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

Claim 1.

The initial covariance. By hypothesis (I) the matrix $\Pi_{0}$ is symmetric. Suppose it were not positive semidefinite, so that $x\cdot(\Pi_{0}x)=-2\eta_{0}<0$ for some $x\in\mathbb{R}^{l}$. Put $\zeta=\eta_{0}/\bigl(1+\sum_{\gamma,\delta}|x^{\gamma}x^{\delta}|\bigr)>0$. By hypothesis (I) and the definition of the limit of a real sequence, applied to each of the finitely many sequences $\bigl(\mathbb{E}[\mathfrak{s}^{\gamma}_{0}\mathfrak{s}^{\delta}_{0}]\bigr)_{N\ge1}$, there is $N'$ such that $\bigl|\mathbb{E}[\mathfrak{s}^{\gamma}_{0}\mathfrak{s}^{\delta}_{0}]-\Pi^{\gamma\delta}_{0}\bigr|\le\zeta$ for all $\gamma,\delta$ and all $N\ge N'$. For such $N$, by the linearity of the integral,

$$\mathbb{E}\bigl[(x\cdot\mathfrak{s}_{0})^{2}\bigr]=\sum_{\gamma,\delta}x^{\gamma}x^{\delta}\,\mathbb{E}[\mathfrak{s}^{\gamma}_{0}\mathfrak{s}^{\delta}_{0}]\ \le\ \sum_{\gamma,\delta}x^{\gamma}x^{\delta}\,\Pi^{\gamma\delta}_{0}+\zeta\sum_{\gamma,\delta}|x^{\gamma}x^{\delta}|\ \le\ -2\eta_{0}+\eta_{0}<0,$$

contradicting the nonnegativity of the expectation of the nonnegative random variable $(x\cdot\mathfrak{s}_{0})^{2}$. Hence $\Pi_{0}$ is positive semidefinite.

The coefficient families. Every entry of $t\mapsto E_{t}=\mathcal{E}_{t}$ is continuous on $[0,T]$ by conclusion (a) of the completion-of-squares theorem. Every entry of $t\mapsto\Theta^{\star}_{t}$ is continuous by clause (c) of the covariance deviation lemma, and every $\Theta^{\star}_{t}$ is positive semidefinite, hence symmetric, by claim 3 of the covariance positivity lemma.

Under (OC) each $\tilde{\Theta}^{\star}_{t}$ is the diagonal matrix with diagonal entries $\tilde{b}^{\upsilon}(S_{t})\ge\tilde{\beta}_{\min}>0$; the diagonal matrix with diagonal entries $1/\tilde{b}^{\upsilon}(S_{t})$ is its inverse, as the index formula for the matrix product shows, and its entries are continuous in $t$ by the continuity of the reciprocal of a nonvanishing continuous function. Consequently

$$\tilde{D}^{\gamma\delta}_{t}=\sum_{\upsilon=1}^{\tilde{l}}\frac{\tilde{\mathcal{E}}^{\upsilon\gamma}_{t}\,\tilde{\mathcal{E}}^{\upsilon\delta}_{t}}{\tilde{b}^{\upsilon}(S_{t})},$$

which is symmetric in $(\gamma,\delta)$ and continuous in $t$ by the continuity of sums and products; and for $x\in\mathbb{R}^{l}$,

$$x\cdot(\tilde{D}_{t}x)=\sum_{\upsilon=1}^{\tilde{l}}\frac{\bigl((\tilde{\mathcal{E}}_{t}x)^{\upsilon}\bigr)^{2}}{\tilde{b}^{\upsilon}(S_{t})}\ \ge\ 0,$$

so every $\tilde{D}_{t}$ is positive semidefinite. The Riccati existence theorem therefore applies on $[0,T]$ with $k$ there equal to $l$, $A=\mathcal{E}$, $C=\Theta^{\star}$, $D=\tilde{D}$ and $P_{0}=\Pi_{0}$, and yields the unique $\Pi$ with continuous entries, every $\Pi_{t}$ being symmetric and satisfying $0\preceq\Pi_{t}$, that is, positive semidefinite.

The weight $\Xi$. Every $\Xi_{t}$ is symmetric positive semidefinite with continuous entries by claim 2 of the cascade filtering lemma. Fix $t$. Under (H1), conclusion (a) of the completion-of-squares theorem gives that $R_{t}$ is symmetric positive definite, so $R_{t}^{-1}$ is symmetric positive definite by Invertibility of Positive Definite Matrices, and by the Cholesky factorisation there is a real matrix $L_{t}$ with $m$ rows and $m$ columns and $R_{t}^{-1}=L_{t}L_{t}^{\top}$. Put $B_{t}=W_{t}L_{t}$, a real matrix with $l$ rows and $m$ columns (this matrix always carries a time subscript and is distinct from the control matrix $\mathsf{B}_{t}$ and from the rate bound $B$ of the common data), and let $b_{t,1},\dots,b_{t,m}\in\mathbb{R}^{l}$ be its columns, so that $(B_{t})_{\gamma j}=b^{\gamma}_{t,j}$. Then

$$\Xi_{t}=W_{t}L_{t}L_{t}^{\top}W_{t}^{\top}=B_{t}B_{t}^{\top},\qquad\text{that is}\qquad\Xi^{\gamma\delta}_{t}=\sum_{j=1}^{m}b^{\gamma}_{t,j}b^{\delta}_{t,j},$$

by the index formula for the matrix product and the transpose. Hence, for every real matrix $\mathcal{M}$ with $l$ rows and $l$ columns,

$$\sum_{\gamma=1}^{l}\sum_{\delta=1}^{l}\Xi^{\gamma\delta}_{t}\,\mathcal{M}^{\gamma\delta}=\sum_{j=1}^{m}\sum_{\gamma,\delta}b^{\gamma}_{t,j}b^{\delta}_{t,j}\mathcal{M}^{\gamma\delta}=\sum_{j=1}^{m}b_{t,j}\cdot\bigl(\mathcal{M}\,b_{t,j}\bigr),$$

called the column identity below. Taking $\mathcal{M}=\Pi_{t}$ shows $\sum_{\gamma,\delta}\Xi^{\gamma\delta}_{t}\Pi^{\gamma\delta}_{t}\ge0$, every $\Pi_{t}$ being positive semidefinite. That function of $t$ is a finite sum of products of continuous functions, hence continuous, and therefore Lebesgue integrable on $[0,T]$ by claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval. This proves claim 1.

Claim 2. Fix $s\in[0,T]$, $c\in\mathbb{R}^{l}$ and $\eta>0$, and put $\kappa=c\cdot(\Pi_{s}c)$, a nonnegative real by claim 1.

The filtering error of $X'_{s}$. Let $N\ge1$ and put $\Delta_{s}=X'_{s}-\mathfrak{s}_{s}=\sqrt{N}\,(S_{s}-\Phi_{s})$. Each component $\Delta^{\gamma}_{s}$ is $\mathcal{G}_{s}$-measurable, being the constant $\sqrt{N}S^{\gamma}_{s}$ minus the $\mathcal{G}_{s}$-measurable random variable $\sqrt{N}\Phi^{\gamma}_{s}$ (claim 3 of the realized-flow adaptedness lemma), and is square-integrable as the difference of the square-integrable $X'^{\gamma}_{s}$ and $\mathfrak{s}^{\gamma}_{s}$. By the second immediate consequence recorded in the conditional-expectation definition, $\Delta^{\gamma}_{s}$ is a conditional expectation of itself given $\mathcal{G}_{s}$. Let $M^{\gamma}_{s}$ be the conditional expectation of $\mathfrak{s}^{\gamma}_{s}$ chosen in the definition of $\varepsilon^{\gamma}_{s}=\mathfrak{s}^{\gamma}_{s}-M^{\gamma}_{s}$. By claim 1 of the conditional expectation properties lemma, $M^{\gamma}_{s}+\Delta^{\gamma}_{s}$ is a conditional expectation of $\mathfrak{s}^{\gamma}_{s}+\Delta^{\gamma}_{s}=X'^{\gamma}_{s}$ given $\mathcal{G}_{s}$, so by the uniqueness part of the existence and uniqueness theorem every conditional expectation $\tilde{M}^{\gamma}$ of $X'^{\gamma}_{s}$ given $\mathcal{G}_{s}$ satisfies $\tilde{M}^{\gamma}=M^{\gamma}_{s}+\Delta^{\gamma}_{s}$ almost surely, whence

$$X'^{\gamma}_{s}-\tilde{M}^{\gamma}\ =\ \mathfrak{s}^{\gamma}_{s}+\Delta^{\gamma}_{s}-M^{\gamma}_{s}-\Delta^{\gamma}_{s}\ =\ \varepsilon^{\gamma}_{s}\qquad\text{almost surely.}$$

This is the first assertion of claim 2. In particular, whenever Localized Filtering Lower Bound from a Mixture-Weight van Trees Certificate is applied below to the tuple $X=X'_{s}$ and the sub-$\sigma$-algebra $\mathcal{G}_{s}$, its filtering error $\varepsilon$ is almost surely equal to $\varepsilon_{s}$, so that every expectation of a function of $\varepsilon$ appearing in its conclusions equals the same expectation with $\varepsilon_{s}$ in place of $\varepsilon$ (almost surely equal integrable random variables have equal expectations), and the same applies to the quantities of its claims 1 and 2 with $\mathbf{1}_{\mathcal{H}}\equiv1$.

The fourth moment. By The Empirical State Measure Deviates from the Realized Mean-Field Flow by at Most the Noise Majorant, $|X'_{s}|=\sqrt{N}\,|\Sigma_{s}-\Phi_{s}|\le\sqrt{N}\,Q$ at every point of $\Omega_{0}$, and $P(\Omega_{0})=1$ by the definition of the solution. Hence $|X'_{s}|^{4}\le N^{2}Q^{4}$ almost surely, and by the monotonicity of the integral (a null set not affecting integrals) and claim 2 of the pre-stopping envelope lemma,

$$\mathbb{E}\bigl[|X'_{s}|^{4}\bigr]\ \le\ N^{2}\,\mathbb{E}\bigl[Q^{4}\bigr]\ \le\ N^{2}\,c_{Q}\,\kappa_{0}\,N^{-2}\ =\ c_{Q}\kappa_{0}\ \le\ c_{Q}\kappa^{\sharp},$$

the last step by hypothesis (I$'$). This is the moment bound of claim 2. In particular $|X'_{s}|^{2}$ is square-integrable.

Apply Adjoint Energy Identity for the Kalman Covariance Riccati Equation with the endpoints $0$ and $T$ in the roles of $a$ and $b$ there, $k$ there equal to $l$, the data $\mathcal{E}$, $\Theta^{\star}$, $\tilde{D}$, $\Pi_{0}$ and $\Pi$ of claim 1, the time $s$ and the vector $x=c$. It produces an assignment $\lambda$ with continuous components and $\lambda(s)=c$, and by its claim 2 the assignment $u\mapsto\Pi_{u}\lambda(u)$ has continuous components and satisfies the integral equation defining $\psi_{\lambda}$ — that lemma writes its integrals as Riemann integrals of continuous functions, which coincide with the Lebesgue integrals over compact intervals used here by claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval; by the uniqueness in claim 3 of the variation of constants theorem, $\psi_{\lambda}(u)=\Pi_{u}\lambda(u)$ for every $u$. Its claim 3 then gives

$$c\cdot\psi_{\lambda}(s)=c\cdot(\Pi_{s}c)=\kappa\qquad\text{and}\qquad\mathcal{A}_{s}(\lambda)=\kappa,$$

the second because the energy displayed there is exactly $\mathcal{A}_{s}(\lambda)$ once $\psi_{\lambda}=\Pi\lambda$ is substituted, its Riemann integral again agreeing with the Lebesgue integral in the definition of $\mathcal{A}_{s}(\lambda)$ by claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval (the integrand is continuous by claim 3 of Adjoint Energy Identity for the Kalman Covariance Riccati Equation).

Suppose first $\kappa=0$. Apply (VT) with this $\lambda$ and with $\epsilon=1$, obtaining $N_{2}$; for $N\ge N_{2}$ the data of (VT) exist, and taking $\varkappa=0$, which satisfies (C4) because $(\alpha\cdot z)^{2}\ge0=0\cdot\bigl(z\cdot(\mathcal{I}z)\bigr)$, they form a van Trees certificate for the data $(X'_{s},\mathcal{G}_{s},\Omega,c)$; claim 1 of Localized Filtering Lower Bound from a Mixture-Weight van Trees Certificate is therefore available (only the presence of a certificate is used here, not its tolerance or its value) and gives, with $\mathbf{1}_{\Omega}\equiv1$ and the almost-sure identification of the filtering errors made above,

$$\mathbb{E}\Bigl[\sum_{\gamma,\delta}c^{\gamma}c^{\delta}\varepsilon^{\gamma}_{s}\varepsilon^{\delta}_{s}\Bigr]=\mathbb{E}\bigl[(c\cdot\varepsilon_{s})^{2}\bigr]\ \ge\ 0=\kappa\ \ge\ \kappa-\eta,$$

the inequality being the nonnegativity of the expectation of a nonnegative random variable. Take $N_{3}=N_{2}$.

Suppose now $\kappa>0$, and put

$$\epsilon=\min\Bigl\{\frac{\kappa}{8},\ \frac{\sqrt{\kappa}}{2},\ \frac{\eta}{3+2\sqrt{\kappa}}\Bigr\}>0,\qquad \varkappa=\frac{(\kappa-\epsilon)^{2}}{\kappa+\epsilon},$$

with $\sqrt{\cdot}$ the nonnegative square root. Apply (VT) with this $\lambda$ and this $\epsilon$, obtaining $N_{2}$, and let $N\ge N_{3}=N_{2}$ and let $d$, $(\mathsf{Y},\mathcal{Y})$, $\varrho_{0}$, $\mathsf{D}$, $\vartheta_{1},\dots,\vartheta_{d}$, $\alpha$, $z$, $G$ be data as provided by (VT). Then

$$\alpha\cdot z\ \ge\ \kappa-\epsilon\ >\ 0,\qquad z\cdot(\mathcal{I}z)\ \le\ \kappa+\epsilon,$$

so $(\alpha\cdot z)^{2}\ge(\kappa-\epsilon)^{2}=\varkappa\,(\kappa+\epsilon)\ge\varkappa\,\bigl(z\cdot(\mathcal{I}z)\bigr)$, which is condition (C4) with this $\varkappa$; here $\varkappa\ge0$ and $z\cdot(\mathcal{I}z)>0$ by claim 1 of Rank-One Lower Bound for the Inverse of a Positive Definite Matrix.

Three elementary estimates follow from $0<\epsilon\le\kappa/8$ and $\epsilon\le\sqrt{\kappa}/2$. First, $(\kappa-3\epsilon)(\kappa+\epsilon)=\kappa^{2}-2\kappa\epsilon-3\epsilon^{2}\le\kappa^{2}-2\kappa\epsilon+\epsilon^{2}=(\kappa-\epsilon)^{2}$, so $\varkappa\ge\kappa-3\epsilon$. Second, $\epsilon\le3\kappa$ gives $\epsilon^{2}\le3\kappa\epsilon$, hence $(\kappa-\epsilon)^{2}=\kappa^{2}-2\kappa\epsilon+\epsilon^{2}\le\kappa^{2}+\kappa\epsilon=\kappa(\kappa+\epsilon)$ and therefore $\varkappa\le\kappa$ and $\sqrt{\varkappa}\le\sqrt{\kappa}$. Third, $\varkappa\ge\kappa-3\epsilon\ge\kappa-3\kappa/8\ge\kappa/4\ge\epsilon^{2}$, so $\sqrt{\varkappa}\ge\epsilon$.

Claim 2 of Localized Filtering Lower Bound from a Mixture-Weight van Trees Certificate, for the data $(X'_{s},\mathcal{G}_{s},\Omega,c)$, is therefore available and gives, together with its claim 1 and the almost-sure identification of the filtering errors made above,

$$\mathbb{E}\Bigl[\sum_{\gamma,\delta}c^{\gamma}c^{\delta}\varepsilon^{\gamma}_{s}\varepsilon^{\delta}_{s}\Bigr]\ \ge\ \bigl(\sqrt{\varkappa}-\epsilon\bigr)^{2}=\varkappa-2\epsilon\sqrt{\varkappa}+\epsilon^{2}\ \ge\ \varkappa-2\epsilon\sqrt{\kappa}\ \ge\ \kappa-3\epsilon-2\epsilon\sqrt{\kappa}\ \ge\ \kappa-\eta,$$

the last step by the choice $\epsilon\le\eta/(3+2\sqrt{\kappa})$. This proves claim 2.

Claim 3. Fix an admissible parameter vector $\pi$ satisfying the absorption condition, fix $\eta>0$, and for $k\in\{0,\dots,K-1\}$ and $N\ge1$ define

$$f^{(k)}_{N}(s)=\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}^{\mathrm{nr}}_{k}(s)}\,u_{s}\cdot R_{s}u_{s}\bigr]\qquad(s\in[t_{k},t_{k+1}]).$$

Each $f^{(k)}_{N}$ is nonnegative, because $u_{s}\cdot R_{s}u_{s}\ge r|u_{s}|^{2}\ge0$ by (H1), and is measurable on $[t_{k},t_{k+1}]$ by hypothesis (MS); its integral over that block is the $k$-th summand appearing in claim 7 of the ledger lemma. Define also the bad sets and their probabilities

$$\mathcal{C}_{k}(s)=\Omega\setminus\mathcal{T}^{\mathrm{nr}}_{k}(s),\qquad p^{(k)}_{N}(s)=P\bigl(\mathcal{C}_{k}(s)\bigr)\qquad(s\in[t_{k},t_{k+1}]).$$

By claim 2 of the ledger lemma $\mathcal{T}^{\mathrm{nr}}_{k}(s)\in\mathcal{G}_{s}$, so $\mathcal{C}_{k}(s)\in\mathcal{G}_{s}$, a $\sigma$-algebra being closed under complements. By the definitions in the ledger lemma, $\mathcal{T}_{k}(s)$ is the disjoint union of $\mathcal{T}^{\mathrm{nr}}_{k}(s)$ and $\mathcal{T}^{\mathrm{fr}}_{k}(s)$, so $\mathcal{C}_{k}(s)$ is the disjoint union of $\Omega\setminus\mathcal{T}_{k}(s)$ and $\mathcal{T}^{\mathrm{fr}}_{k}(s)$ and

$$p^{(k)}_{N}(s)=\mathbb{E}\bigl[1-\mathbf{1}_{\mathcal{T}_{k}(s)}\bigr]+\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}^{\mathrm{fr}}_{k}(s)}\bigr],$$

a function measurable on $[t_{k},t_{k+1}]$ by hypothesis (MS) and claim 2 of the arithmetic of measurable functions, with values in $[0,1]$. Its integral over the block is the sum of the integrals of the two summands, by the linearity of the integral (both summands are nonnegative, measurable and bounded by $1$, so by monotonicity their integrals are at most $t_{k+1}-t_{k}$, the measure of the block by claim 1 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval); summing over the blocks and using claims 1(c) and 1(d) of the ledger lemma,

$$\sum_{k=0}^{K-1}\int_{[t_{k},t_{k+1}]}p^{(k)}_{N}(s)\,ds\ \le\ T\,\mathcal{P}+\frac{\mathcal{Z}}{N\varrho^{2}}\ \le\ \bigl(T\Lambda_{\star}\Upsilon_{\mathrm{lev}}+\varrho^{-2}\bigr)\frac{\mathcal{Z}}{N},$$

with $\mathcal{P}\le\Lambda_{\star}\mathcal{Z}\,\Upsilon_{\mathrm{lev}}N^{-1}$ from claim 1(d). Since $\pi$ satisfies the absorption condition, part (a) of the asymptotic lower bound theorem provides $N_{1}$ with $\mathcal{Z}\le\mathcal{Z}^{\sharp}(\pi)$ for $N\ge N_{1}$. Put $C_{\pi}=(T\Lambda_{\star}\Upsilon_{\mathrm{lev}}+\varrho^{-2})\,\mathcal{Z}^{\sharp}(\pi)$, a real number that does not depend on $N$: $\varrho$ is a component of $\pi$; $\Lambda_{\star}$ and the level sum $\Upsilon_{\mathrm{lev}}=\sum_{k=0}^{K-1}L_{k}^{-2}$ of the ledger lemma are given by the explicit formulas $\Lambda_{\star}=\max(4C_{a}^{2}K_{2}^{2}T_{0},\lambda_{c},\lambda_{o})$ and $L_{k}=(2C_{a})^{k+1-K}\varepsilon_{1}$ of the block cascade lemma, in which $K$ is determined by $T_{0}$ and $T$, the constant $C_{a}$ by $l$, $\Lambda_{b}$ and $T$, and $K_{2}$ is one of the constants of the ledger lemma that the asymptotic lower bound theorem records as being the same for every $N$; so neither depends on $N$; and $\mathcal{Z}^{\sharp}(\pi)$ is the real number of part (a) of that theorem. Every summand on the left being nonnegative,

$$\int_{[t_{k},t_{k+1}]}p^{(k)}_{N}(s)\,ds\ \le\ \frac{C_{\pi}}{N}\qquad(k\in\{0,\dots,K-1\},\ N\ge N_{1}). \tag{3.1}$$

The bad-set estimate. Fix $k$, $s\in[t_{k},t_{k+1}]$, $c\in\mathbb{R}^{l}$ and $N\ge1$, write $\mathcal{C}=\mathcal{C}_{k}(s)$ and $p=p^{(k)}_{N}(s)$, and let $\tilde{M}^{\gamma}$ be conditional expectations of $X'^{\gamma}_{s}$ given $\mathcal{G}_{s}$, so that $c\cdot\varepsilon_{s}=c\cdot X'_{s}-c\cdot\tilde{M}$ almost surely by claim 2, where $c\cdot\tilde{M}=\sum_{\gamma}c^{\gamma}\tilde{M}^{\gamma}$. By claim 1 of the conditional expectation properties lemma, $c\cdot\tilde{M}$ is a conditional expectation of $c\cdot X'_{s}$ given $\mathcal{G}_{s}$; by claim 4 of that lemma, applied with the bounded $\mathcal{G}_{s}$-measurable $Z=\mathbf{1}_{\mathcal{C}}$, the random variables $\mathbf{1}_{\mathcal{C}}(c\cdot X'_{s})$ and $\mathbf{1}_{\mathcal{C}}(c\cdot\tilde{M})$ are square-integrable and the latter is a conditional expectation of the former given $\mathcal{G}_{s}$; and by claim 6 of that lemma, $\lVert\mathbf{1}_{\mathcal{C}}(c\cdot\tilde{M})\rVert_{2}\le\lVert\mathbf{1}_{\mathcal{C}}(c\cdot X'_{s})\rVert_{2}$. Since $\mathbf{1}_{\mathcal{C}}(c\cdot\varepsilon_{s})=\mathbf{1}_{\mathcal{C}}(c\cdot X'_{s})-\mathbf{1}_{\mathcal{C}}(c\cdot\tilde{M})$ almost surely, and almost surely equal random variables have the same mean-square norm, claim 2 of the triangle inequality for the mean-square norm gives $\lVert\mathbf{1}_{\mathcal{C}}(c\cdot\varepsilon_{s})\rVert_{2}\le2\lVert\mathbf{1}_{\mathcal{C}}(c\cdot X'_{s})\rVert_{2}$, hence

$$\mathbb{E}\bigl[\mathbf{1}_{\mathcal{C}}(c\cdot\varepsilon_{s})^{2}\bigr]\ \le\ 4\,\mathbb{E}\bigl[\mathbf{1}_{\mathcal{C}}(c\cdot X'_{s})^{2}\bigr]\ \le\ 4|c|^{2}\,\mathbb{E}\bigl[\mathbf{1}_{\mathcal{C}}\,|X'_{s}|^{2}\bigr]\ \le\ 4|c|^{2}\,\lVert\mathbf{1}_{\mathcal{C}}\rVert_{2}\,\bigl\lVert|X'_{s}|^{2}\bigr\rVert_{2}\ \le\ 4|c|^{2}\,C_{4}\,\sqrt{p}, \tag{3.2}$$

where $C_{4}=(c_{Q}\kappa^{\sharp})^{1/2}$: the second step is $(c\cdot X'_{s})^{2}\le|c|^{2}|X'_{s}|^{2}$ pointwise, by the Cauchy–Schwarz inequality for the dot product, together with the monotonicity of the integral; the third is claim 1 of the Cauchy–Schwarz inequality for the mean-square norm, applied to the square-integrable random variables $\mathbf{1}_{\mathcal{C}}$ and $|X'_{s}|^{2}$ (the latter by the moment bound of claim 2); the fourth uses $\lVert\mathbf{1}_{\mathcal{C}}\rVert_{2}^{2}=\mathbb{E}[\mathbf{1}_{\mathcal{C}}]=p$ and $\lVert|X'_{s}|^{2}\rVert_{2}^{2}=\mathbb{E}[|X'_{s}|^{4}]\le c_{Q}\kappa^{\sharp}$ from claim 2, the nonnegative square root being nondecreasing.

Lower bound for the tracked density. For $s\in[0,T]$ put $\tau_{s}=\sum_{\gamma}\Xi^{\gamma\gamma}_{s}$; the function $s\mapsto\tau_{s}$ is continuous on $[0,T]$ by claim 1, hence by the extreme value theorem there is a real $\tau^{\sharp}$ with $\tau_{s}\le\tau^{\sharp}$ for every $s\in[0,T]$, and $\tau^{\sharp}\ge0$ because $\tau_{s}=\sum_{j=1}^{m}\sum_{\gamma}(b^{\gamma}_{s,j})^{2}\ge0$ by the factorisation $\Xi^{\gamma\delta}_{s}=\sum_{j}b^{\gamma}_{s,j}b^{\delta}_{s,j}$ of claim 1. Now fix $k$ and $s\in[t_{k},t_{k+1}]$, and write $\mathcal{H}=\mathcal{T}^{\mathrm{nr}}_{k}(s)$ and $\mathcal{C}=\mathcal{C}_{k}(s)$, so that $\mathbf{1}_{\mathcal{H}}=1-\mathbf{1}_{\mathcal{C}}$. Since $\mathcal{H}\in\mathcal{G}_{s}$, claim 3 of the cascade filtering lemma gives

$$f^{(k)}_{N}(s)\ \ge\ \sum_{\gamma,\delta}\Xi^{\gamma\delta}_{s}\,\mathbb{E}\bigl[\mathbf{1}_{\mathcal{H}}\varepsilon^{\gamma}_{s}\varepsilon^{\delta}_{s}\bigr]=\sum_{j=1}^{m}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{H}}(b_{s,j}\cdot\varepsilon_{s})^{2}\bigr]=\sum_{j=1}^{m}\Bigl(\mathbb{E}\bigl[(b_{s,j}\cdot\varepsilon_{s})^{2}\bigr]-\mathbb{E}\bigl[\mathbf{1}_{\mathcal{C}}(b_{s,j}\cdot\varepsilon_{s})^{2}\bigr]\Bigr),$$

the first equality by the column identity of claim 1 applied to the matrix $\mathcal{M}^{\gamma\delta}=\mathbb{E}[\mathbf{1}_{\mathcal{H}}\varepsilon^{\gamma}_{s}\varepsilon^{\delta}_{s}]$, by the pointwise identity $(b\cdot\varepsilon_{s})^{2}=\sum_{\gamma,\delta}b^{\gamma}b^{\delta}\varepsilon^{\gamma}_{s}\varepsilon^{\delta}_{s}$ and by the linearity of the integral, the second by $\mathbf{1}_{\mathcal{H}}=1-\mathbf{1}_{\mathcal{C}}$ and linearity again. Now $\sum_{j=1}^{m}|b_{s,j}|^{2}=\sum_{j}\sum_{\gamma}(b^{\gamma}_{s,j})^{2}=\tau_{s}\le\tau^{\sharp}$. Applying (3.2) to each $c=b_{s,j}$ and summing,

$$f^{(k)}_{N}(s)\ \ge\ \sum_{j=1}^{m}\mathbb{E}\bigl[(b_{s,j}\cdot\varepsilon_{s})^{2}\bigr]-4C_{4}\tau^{\sharp}\sqrt{p^{(k)}_{N}(s)}\qquad(s\in[t_{k},t_{k+1}],\ N\ge1). \tag{3.3}$$

Passage to the limit. For $N\ge1$ define on $[t_{k},t_{k+1}]$

$$g^{(k)}_{N}(s)=f^{(k)}_{N}(s)+4C_{4}\tau^{\sharp}\sqrt{p^{(k)}_{N}(s)} ,$$

which is measurable: the nonnegative square root is sequentially continuous on $E=[0,1]$, because for $x\ge y\ge0$ one has $(\sqrt{y}+\sqrt{x-y})^{2}=x+2\sqrt{y}\sqrt{x-y}\ge x$, hence $|\sqrt{x}-\sqrt{y}|\le\sqrt{|x-y|}$ for all $x,y\ge0$, so that $|x_{n}-x|<\zeta^{2}$ forces $|\sqrt{x_{n}}-\sqrt{x}|<\zeta$; therefore $\sqrt{p^{(k)}_{N}}$ is measurable as the composition of the measurable function $p^{(k)}_{N}$, with values in $E$, with a sequentially continuous function on $E$ (sequentially continuous functions of measurable maps are measurable, with $d=1$), and sums and scalar multiples of measurable functions are measurable (claim 2 of the arithmetic of measurable functions); and which is nonnegative by (3.3), since $g^{(k)}_{N}(s)\ge\sum_{j}\mathbb{E}[(b_{s,j}\cdot\varepsilon_{s})^{2}]\ge0$. Fix $s\in[t_{k},t_{k+1}]$ and let $\eta'>0$. Applying claim 2 to each of the $m$ vectors $c=b_{s,j}$ with tolerance $\eta'/m$ and taking the largest of the resulting thresholds, there is $N''$ such that for every $N\ge N''$

$$g^{(k)}_{N}(s)\ \ge\ \sum_{j=1}^{m}\Bigl(b_{s,j}\cdot(\Pi_{s}b_{s,j})-\frac{\eta'}{m}\Bigr)=\sum_{\gamma,\delta}\Xi^{\gamma\delta}_{s}\Pi^{\gamma\delta}_{s}-\eta',$$

again by the column identity. As $\eta'>0$ was arbitrary,

$$\liminf_{N\to\infty}g^{(k)}_{N}(s)\ \ge\ \sum_{\gamma,\delta}\Xi^{\gamma\delta}_{s}\Pi^{\gamma\delta}_{s}\qquad(s\in[t_{k},t_{k+1}]).$$

By Fatou's lemma, applied to the nonnegative measurable functions $g^{(k)}_{N}$ on $[t_{k},t_{k+1}]$ with the restricted Lebesgue measure, and then by the monotonicity of the integral together with claim 1,

$$\liminf_{N\to\infty}\int_{[t_{k},t_{k+1}]}g^{(k)}_{N}(s)\,ds\ \ge\ \int_{[t_{k},t_{k+1}]}\liminf_{N\to\infty}g^{(k)}_{N}(s)\,ds\ \ge\ \int_{[t_{k},t_{k+1}]}\sum_{\gamma,\delta}\Xi^{\gamma\delta}_{s}\Pi^{\gamma\delta}_{s}\,ds .$$

The correction term vanishes in the limit: for every real $\theta>0$ and every real $x\ge0$ one has $2\theta\sqrt{x}\le\theta^{2}+x$, because $(\sqrt{x}-\theta)^{2}\ge0$; hence, by the monotonicity and linearity of the integral and by (3.1), for $N\ge N_{1}$

$$\int_{[t_{k},t_{k+1}]}\sqrt{p^{(k)}_{N}(s)}\,ds\ \le\ \frac{\theta\,(t_{k+1}-t_{k})}{2}+\frac{1}{2\theta}\int_{[t_{k},t_{k+1}]}p^{(k)}_{N}(s)\,ds\ \le\ \frac{\theta T}{2}+\frac{C_{\pi}}{2\theta N},$$

the integrals being finite because the integrands are nonnegative, measurable and bounded by $1$, so that by monotonicity each is at most the measure $t_{k+1}-t_{k}$ of the block (claim 1 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval). Thus $\limsup_{N\to\infty}\int_{[t_{k},t_{k+1}]}\sqrt{p^{(k)}_{N}}\,ds\le\theta T/2$ for every $\theta>0$, so this sequence of nonnegative reals converges to $0$. Since $\int g^{(k)}_{N}\,ds=\int f^{(k)}_{N}\,ds+4C_{4}\tau^{\sharp}\int\sqrt{p^{(k)}_{N}}\,ds$ over $[t_{k},t_{k+1}]$ by the linearity of the integral of nonnegative measurable functions (claim 1 of the linearity theorem, an identity in $[0,\infty]$), the second summand is finite, and the sequence of second summands converges to $0$ — so that the two sequences $\int f^{(k)}_{N}\,ds$ and $\int g^{(k)}_{N}\,ds$, with values in $[0,\infty]$, have the same limit inferior in $[0,\infty]$ —

$$\liminf_{N\to\infty}\int_{[t_{k},t_{k+1}]}f^{(k)}_{N}(s)\,ds\ =\ \liminf_{N\to\infty}\int_{[t_{k},t_{k+1}]}g^{(k)}_{N}(s)\,ds\ \ge\ \int_{[t_{k},t_{k+1}]}\sum_{\gamma,\delta}\Xi^{\gamma\delta}_{s}\Pi^{\gamma\delta}_{s}\,ds .$$

Hence for each $k$ there is $N^{(k)}$ with $\int_{[t_{k},t_{k+1}]}f^{(k)}_{N}\,ds\ge\int_{[t_{k},t_{k+1}]}\sum_{\gamma,\delta}\Xi^{\gamma\delta}_{s}\Pi^{\gamma\delta}_{s}\,ds-\eta/K$ for every $N\ge N^{(k)}$. Let $N_{4}$ be the largest of $N_{1},N^{(0)},\dots,N^{(K-1)}$ and sum over $k$. Since $t_{0}=0$, $t_{K}=T$ and $t_{k}<t_{k+1}$ for $k\le K-1$ (the number of blocks $K$ being least with $KT_{0}\ge T$), the intervals $[t_{k},t_{k+1}]$ are adjacent with union $[0,T]$, so by the additivity of the Riemann integral of a continuous function over adjacent compact intervals (Additivity of the Riemann Integral on Adjacent Intervals), the Lebesgue integral of a continuous function over a compact interval being its Riemann integral by claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval, the sum of the right-hand sides is $\int_{[0,T]}\sum_{\gamma,\delta}\Xi^{\gamma\delta}_{s}\Pi^{\gamma\delta}_{s}\,ds-\eta$. This proves claim 3.

Claim 4. Let $\varepsilon''>0$. By conclusion (b) of the asymptotic lower bound theorem, applied with $\varepsilon'=\varepsilon''/2$, there are an admissible parameter vector $\pi$ satisfying the absorption condition of part (a) of that theorem and a natural number $N_{0}$ such that for every $N\ge N_{0}$

$$\mathcal{J}_{N}\ \ge\ V_{0}+\sum_{k=0}^{K-1}\int_{[t_{k},t_{k+1}]}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}^{\mathrm{nr}}_{k}(s)}u_{s}\cdot R_{s}u_{s}\bigr]\,ds-\frac{\varepsilon''}{2},$$

the blocks and near-field tracked events being those formed from $\pi$. Apply claim 3 to this $\pi$ (which satisfies the absorption condition) with $\eta=\varepsilon''/2$, obtaining $N_{4}$, and put $N_{5}=\max(N_{0},N_{4})$. For $N\ge N_{5}$ the two displays combine to

$$\mathcal{J}_{N}\ \ge\ V_{0}+\int_{[0,T]}\sum_{\gamma,\delta}\Xi^{\gamma\delta}_{s}\Pi^{\gamma\delta}_{s}\,ds-\frac{\varepsilon''}{2}-\frac{\varepsilon''}{2},$$

which is claim 4.