Asymptotic Lower Bound for the Recentred N-Agent Cost by the Fluctuation LQG Value, under Law-Transported Injection Certificates

Adopt the common data, the family of solutions, the standing hypotheses, hypotheses (I), (I$'$) and (CB), the notion of an admissible parameter vector, and the real number $V_{0}$ of the asymptotic lower bound theorem: in particular the stationary mean-field triple $(S,A,P)$, the aggregate fluctuation covariance $\Theta$ with $\Theta^{\star}_{t}=\Theta(S_{t},A_{t})$, the matrices $E_{t}$, $\mathsf{B}_{t}$, $V_{t}$, $R_{t}$ and $W_{t}=Z_{t}\mathsf{B}_{t}+\tfrac{1}{2}V_{t}$ of the completion-of-squares theorem together with its hypotheses (H1), with constant $r>0$, and (H2), with Riccati family $Z$, and, for each natural number $N\ge1$, the solution, carried by the $N$-agent driving system of the family, written $(\Omega^{\mathrm{ag}},\mathcal{F}^{\mathrm{ag}},P^{\mathrm{ag}})$ here, with expectation $\mathbb{E}$ (its probability measure is written $P^{\mathrm{ag}}$ because $P$ is the co-state of the triple), with regular event $\Omega_{0}$, empirical state measure $\Sigma$, observation record $W$ with values in the observation record space $(\mathbf{R}_{T},\mathcal{R}_{T})$ with horizon $T$ and $\tilde{l}$ channels, $\tilde{l}$ being the number of channels of the observation-rate family $\tilde{\beta}$ of the common data, observation filtration $(\mathcal{G}_{t})_{t\in[0,T]}$, fluctuation processes $\mathfrak{s}$ and $\mathfrak{a}$, and recentred cost $\mathcal{J}_{N}$. Assume:

(I0) (Vanishing initial covariance.) The matrix $\Pi_{0}$ of hypothesis (I) is the zero matrix.

Adopt also, from the cascade filtering lemma, the matrices

$$\Xi_{t}=W_{t}R_{t}^{-1}W_{t}^{\top}\qquad(t\in[0,T])$$

and, for $t\in[0,T]$, the filtering error $\varepsilon_{t}$ with components $\varepsilon^{\gamma}_{t}=\mathfrak{s}^{\gamma}_{t}-\mathbb{E}[\mathfrak{s}^{\gamma}_{t}\mid\mathcal{G}_{t}]$, formed from any choice of conditional expectations; each $\mathfrak{s}^{\gamma}_{t}$ is bounded in absolute value by $2\sqrt{N}$, any two points of the probability simplex being at Euclidean distance at most $2$, and is therefore square-integrable. For an admissible parameter vector $\pi$ and the $N$-th solution write $K$, $t_{0},\dots,t_{K}$, $u_{t}=\mathfrak{a}_{t}+R_{t}^{-1}W_{t}^{\top}\mathfrak{s}_{t}$, $\mathcal{T}_{k}(s)$, $\mathcal{T}^{\mathrm{nr}}_{k}(s)$ and $\mathcal{T}^{\mathrm{fr}}_{k}(s)$ for the block count, the block endpoints, the process $u$, the tracked events, the near-field tracked events and the far-field tracked events of the ledger lemma formed from $\pi$, and adopt from that lemma also its constant $\Lambda_{\star}$, the near-field radius $\varrho$ (a component of $\pi$), the level sum $\Upsilon_{\mathrm{lev}}$, the tracked energy $\mathcal{Z}$ of its setting paragraph and the quantity $\mathcal{P}$ of its derived-quantities paragraph, both finite by its claim 1; adopt from part (a) of the asymptotic lower bound theorem the real number $\mathcal{Z}^{\sharp}(\pi)$. By claim 2 of the ledger lemma $\mathcal{T}^{\mathrm{nr}}_{k}(s)\in\mathcal{G}_{s}$ for $k\in\{0,\dots,K-1\}$ and $s\in[t_{k},T]$. Adopt from the ledger lemma also the realized mean-field flow $\Phi$ and the noise majorant $Q\ge0$ with $\mathbb{E}[Q^{4}]\le c_{Q}\kappa_{0}N^{-2}$. These are the objects of the pre-stopping envelope lemma for the instance fixed in the block cascade lemma, which is the instance of the extended good-set stopping-time lemma with base point $x_{0}=S_{0}$: there $\Phi_{t}(\omega)=S_{t}(x_{0},\hat{\alpha}(\omega))$ for the realized control $\hat{\alpha}$ of the $N$-th solution and the two-argument mean-field flow $S_{t}(\cdot,\cdot)$ of the flow stability lemma, and $Q=\overline{M}+\Lambda_{b}e^{\Lambda_{b}T}I+e^{\Lambda_{b}T}N^{-1/2}|\mathfrak{s}_{0}|$ is formed from the martingale part of the $N$-th solution, its integral $I$, the initial fluctuation $\mathfrak{s}_{0}$, the constant $\Lambda_{b}$ and $T$. Neither formula involves a component of $\pi$, so $\Phi$ and $Q$ are the same for every admissible $\pi$. For $t\in[0,T]$ define the observation-centred fluctuation

$$X'_{t}=\sqrt{N}\,(\Sigma_{t}-\Phi_{t})=\mathfrak{s}_{t}-\sqrt{N}\,(\Phi_{t}-S_{t}),$$

with components $X'^{\gamma}_{t}=\mathfrak{s}^{\gamma}_{t}-\sqrt{N}(\Phi^{\gamma}_{t}-S^{\gamma}_{t})$. Each $X'^{\gamma}_{t}$ is a random variable, $\mathfrak{s}^{\gamma}_{t}$ being one and $\Phi^{\gamma}_{t}$ being $\mathcal{G}_{t}$-measurable by claim 3 of the realized-flow adaptedness lemma, and is square-integrable: by The Empirical State Measure Deviates from the Realized Mean-Field Flow by at Most the Noise Majorant one has $(X'^{\gamma}_{t})^{2}\le|X'_{t}|^{2}\le N\,Q^{2}\le N\,(1+Q^{4})$ on $\Omega_{0}$, hence almost surely, $\Omega_{0}$ having probability one, and $\mathbb{E}[Q^{4}]\le c_{Q}\kappa_{0}N^{-2}<\infty$, so that $\mathbb{E}[(X'^{\gamma}_{t})^{2}]\le N(1+\mathbb{E}[Q^{4}])$ by monotonicity of the integral (a crude bound, used only to establish square-integrability; claim 2 gives an $N$-uniform bound).

Record prefixes. For a real number $s$ with $0<s\le T$ let $(\mathbf{R}_{s},\mathcal{R}_{s})$ be the observation record space with horizon $s$ and $\tilde{l}$ channels (for $s=T$ this is the space named above), let $\pi_{s}:\mathbf{R}_{T}\to\mathbf{R}_{s}$ be the prefix map when $s<T$ and the identity map of $\mathbf{R}_{T}$ when $s=T$, and put $W^{(s)}=\pi_{s}\circ W$, the record prefix, a map from $\Omega^{\mathrm{ag}}$ to $\mathbf{R}_{s}$. Write $\mathcal{B}(\mathbb{R})$ for the Borel $\sigma$-algebra of the real line, $\otimes$ for the product $\sigma$-algebra, $\lVert\cdot\rVert_{2}$ for the mean-square norm on the probability space indicated, and measurability and image measures as in those items. For a map $\mathsf{D}'$ from a set $\Omega'$ into $\mathbf{R}_{s}$ write $\sigma(\mathsf{D}')=\{\mathsf{D}'^{-1}(E):E\in\mathcal{R}_{s}\}$, as in Factorisation of Random Variables Through a Measurable Map, the Variational Form of the Mean-Square Filtering Error, and Its Invariance Under the Joint Law.

Notational points: the near-field radius $\varrho$ is unrelated to the certificate measure $\varrho_{0}$ of hypothesis (VT$'$) below; the tolerances $\epsilon$ of (VT$'$) and $\eta$ of claims 2 and 3 are free real numbers, not the components $\epsilon$ and $\eta$ of $\pi$ (of the components of $\pi$, only the near-field radius $\varrho$ is named in this statement); the certificate vectors $\alpha$, $z$, $a_{1},\dots,a_{d}$ of (VT$'$) are unrelated to the realized control $\hat{\alpha}$ and to the control process; the certificate random variables of (VT$'$) are written $\vartheta_{1},\dots,\vartheta_{d}$, so that the letter $\Theta$ is reserved for the covariance matrices $\Theta$, $\Theta^{\star}$ and $\tilde{\Theta}^{\star}$ of the fluctuation LQG data; the certificate map $G$ of (VT$'$) is unrelated to the terminal cost $G$ of the population cost data; the certificate density $p$ of (VT$'$) is a function on $\mathbb{R}^{d}\times\mathsf{Y}$, unrelated to the co-state $P$; the certificate probability space $(\Omega',\mathcal{F}',P')$ of (VT$'$) is a space distinct from the driving system, with its own expectation and mean-square norm; $\pi_{s}$ with a time subscript is a prefix map, unrelated to the parameter vector $\pi$; $W_{t}$ with a time subscript is the matrix $Z_{t}\mathsf{B}_{t}+\tfrac12V_{t}$, while $W$ and $W^{(s)}$ without a time subscript are the observation record and its prefix; $P_{0}$ in the instantiation of the Riccati theorem below is the name of the initial-value datum of that theorem, not the co-state at time $0$; the letter $\mathcal{G}$ of the van Trees items cited in (C2$'$) denotes their data-space $\sigma$-algebra (played here by $\mathcal{Y}$), not the observation filtration; and the variable written $X'$ in Factorisation of Random Variables Through a Measurable Map, the Variational Form of the Mean-Square Filtering Error, and Its Invariance Under the Joint Law is that lemma's second-space random variable, unrelated to the observation-centred fluctuation $X'_{t}$. Write $\mathbf{1}_{\mathcal{H}}$ for the indicator of an event $\mathcal{H}$, $x\cdot y$ for the dot product on the Euclidean space $\mathbb{R}^{l}$, $Mx$ for the matrix-vector product, $MM'$ for the matrix product, $M^{\top}$ for the transpose, and $\int_{I}\cdot\,ds$ for the Lebesgue integral over a compact interval.

Observation data. Let $(\tilde{U},\bar{\tilde{\beta}})$ be a twice continuously differentiable extension of the observation-rate family $\tilde{\beta}$ of the common data, with $\tilde{l}$ observation channels, let $\tilde{b}$ be the aggregate observation drift of $\tilde{\beta}$, and let $\tilde{\mathcal{E}}_{t}$ and $\tilde{\Theta}^{\star}_{t}$ be the observation matrix and the observation noise covariance of the fluctuation LQG data of $(S,A,P)$ relative to these extensions, so that $\tilde{\Theta}^{\star}_{t}$ is the diagonal matrix with diagonal entries $\tilde{b}^{\upsilon}(S_{t})$, $\upsilon\in\{1,\dots,\tilde{l}\}$. The state matrix $\mathcal{E}_{t}$ of those data coincides with $E_{t}$, and their state noise covariance with $\Theta^{\star}_{t}$, by the identical defining formulas. Assume:

(OC) (Observation regularity and positivity.) Every entry of $t\mapsto\tilde{\mathcal{E}}_{t}$ and every function $t\mapsto\tilde{b}^{\upsilon}(S_{t})$ is continuous on $[0,T]$, and there is a real $\tilde{\beta}_{\min}>0$ with $\tilde{b}^{\upsilon}(S_{t})\ge\tilde{\beta}_{\min}$ for every $t\in[0,T]$ and every $\upsilon$. (Throughout, a real-valued function on a subinterval $I$ of the real numbers $\mathbb{R}$ is called continuous on $I$ when it is continuous relative to $I$, both $I$ and the codomain $\mathbb{R}$ carrying the metric of the real line.)

Under (OC) each $\tilde{\Theta}^{\star}_{t}$ is invertible and

$$\tilde{D}_{t}=\tilde{\mathcal{E}}_{t}^{\top}\,(\tilde{\Theta}^{\star}_{t})^{-1}\,\tilde{\mathcal{E}}_{t}\qquad(t\in[0,T])$$

is a real matrix with $l$ rows and $l$ columns. Let $\Pi=(\Pi_{t})_{t\in[0,T]}$ be the solution of the Kalman covariance Riccati equation on $[0,T]$ with the data $A=\mathcal{E}$, $C=\Theta^{\star}$, $D=\tilde{D}$ and initial value $P_{0}=\Pi_{0}$, the matrix of hypothesis (I); claim 1 below verifies that this theorem applies.

Injection profiles. Let $\lambda$ be an assignment of a vector $\lambda(u)\in\mathbb{R}^{l}$ to each $u\in[0,T]$, with continuous components (a profile). Let $\psi_{\lambda}$ be the unique assignment with continuous components such that

$$\psi_{\lambda}(u)=\Pi_{0}\lambda(0)+\int_{[0,u]}\bigl(\mathcal{E}_{r}\psi_{\lambda}(r)+\Theta^{\star}_{r}\lambda(r)\bigr)\,dr\qquad(0\le u\le T),$$

which exists and is unique by claim 3 of the variation of constants theorem, and for $s\in[0,T]$ put

$$\mathcal{A}_{s}(\lambda)=\lambda(0)\cdot\bigl(\Pi_{0}\lambda(0)\bigr)+\int_{[0,s]}\Bigl(\lambda(r)\cdot\bigl(\Theta^{\star}_{r}\lambda(r)\bigr)+\psi_{\lambda}(r)\cdot\bigl(\tilde{D}_{r}\psi_{\lambda}(r)\bigr)\Bigr)\,dr .$$

Under (I0) the terms $\Pi_{0}\lambda(0)$ and $\lambda(0)\cdot(\Pi_{0}\lambda(0))$ vanish; they are displayed for uniformity of notation with the general case.

Assume finally:

(VT$'$) (Law-transported injection certificates.) For every $N\ge1$ and every real number $s$ with $0<s\le T$, the observation filtration satisfies:

(C0) (The conditioning $\sigma$-algebra is generated by the record prefix.) $\mathcal{G}_{s}$ is $W^{(s)}$-generated up to null sets in $(\Omega^{\mathrm{ag}},\mathcal{F}^{\mathrm{ag}},P^{\mathrm{ag}})$, in the sense of Factorisation of Random Variables Through a Measurable Map, the Variational Form of the Mean-Square Filtering Error, and Its Invariance Under the Joint Law applied with the measurable space $(\mathbf{R}_{s},\mathcal{R}_{s})$ and the map $W^{(s)}$.

Moreover, for every real number $s$ with $0<s\le T$, every $c\in\mathbb{R}^{l}$, every profile $\lambda$ and every real number $\epsilon>0$ there is a natural number $N_{2}$ such that for every natural number $N\ge N_{2}$ there exist: a probability space $(\Omega',\mathcal{F}',P')$; a set $\mathsf{Y}\in\mathcal{R}_{s}$ with its trace $\sigma$-algebra $\mathcal{Y}=\{E\in\mathcal{R}_{s}:E\subseteq\mathsf{Y}\}$ (a $\sigma$-algebra on $\mathsf{Y}$ by claim 1 of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions) and a $\sigma$-finite measure $\varrho_{0}$ on $(\mathsf{Y},\mathcal{Y})$; a natural number $d\ge1$; square-integrable random variables $\vartheta_{1},\dots,\vartheta_{d}$ on $(\Omega',\mathcal{F}',P')$, collected in $\vartheta=(\vartheta_{1},\dots,\vartheta_{d})$; a map $\mathsf{D}':\Omega'\to\mathsf{Y}$ measurable with respect to $\mathcal{F}'$ and $\mathcal{Y}$; a square-integrable random variable $X^{\dagger}$ on $(\Omega',\mathcal{F}',P')$; a function $p:\mathbb{R}^{d}\times\mathsf{Y}\to[0,\infty)$; a map $G:\mathsf{Y}\to\mathbb{R}$; and vectors $\alpha$, $z$, $a_{1},\dots,a_{d}$ in Euclidean space $\mathbb{R}^{d}$, such that:

(a) (Law identity.) The pair $(X^{\dagger},\mathsf{D}')$, with $\mathsf{D}'$ regarded as $\mathbf{R}_{s}$-valued, is measurable with respect to $\mathcal{F}'$ and $\mathcal{B}(\mathbb{R})\otimes\mathcal{R}_{s}$, the pair $(c\cdot X'_{s},W^{(s)})$ is measurable with respect to $\mathcal{F}^{\mathrm{ag}}$ and $\mathcal{B}(\mathbb{R})\otimes\mathcal{R}_{s}$, $c\cdot X'_{s}$ is square-integrable on $(\Omega^{\mathrm{ag}},\mathcal{F}^{\mathrm{ag}},P^{\mathrm{ag}})$, and the image measure of $P'$ under $(X^{\dagger},\mathsf{D}')$ equals the image measure of $P^{\mathrm{ag}}$ under $(c\cdot X'_{s},W^{(s)})$ on $\mathcal{B}(\mathbb{R})\otimes\mathcal{R}_{s}$.

(C1) (Factorisation through the record.) $\sigma(\mathsf{D}')=\{\mathsf{D}'^{-1}(E):E\in\mathcal{Y}\}$, and every random variable on $(\Omega',\mathcal{F}',P')$ that is measurable with respect to $\sigma(\mathsf{D}')$ and square-integrable equals $g\circ\mathsf{D}'$ for some $g:\mathsf{Y}\to\mathbb{R}$ measurable with respect to $\mathcal{Y}$ and $\mathcal{B}(\mathbb{R})$, for which $g\circ\mathsf{D}'$ is then square-integrable.

(C2$'$) (Van Trees regularity and mixture-weight information.) The probability space $(\Omega',\mathcal{F}',P')$, the natural number $d$ (in the role of $l$), the measurable space $(\mathsf{Y},\mathcal{Y})$ carrying $\varrho_{0}$, the random variables $\vartheta_{1},\dots,\vartheta_{d}$, the map $\mathsf{D}'$ and the density $p$ satisfy hypotheses (i), (ii) and (iii) of The Multivariate van Trees Inequality and hypothesis (iv$'$) of Score Identities and the Mixture-Weight Directional van Trees Inequality; write $\mathcal{I}_{z}\in[0,\infty)$ for the mixture-weight information of that lemma in the direction $z$ for the shifts $a_{1},\dots,a_{d}$ (with $n=d$).

(C3) (Approximation of the estimand.) $G$ is measurable with respect to $\mathcal{Y}$ and $\mathcal{B}(\mathbb{R})$, $G(\mathsf{D}')=G\circ\mathsf{D}'$ is square-integrable, and, with $\alpha\cdot\vartheta=\sum_{j=1}^{d}\alpha_{j}\vartheta_{j}$,

$$\bigl\lVert\alpha\cdot\vartheta+G(\mathsf{D}')-X^{\dagger}\bigr\rVert_{2}\le\epsilon .$$

(C4$'$) (Pairing, information and shift bounds.)

$$\alpha\cdot z\ \ge\ c\cdot\psi_{\lambda}(s)-\epsilon,\qquad \mathcal{I}_{z}\ \le\ \mathcal{A}_{s}(\lambda)+\epsilon,\qquad \max_{1\le j\le d}|\alpha\cdot a_{j}|\ \le\ \epsilon ,$$

with $|\cdot|$ the absolute value.

Hypothesis (VT$'$) makes no reference to $\pi$ or to the blocks: it certifies, on a copy $(\Omega',\mathcal{F}',P')$ carrying the joint law of $(c\cdot X'_{s},W^{(s)})$, the estimand $c\cdot X'_{s}$ on the whole sample space, and the localization to the near-field tracked events is carried out in the proof of claim 3 by means of the fourth-moment bound of claim 2. The time $s=0$ is excluded from (VT$'$); at $s=0$ claim 2 rests on (I0) alone.

(MS) (Measurability of the tracked energy density.) For every natural number $N\ge1$, every admissible parameter vector $\pi$ and every $k\in\{0,\dots,K-1\}$, the function

$$s\ \longmapsto\ \mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}^{\mathrm{nr}}_{k}(s)}\,u_{s}\cdot R_{s}u_{s}\bigr]$$

is measurable on $[t_{k},t_{k+1}]$ for the trace Borel $\sigma$-algebra of that interval, and so are the functions $s\mapsto\mathbb{E}[1-\mathbf{1}_{\mathcal{T}_{k}(s)}]$ and $s\mapsto\mathbb{E}[\mathbf{1}_{\mathcal{T}^{\mathrm{fr}}_{k}(s)}]$. The integrals of these three functions over $[t_{k},t_{k+1}]$ are the $k$-th summands of claim 7 and of claims 1(d) and 1(c) of the ledger lemma, so (MS) records what the writing of those claims already presupposes.

Then the following hold.

1. (The Kalman covariance is well defined.) $\Pi_{0}$ is symmetric positive semidefinite; every entry of $\mathcal{E}$, of $\Theta^{\star}$ and of $\tilde{D}$ is continuous on $[0,T]$; and every $\Theta^{\star}_{t}$ and every $\tilde{D}_{t}$ is positive semidefinite. Consequently $\Pi$ exists, is unique, has continuous entries, and every $\Pi_{t}$ is symmetric positive semidefinite. Moreover every $\Xi_{t}$ is symmetric positive semidefinite with entries continuous on $[0,T]$, and the function

$$s\ \longmapsto\ \sum_{\gamma=1}^{l}\sum_{\delta=1}^{l}\Xi^{\gamma\delta}_{s}\,\Pi^{\gamma\delta}_{s}$$

is continuous, nonnegative and Lebesgue integrable on $[0,T]$.

2. (Pointwise filtering bound.) Let $s\in[0,T]$. Then for every $N\ge1$ the components of $X'_{s}-\mathfrak{s}_{s}$ are $\mathcal{G}_{s}$-measurable, so that $\varepsilon^{\gamma}_{s}$ is almost surely equal to $X'^{\gamma}_{s}-\mathbb{E}[X'^{\gamma}_{s}\mid\mathcal{G}_{s}]$ for every choice of the conditional expectations, and

$$\mathbb{E}\bigl[|X'_{s}|^{4}\bigr]\ \le\ c_{Q}\,\kappa^{\sharp},$$

with $|X'_{s}|^{2}=\sum_{\gamma}(X'^{\gamma}_{s})^{2}$. Moreover, for every $c\in\mathbb{R}^{l}$ and every real $\eta>0$ there is a natural number $N_{3}$ such that for every $N\ge N_{3}$

$$\mathbb{E}\Bigl[\sum_{\gamma=1}^{l}\sum_{\delta=1}^{l}c^{\gamma}c^{\delta}\,\varepsilon^{\gamma}_{s}\varepsilon^{\delta}_{s}\Bigr]\ \ge\ c\cdot\bigl(\Pi_{s}c\bigr)-\eta .$$

3. (The filtering term.) Let $\pi$ be an admissible parameter vector satisfying the absorption condition of part (a) of the asymptotic lower bound theorem, and let $\eta>0$ be real. Then there is a natural number $N_{4}$ such that for every $N\ge N_{4}$

$$\sum_{k=0}^{K-1}\int_{[t_{k},t_{k+1}]}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}^{\mathrm{nr}}_{k}(s)}\,u_{s}\cdot R_{s}u_{s}\bigr]\,ds\ \ge\ \int_{[0,T]}\sum_{\gamma=1}^{l}\sum_{\delta=1}^{l}\Xi^{\gamma\delta}_{s}\,\Pi^{\gamma\delta}_{s}\,ds\ -\ \eta ,$$

the entry pairing $x\cdot My=\sum_{p,q}M^{pq}x^{p}y^{q}$ of the completion-of-squares theorem being used on the left.

4. (Asymptotic lower bound by the fluctuation LQG value.) For every real $\varepsilon''>0$ there is a natural number $N_{5}$ such that

$$\mathcal{J}_{N}\ \ge\ V_{0}+\int_{[0,T]}\sum_{\gamma=1}^{l}\sum_{\delta=1}^{l}\Xi^{\gamma\delta}_{s}\,\Pi^{\gamma\delta}_{s}\,ds\ -\ \varepsilon''\qquad\text{for every }N\ge N_{5};$$

that is, $\liminf_{N\to\infty}\mathcal{J}_{N}\ \ge\ V_{0}+\int_{[0,T]}\sum_{\gamma,\delta}\Xi^{\gamma\delta}_{s}\Pi^{\gamma\delta}_{s}\,ds$.