Asymptotic Lower Bound for the Recentred N-Agent Cost by the Fluctuation LQG Value, under 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 with regular event $\Omega_{0}$, empirical state measure $\Sigma$, observation filtration $(\mathcal{G}_{t})_{t\in[0,T]}$, fluctuation processes $\mathfrak{s}$ and $\mathfrak{a}$, and recentred cost $\mathcal{J}_{N}$.

Adopt also, from the cascade filtering lemma, the matrices

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 $K$, $t_{0},\dots,t_{K}$, $u_{t}=\mathfrak{a}_{t}+R_{t}^{-1}W_{t}^{\top}\mathfrak{s}_{t}$ and $\mathcal{T}^{\mathrm{nr}}_{k}(s)$ for the block count, the block endpoints, the process $u$ and the near-field tracked events of the ledger lemma formed from $\pi$; by claim 2 of that lemma $\mathcal{T}^{\mathrm{nr}}_{k}(s)\in\mathcal{G}_{s}$ for $k\in\{0,\dots,K-1\}$ and $s\in[t_{k},T]$. Write $\mathbb{E}$ for the expectation, $\mathbf{1}_{\mathcal{H}}$ 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

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. Let $\psi_{\lambda}$ be the unique assignment with continuous components such that

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

Assume finally:

(VT) (Injection certificates.) For every $s\in[0,T]$, every $c\in\mathbb{R}^{l}$, every $\lambda$ as above, every admissible parameter vector $\pi$, every $k\in\{0,\dots,K-1\}$ with $s\in[t_{k},t_{k+1}]$, and every real $\epsilon>0$, there is a natural number $N_{2}$ such that for every $N\ge N_{2}$ there are, on the probability space of the $N$-th solution, data $d$, $(\mathsf{Y},\mathcal{Y})$, $\varrho_{0}$, $\mathsf{D}$, $\Theta_{1},\dots,\Theta_{d}$, $\alpha$ and $z$, with $z$ nonzero, satisfying conditions (C1), (C2) and (C3) of Localized Filtering Lower Bound from a van Trees Certificate for the tuple $X=\mathfrak{s}_{s}$, the sub-$\sigma$-algebra $\mathcal{G}_{s}$, the event $\mathcal{H}=\mathcal{T}^{\mathrm{nr}}_{k}(s)$, the vector $c$ and the tolerance $\epsilon$, and satisfying moreover, with $\mathcal{I}$ the van Trees information matrix of (C2),

(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

is measurable on $[t_{k},t_{k+1}]$ for the trace Borel $\sigma$-algebra of that interval. Its integral over $[t_{k},t_{k+1}]$ is the $k$-th summand of claim 7 of the ledger lemma, so (MS) records what the writing of that claim 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

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

2. (Pointwise filtering bound.) Let $\pi$ be an admissible parameter vector, let $k\in\{0,\dots,K-1\}$, let $s\in[t_{k},t_{k+1}]$, let $c\in\mathbb{R}^{l}$ and let $\eta>0$ be real. Then there is a natural number $N_{3}$ such that for every $N\ge N_{3}$

3. (The filtering term.) Let $\pi$ be an admissible parameter vector and let $\eta>0$ be real. Then there is a natural number $N_{4}$ such that for every $N\ge N_{4}$

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

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