfies every hypothesis that Asymptotic Lower Bound for the Recentred N-Agent Cost without Uniform Control-Moment Hypotheses imposes on a family — (I), (I$'$), (CB) — beyond the common data and the standing hypotheses.

The setting of the certificate lemma. The certificate lemma adopts the setting, notation and standing hypotheses of The Path-Closeness Event under the Cost Bound: Closeness of the Empirical State Measure to the Mean-Field Trajectory and of the Record-Frozen Control to the Mean-Field Control on an Event of Probability $1-O(N^{-1/2})$, hence of Closeness of the Realized Control and the Realized Mean-Field Flow on a High-Probability Event under the Cost Bound and of Asymptotic Lower Bound for the Recentred N-Agent Cost without Uniform Control-Moment Hypotheses: the common data and the standing hypotheses of the latter theorem, which are those adopted in the statement; a family, indexed by $N\ge1$, of solutions of the controlled $N$-agent dynamics on $[0,T]$ for $\mathcal{A}$-valued observation-driven control policies $h^{(N)}$ on $N$-agent driving systems, with the objects named in the statement — which the given family provides; and the hypotheses of Asymptotic Lower Bound for the Recentred N-Agent Cost without Uniform Control-Moment Hypotheses on the family — (I$'$) with the bound $\kappa^{\sharp}$ and (CB) with the bound $\mathcal{J}^{\sharp}$, which hold by admissibility, and (I), which has just been verified directly, so that the appeal to the certificate lemma is not circular on any reading of its adopted setting (its claim 1 re-derives (I) with the zero matrix from (D0)). The remaining objects of that setting are auxiliary and exist: the constants of Closeness of the Realized Control and the Realized Mean-Field Flow on a High-Probability Event under the Cost Bound and of The Path-Closeness Event under the Cost Bound: Closeness of the Empirical State Measure to the Mean-Field Trajectory and of the Record-Frozen Control to the Mean-Field Control on an Event of Probability $1-O(N^{-1/2})$ are furnished by their claims; a natural number $N_{\mathrm{cl}}\ge\max(N_{1},C_{\mathrm{esc}}^{2})$ exists; the realized controls, records, record spaces, prefix maps, record prefixes, truncated policies and observation-centred fluctuations are defined from the family; a dense sequence in $L^{2}([0,s];\mathbb{R}^{m})$ exists for every $s\in(0,T]$, as the certificate lemma's setting records; and reconstruction data exist by Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records, as that setting records. Fix such choices. Finally its hypotheses (D0), (OC) and (X$'$) are the hypothesis (D0) of admissibility (with the points $\mathsf{x}^{N}_{0}$) and the hypotheses (OC) and (X$'$) of the statement. Hence the certificate lemma applies to the given family, and its claims 1 and 2 hold for it.

Hypothesis (OC) of the LQG lower bound theorem. By claim 1 of the certificate lemma, hypothesis (OC) of Asymptotic Lower Bound for the Recentred N-Agent Cost by the Fluctuation LQG Value, under Law-Transported Injection Certificates holds with $\tilde\beta_{\min}=\underline{b}$ for the triple $(S,A,P)$ and the observation extension $(\tilde{U},\bar{\tilde\beta})$ of (X$'$). Hypothesis (OC) of the LQG lower bound theorem is stated in the same words for the same objects — continuity on $[0,T]$ of every entry of $t\mapsto\tilde{\mathcal{E}}_{t}$ and of every $t\mapsto\tilde{b}^{\upsilon}(S_{t})$, and a uniform positive lower bound $\tilde\beta_{\min}$ on the $\tilde{b}^{\upsilon}(S_{t})$ — so it holds with $\tilde\beta_{\min}=\underline{b}$.

The profile responses and the information functionals. The LQG lower bound theorem defines, for a profile $\lambda$, its $\psi_{\lambda}$ as the unique assignment with continuous components satisfying $\psi_{\lambda}(u)=\Pi_{0}\lambda(0)+\int_{[0,u]}(\mathcal{E}_{r}\psi_{\lambda}(r)+\Theta^{\star}_{r}\lambda(r))\,dr$ and its $\mathcal{A}_{s}(\lambda)$ as $\lambda(0)\cdot(\Pi_{0}\lambda(0))+\int_{[0,s]}(\lambda(r)\cdot(\Theta^{\star}_{r}\lambda(r))+\psi_{\lambda}(r)\cdot(\tilde{D}_{r}\psi_{\lambda}(r)))\,dr$, with $\Pi_{0}$ the matrix of hypothesis (I) — here the zero matrix — and with the matrices $\mathcal{E}_{r}$, $\Theta^{\star}_{r}$ and $\tilde{D}_{r}$ of the fluctuation LQG data relative to the extensions of the statement. These are the displayed formulas by which Asymptotic Lower Bound for the Recentred N-Agent Cost by the Fluctuation LQG Value, under Law-Transported Injection Certificates defines its $\psi_{\lambda}$ and $\mathcal{A}_{s}(\lambda)$, evaluated with the zero matrix as $\Pi_{0}$ and with the observation extension $(\tilde{U},\bar{\tilde\beta})$; by claim 1 of the certificate lemma these coincide with the profile response $\psi_{\lambda}$ and the information functional $\mathcal{A}_{s}(\lambda)$ of the certificate lemma.

Hypothesis (VT$'$). The objects named in hypothesis (VT$'$) of the LQG lower bound theorem — the driving system $(\Omega^{\mathrm{ag}},\mathcal{F}^{\mathrm{ag}},P^{\mathrm{ag}})$, the record $W$, the record spaces $(\mathbf{R}_{s},\mathcal{R}_{s})$, the prefix maps $\pi_{s}$, the record prefixes $W^{(s)}=\pi_{s}\circ W$, the observation filtration $(\mathcal{G}_{t})$, and the observation-centred fluctuation $X'_{s}=\sqrt{N}(\Sigma_{s}-\Phi_{s})$ with the realized mean-field flow $\Phi_{t}=\mathsf{S}_{t}(S_{0},\hat\alpha)$, formed from the realized control $\hat\alpha$ and the two-argument mean-field flow $\mathsf{S}$ with the base point $S_{0}$ — are the objects so named in the setting of the certificate lemma, which adopts them from the same sources (its $X'_{s}$ being, as it records, the observation-centred fluctuation of Asymptotic Lower Bound for the Recentred N-Agent Cost by the Fluctuation LQG Value, under Law-Transported Injection Certificates formed with the base point $S_{0}$, and the LQG lower bound theorem defining $X'_{s}$ by the same formula from the same realized flow). Claim 2 of the certificate lemma asserts, for these objects and for its $\psi_{\lambda}$ and $\mathcal{A}_{s}(\lambda)$, exactly the statement (VT$'$) of the LQG lower bound theorem — the condition (C0) for every $N\ge1$ and $s\in(0,T]$, and, for every $s\in(0,T]$, every vector of $\mathbb{R}^{l}$ (written $\mathbf{c}$ there and $c$ here), every profile $\lambda$ and every $\epsilon>0$, the existence of $N_{2}$ and of the certificate data with (a), (C1), (C2$'$), (C3), (C4$'$), stated in the same words. Since the $\psi_{\lambda}$ and $\mathcal{A}_{s}(\lambda)$ of the two items coincide, hypothesis (VT$'$) holds.

Hypothesis (MS). Fix $N\ge1$, an admissible parameter vector $\pi$ in the sense of Asymptotic Lower Bound for the Recentred N-Agent Cost without Uniform Control-Moment Hypotheses and a block index $k\in\{0,\dots,K_{\pi}-1\}$, $K_{\pi}$ being the number of blocks determined by $\pi$ (written $K$ in the ledger lemma; the letter $K$ is the derivative bound here). By the definition of admissibility of $\pi$, when the block cascade and the near-field data are formed from $\pi$ for the $N$-th solution, every parameter-dependent requirement of Ledger Decomposition of the Recentred N-Agent Cost over the Block Cascade and Its Near-Field Filtering Lower Bound holds; its remaining requirements are the common data, the standing hypotheses, and the $N$-th solution with its data, all in force. So the setting of the ledger lemma, and with it that of the measurability lemma, is instantiated by the common data, the $N$-th solution and $\pi$, and claim 2 of the measurability lemma gives that the three functions $s\mapsto\mathbb{E}[\mathbf{1}_{\mathcal{T}^{\mathrm{nr}}_{k}(s)}u_{s}\cdot R_{s}u_{s}]$, $s\mapsto\mathbb{E}[1-\mathbf{1}_{\mathcal{T}_{k}(s)}]$ and $s\mapsto\mathbb{E}[\mathbf{1}_{\mathcal{T}^{\mathrm{fr}}_{k}(s)}]$, restricted to $[t_{k},t_{k+1}]$, are measurable with respect to the trace Borel $\sigma$-algebra of that block. This is hypothesis (MS) of the LQG lower bound theorem for the family.

The remaining hypotheses; the theorem applies. The common data, the family of solutions, the standing hypotheses, hypotheses (I$'$) and (CB), the admissible parameter vectors and the number $V_{0}$ are adopted by the LQG lower bound theorem from Asymptotic Lower Bound for the Recentred N-Agent Cost without Uniform Control-Moment Hypotheses; hypothesis (I) holds as shown; hypothesis (H2) of the completion-of-squares theorem, with the Riccati family $Z$, is part of the common data; hypothesis (H1) holds with $r=c_{J}$; the observation data of the LQG lower bound theorem are the extension $(\tilde{U},\bar{\tilde\beta})$ of (X$'$) and the fluctuation LQG data relative to it; and (I0), (OC), (VT$'$) and (MS) hold as shown. Hence every hypothesis of the LQG lower bound theorem holds for the common data and the family. Its $\Pi$ is the solution of the Riccati theorem on $[0,T]$ with the data $A=\mathcal{E}$, $C=\Theta^{\star}$, $D=\tilde{D}$ and initial value the matrix $\Pi_{0}$ of hypothesis (I), that is, the zero matrix; this is the $\Pi$ of the statement. Its matrices $\Xi_{t}=W_{t}R_{t}^{-1}W_{t}^{\top}$, adopted from the cascade filtering lemma, are formed from the same $W_{t}=Z_{t}\mathsf{B}_{t}+\tfrac12V_{t}$ and $R_{t}$ as the $\Xi_{t}$ of the statement, and so coincide with them.

Conclusions. By the paragraph The limit value of Asymptotic Lower Bound for the Recentred N-Agent Cost without Uniform Control-Moment Hypotheses, each product $s\mapsto Z^{\gamma\delta}_{s}\Theta^{\star\gamma\delta}_{s}$ is continuous on $[0,T]$ with finite Lebesgue integral, hence so is the finite sum $s\mapsto\sum_{\gamma,\delta}Z^{\gamma\delta}_{s}\Theta^{\star\gamma\delta}_{s}$ by the continuity of sums and the linearity of the integral, and

the first sum vanishing because $\Pi_{0}$ is the zero matrix. Claim 1 of the LQG lower bound theorem gives that $s\mapsto\sum_{\gamma,\delta}\Xi^{\gamma\delta}_{s}\Pi^{\gamma\delta}_{s}$ is continuous, nonnegative and Lebesgue integrable on $[0,T]$. Hence, by the linearity of the integral (both summands being integrable),

The recentred cost $\mathcal{J}_{N}$ of the LQG lower bound theorem is the recentred cost of First-Order Expansion of the Recentred N-Agent Cost about a Stationary Mean-Field Triple and Its Coercive Lower Bound adopted from Asymptotic Lower Bound for the Recentred N-Agent Cost without Uniform Control-Moment Hypotheses, which is the quantity displayed in the statement. Claim 4 of the LQG lower bound theorem therefore gives: for every real $\varepsilon>0$ there is a natural number $N_{5}$ with $\mathcal{J}_{N}\ge V_{0}+\int_{[0,T]}\sum_{\gamma,\delta}\Xi^{\gamma\delta}_{s}\Pi^{\gamma\delta}_{s}\,ds-\varepsilon=V^{*}-\varepsilon$ for every $N\ge N_{5}$; take $N_{\varepsilon}=N_{5}$. This proves claim 2.

Claim 3.

The Kalman family is a family of solutions. By conclusion 1 of the attainment theorem, available by claim 1, each $h^{\mathrm{K},N}$ is an $\mathcal{A}$-valued observation-driven control policy with horizon $T$, control dimension $m$ and $\tilde{l}$ channels, and the projected solution fixed in the statement is a solution of the controlled $N$-agent dynamics on $[0,T]$ for $\beta$, $\tilde\beta$, the driving system $(\Omega^{\mathrm{K}},\mathcal{F}^{\mathrm{K}},P^{\mathrm{K}})$ and $h^{\mathrm{K},N}$. So the Kalman family is a family of solutions for the common data, with $\mathcal{J}^{\mathrm{K}}_{N}=N(J^{N}[h^{\mathrm{K},N}]-J^{MF})+\sum_{\gamma}P^{\gamma}_{0}\zeta^{\mathrm{K},\gamma}_{N}$, $\zeta^{\mathrm{K}}_{N}=N(\mathbb{E}^{\mathrm{K}}[\Sigma^{\mathrm{K}}_{0}]-S_{0})$.

The initial conditions. Fix $N\ge1$ and write $\Omega^{\mathrm{K}}_{*}=\{\Sigma^{\mathrm{K}}_{0}=\mathsf{y}^{N}_{0}\}$, an event of probability one by (DK) (it is an event, each component of $\Sigma^{\mathrm{K}}_{0}$ being a random variable by the definition of a solution). The empirical state measure takes its values in $\Delta^{l}$ by that definition, and any two points $x,y\in\Delta^{l}$ satisfy $|x-y|\le2$: each $|x^{\gamma}-y^{\gamma}|\le1$, so $(x^{\gamma}-y^{\gamma})^{2}\le|x^{\gamma}-y^{\gamma}|\le x^{\gamma}+y^{\gamma}$, whence $|x-y|^{2}\le\sum_{\gamma}(x^{\gamma}+y^{\gamma})=2\le4$. Hence $|\mathfrak{s}^{\mathrm{K}}_{0}|=\sqrt{N}\,|\Sigma^{\mathrm{K}}_{0}-S_{0}|\le2\sqrt{N}$, and every component of $\mathfrak{s}^{\mathrm{K}}_{0}$ is bounded in absolute value by $2\sqrt{N}$, at every point of $\Omega^{\mathrm{K}}$; so the random variables $\mathfrak{s}^{\mathrm{K},\gamma}_{0}\mathfrak{s}^{\mathrm{K},\delta}_{0}$ and $|\mathfrak{s}^{\mathrm{K}}_{0}|^{4}$ are bounded in absolute value by $4N$ and by $16N^{2}$ respectively, everywhere on $\Omega^{\mathrm{K}}$. On $\Omega^{\mathrm{K}}_{*}$ one has $\mathfrak{s}^{\mathrm{K}}_{0}=\sqrt{N}(\mathsf{y}^{N}_{0}-S_{0})$, so there $\mathfrak{s}^{\mathrm{K},\gamma}_{0}\mathfrak{s}^{\mathrm{K},\delta}_{0}=N(\mathsf{y}^{N,\gamma}_{0}-S^{\gamma}_{0})(\mathsf{y}^{N,\delta}_{0}-S^{\delta}_{0})$ and $|\mathfrak{s}^{\mathrm{K}}_{0}|^{4}=N^{2}|\mathsf{y}^{N}_{0}-S_{0}|^{4}$, constants which are themselves bounded in absolute value by $4N$ and by $16N^{2}$, since $\mathsf{y}^{N}_{0}$ and $S_{0}$ lie in $\Delta^{l}$. By claim 2 of Almost Sure Inequalities Between Bounded Random Variables Pass to Expectations (applied with $\Omega_{*}=\Omega^{\mathrm{K}}_{*}$, the common bounds $4N$, respectively $16N^{2}$, and the constant random variables),

the expectation of a constant random variable on a probability space being that constant by the definition of the expectation; these are the two identities of claim 3. Put $e_{N}=\sqrt{N}|\mathsf{y}^{N}_{0}-S_{0}|\ge0$, a sequence with limit $0$ by (DK). Since $|u^{\gamma}u^{\delta}|\le|u|^{2}$ for $u\in\mathbb{R}^{l}$ (each $|u^{\gamma}|\le|u|$), the first expectation is bounded in absolute value by $N|\mathsf{y}^{N}_{0}-S_{0}|^{2}=e_{N}^{2}$, which has limit $0$ by claim 2 of the arithmetic of limits; by claim 3 of the order properties of limits the sequence $(\mathbb{E}^{\mathrm{K}}[\mathfrak{s}^{\mathrm{K},\gamma}_{0}\mathfrak{s}^{\mathrm{K},\delta}_{0}])_{N\ge1}$ has limit $0=\Pi^{\gamma\delta}_{0}$, which is hypothesis (I1) of Cost Limit Along the Approximate Kalman Policy with the zero matrix as $\Pi_{0}$. The second expectation is $e_{N}^{4}$, which has limit $0$ by the same arithmetic of limits; by clause (d) of claim 3 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities there is $N_{0}$ with $e_{N}^{4}<1$ for every $N\ge N_{0}$, so with $\kappa^{\sharp,\mathrm{K}}=1+\max\{1,e_{1}^{4},\dots,e_{N_{0}}^{4}\}$ one has $\sup\{\mathbb{E}^{\mathrm{K}}[|\mathfrak{s}^{\mathrm{K}}_{0}|^{4}]:N\ge1\}\le\kappa^{\sharp,\mathrm{K}}-1<\infty$, which is hypothesis (I2).

Attainment. Conclusion 2 of the attainment theorem, applied to the driving systems and projected solutions fixed in the statement, whose empirical state measures are $\Sigma^{\mathrm{K}}$ and whose state fluctuations are $\mathfrak{s}^{\mathrm{K}}$, states that each $N(J^{N}[h^{\mathrm{K},N}]-J^{MF})+\sum_{\gamma}P^{\gamma}_{0}\zeta^{\mathrm{K},\gamma}_{N}$, with $\zeta^{\mathrm{K}}_{N}=N(\mathbb{E}^{\mathrm{K}}[\Sigma^{\mathrm{K}}_{0}]-S_{0})$ and with the $N$-agent cost $J^{N}[h^{\mathrm{K},N}]$ of the solution at level $N$ and the mean-field cost $J^{MF}$ of $(S,A)$ — that is, each $\mathcal{J}^{\mathrm{K}}_{N}$ — is a well-defined real number and that the sequence $(\mathcal{J}^{\mathrm{K}}_{N})_{N\ge1}$ has limit the $V^{*}$ of that theorem, which is the present $V^{*}$ by claim 1.

Admissibility. (I$'$): $\kappa_{0}=1+\mathbb{E}^{\mathrm{K}}[|\mathfrak{s}^{\mathrm{K}}_{0}|^{4}]=1+e_{N}^{4}\le\kappa^{\sharp,\mathrm{K}}$ for every $N\ge1$, with $\kappa^{\sharp,\mathrm{K}}\ge1$ as above, which is (I$'$) with the bound $\kappa^{\sharp,\mathrm{K}}$. (CB): by the definition of the limit there is $N_{\star}$ with $|\mathcal{J}^{\mathrm{K}}_{N}-V^{*}|<1$, hence $\mathcal{J}^{\mathrm{K}}_{N}\le V^{*}+1$, for every $N\ge N_{\star}$; put $\mathcal{J}^{\sharp,\mathrm{K}}=\max\{|V^{*}|+1,|\mathcal{J}^{\mathrm{K}}_{1}|,\dots,|\mathcal{J}^{\mathrm{K}}_{N_{\star}}|\}\ge0$; then $\mathcal{J}^{\mathrm{K}}_{N}\le\mathcal{J}^{\sharp,\mathrm{K}}$ for every $N\ge1$, which is (CB) with the bound $\mathcal{J}^{\sharp,\mathrm{K}}$. (D0): the points $\mathsf{x}^{N}_{0}=\mathsf{y}^{N}_{0}\in\mathbb{G}_{N}$ satisfy $P^{\mathrm{K}}(\Sigma^{\mathrm{K}}_{0}=\mathsf{x}^{N}_{0})=1$ and $\sqrt{N}|\mathsf{x}^{N}_{0}-S_{0}|\to0$ by (DK). Hence the Kalman family is admissible, and $(\mathcal{J}^{\mathrm{K}}_{N})_{N\ge1}\in\mathcal{C}$. This proves claim 3.

Claim 4.

$V^{*}$ is the asymptotically optimal value. Condition (i) of Asymptotically Optimal Value of a Set of Real Sequences for $V^{*}$: every sequence in $\mathcal{C}$ is the sequence $(\mathcal{J}_{N})_{N\ge1}$ of an admissible family, so by claim 2, for every $\varepsilon>0$ there is $N_{\varepsilon}$ with $\mathcal{J}_{N}\ge V^{*}-\varepsilon$ for every $N\ge N_{\varepsilon}$. Condition (ii): by claim 3 the sequence $(\mathcal{J}^{\mathrm{K}}_{N})_{N\ge1}$ belongs to $\mathcal{C}$ and converges to $V^{*}$. So $V^{*}$ is an asymptotically optimal value of $\mathcal{C}$, which is nonempty by claim 3.

Uniqueness. Let a real number $v$ be an asymptotically optimal value of $\mathcal{C}$. Suppose $v<V^{*}$ and put $\varepsilon=(V^{*}-v)/2>0$. By condition (ii) for $v$ there is a sequence $(x_{N})_{N\ge1}$ in $\mathcal{C}$ converging to $v$, so there is $N'$ with $|x_{N}-v|<\varepsilon$, hence $x_{N}<v+\varepsilon=V^{*}-\varepsilon$, for every $N\ge N'$; by condition (i) for $V^{*}$ there is $N''$ with $x_{N}\ge V^{*}-\varepsilon$ for every $N\ge N''$; at $N=\max(N',N'')$ these contradict each other. So $v\ge V^{*}$. Suppose $v>V^{*}$ and put $\varepsilon=(v-V^{*})/2>0$. By condition (ii) for $V^{*}$ (claim 3) the sequence $(\mathcal{J}^{\mathrm{K}}_{N})_{N\ge1}$ in $\mathcal{C}$ converges to $V^{*}$, so there is $N'$ with $\mathcal{J}^{\mathrm{K}}_{N}<V^{*}+\varepsilon=v-\varepsilon$ for every $N\ge N'$; by condition (i) for $v$ there is $N''$ with $\mathcal{J}^{\mathrm{K}}_{N}\ge v-\varepsilon$ for every $N\ge N''$; again a contradiction at $N=\max(N',N'')$. So $v\le V^{*}$, and $v=V^{*}$. Hence $V^{*}$ is the unique asymptotically optimal value of $\mathcal{C}$, as claimed.

The LQG minimum. Conclusion 3 of the attainment theorem, available by claim 1, states that for every linear-Gaussian state-observation model matched to the fluctuation LQG data as described there, the linear-quadratic-Gaussian cost $J$ formed there attains a minimum over the extended admissible controls with values in $\mathbb{R}^{m}$ and $\min J=V^{*}$, with the $V^{*}$ of that theorem, which is the present $V^{*}$. This proves claim 4.