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

Claim 1.

The initial covariance. By hypothesis (I0) the matrix $\Pi_{0}$ is the zero matrix, which is symmetric and positive semidefinite, $x\cdot(\Pi_{0}x)=0$ for every $x\in\mathbb{R}^{l}$.

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.

The quadratic form of the filtering error. At every point of $\Omega^{\mathrm{ag}}$ one has $\sum_{\gamma,\delta}c^{\gamma}c^{\delta}\varepsilon^{\gamma}_{s}\varepsilon^{\delta}_{s}=(c\cdot\varepsilon_{s})^{2}$, where $c\cdot\varepsilon_{s}=\sum_{\gamma}c^{\gamma}\varepsilon^{\gamma}_{s}$; each product $\varepsilon^{\gamma}_{s}\varepsilon^{\delta}_{s}$ is integrable, the factors being square-integrable (closure properties of square-integrability), so by the linearity of the integral

$$\mathbb{E}\Bigl[\sum_{\gamma,\delta}c^{\gamma}c^{\delta}\varepsilon^{\gamma}_{s}\varepsilon^{\delta}_{s}\Bigr]=\mathbb{E}\bigl[(c\cdot\varepsilon_{s})^{2}\bigr]\ \ge\ 0, \tag{2.1}$$

the inequality being the nonnegativity of the expectation of a nonnegative random variable. Let $\tilde{M}^{\gamma}$ be conditional expectations of $X'^{\gamma}_{s}$ given $\mathcal{G}_{s}$ and put $c\cdot\tilde{M}=\sum_{\gamma}c^{\gamma}\tilde{M}^{\gamma}$. By the first assertion of claim 2, $c\cdot\varepsilon_{s}=c\cdot X'_{s}-c\cdot\tilde{M}$ almost surely, and by claim 1 of the conditional expectation properties lemma, applied finitely many times, $c\cdot\tilde{M}$ is a conditional expectation of the square-integrable random variable $c\cdot X'_{s}$ given $\mathcal{G}_{s}$. Almost surely equal integrable random variables have equal expectations, so

$$\mathbb{E}\bigl[(c\cdot\varepsilon_{s})^{2}\bigr]=\mathbb{E}\Bigl[\bigl(c\cdot X'_{s}-\mathbb{E}[c\cdot X'_{s}\mid\mathcal{G}_{s}]\bigr)^{2}\Bigr], \tag{2.2}$$

the right-hand side being independent of the choice of the conditional expectation by the uniqueness part of the existence and uniqueness theorem.

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 $P^{\mathrm{ag}}(\Omega_{0})=1$ 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 $\kappa=0$; this is the case whenever $s=0$, because $\Pi_{0}$ is the zero matrix by (I0). Then (2.1) gives, for every $N\ge1$,

$$\mathbb{E}\Bigl[\sum_{\gamma,\delta}c^{\gamma}c^{\delta}\varepsilon^{\gamma}_{s}\varepsilon^{\delta}_{s}\Bigr]\ \ge\ 0=\kappa\ \ge\ \kappa-\eta ,$$

and $N_{3}=1$ serves.

Suppose now $\kappa>0$; then $s>0$, since $\kappa=0$ when $s=0$. Put

$$\epsilon=\min\Bigl\{\frac{\kappa}{8},\ \frac{\sqrt{\kappa}}{8},\ \frac{\eta}{3+4\sqrt{\kappa}}\Bigr\}>0,\qquad q=\frac{\kappa-\epsilon}{\sqrt{\kappa+\epsilon}}>0$$

(positive because $\epsilon\le\kappa/8<\kappa$), with $\sqrt{\cdot}$ the nonnegative square root, which is nondecreasing on $[0,\infty)$ (for $0\le x\le y$ one has $\sqrt{x}\le\sqrt{y}$, since $\sqrt{x}>\sqrt{y}\ge0$ would give $x>y$ on squaring). Three elementary estimates follow from $0<\epsilon\le\kappa/8$ and $\epsilon\le\sqrt{\kappa}/8$. 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, dividing by $\kappa+\epsilon>0$, $q^{2}\ge\kappa-3\epsilon$. Second, $\epsilon\le3\kappa$ gives $\epsilon^{2}\le3\kappa\epsilon$, hence $(\kappa-\epsilon)^{2}=\kappa^{2}-2\kappa\epsilon+\epsilon^{2}\le\kappa^{2}+\kappa\epsilon=\kappa(\kappa+\epsilon)$, so $q^{2}\le\kappa$ and $q\le\sqrt{\kappa}$. Third, $(2\sqrt{\kappa})^{2}=4\kappa\ge\kappa+\epsilon$, so $2\sqrt{\kappa}\ge\sqrt{\kappa+\epsilon}$, whence, using $\kappa-\epsilon\ge7\kappa/8$ and $\kappa/\sqrt{\kappa}=\sqrt{\kappa}$,

$$q\ \ge\ \frac{\kappa-\epsilon}{2\sqrt{\kappa}}\ \ge\ \frac{7\kappa}{16\sqrt{\kappa}}\ =\ \frac{7}{16}\sqrt{\kappa}\ \ge\ 2\epsilon ,$$

the last step because $2\epsilon\le\sqrt{\kappa}/4\le\tfrac{7}{16}\sqrt{\kappa}$ by $\epsilon\le\sqrt{\kappa}/8$. Now put $\varkappa=(q-\epsilon)^{2}$; since $q-\epsilon\ge\epsilon>0$, the uniqueness of the nonnegative square root gives $\sqrt{\varkappa}=q-\epsilon\ge\epsilon$, and $\varkappa>0$. Apply (VT$'$) with this $s$, this $c$, this profile $\lambda$ and this $\epsilon$, obtaining $N_{2}$; put $N_{3}=N_{2}$, let $N\ge N_{3}$, and let $(\Omega',\mathcal{F}',P')$, $\mathsf{Y}$, $\mathcal{Y}$, $\varrho_{0}$, $d$, $\vartheta_{1},\dots,\vartheta_{d}$, $\mathsf{D}'$, $X^{\dagger}$, $p$, $G$, $\alpha$, $z$, $a_{1},\dots,a_{d}$ be data as provided by (VT$'$), satisfying its conditions (a), (C1), (C2$'$), (C3) and (C4$'$). Write $\mathbb{E}'$ for the expectation on $(\Omega',\mathcal{F}',P')$.

Transfer of the filtering error to the certificate space. By (a), the map $(c\cdot X'_{s},W^{(s)})$ is measurable with respect to $\mathcal{F}^{\mathrm{ag}}$ and $\mathcal{B}(\mathbb{R})\otimes\mathcal{R}_{s}$. Hence $W^{(s)}$ is measurable with respect to $\mathcal{F}^{\mathrm{ag}}$ and $\mathcal{R}_{s}$: for $E\in\mathcal{R}_{s}$ the set $\mathbb{R}\times E$ belongs to the product $\sigma$-algebra $\mathcal{B}(\mathbb{R})\otimes\mathcal{R}_{s}$, being a product of measurable sets, and its preimage under $(c\cdot X'_{s},W^{(s)})$ is $(W^{(s)})^{-1}(E)$. In the same way $\mathsf{D}'$, regarded as $\mathbf{R}_{s}$-valued, is measurable with respect to $\mathcal{F}'$ and $\mathcal{R}_{s}$, by the first assertion of (a). Put $\mathcal{G}'=\sigma(\mathsf{D}')=\{\mathsf{D}'^{-1}(E):E\in\mathcal{R}_{s}\}$. By claim 1 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 on $(\Omega',\mathcal{F}',P')$ with the measurable space $(\mathbf{R}_{s},\mathcal{R}_{s})$ and the map $\mathsf{D}'$, $\mathcal{G}'$ is a $\sigma$-algebra on $\Omega'$ with $\mathcal{G}'\subseteq\mathcal{F}'$; it is $\mathsf{D}'$-generated up to null sets in the sense of that lemma, since $\sigma(\mathsf{D}')\subseteq\mathcal{G}'\subseteq\mathcal{F}'$ and every $A\in\mathcal{G}'$ admits $A'=A\in\sigma(\mathsf{D}')$, whose symmetric difference with $A$ is empty. By (C0), $\mathcal{G}_{s}$ is $W^{(s)}$-generated up to null sets in $(\Omega^{\mathrm{ag}},\mathcal{F}^{\mathrm{ag}},P^{\mathrm{ag}})$ for the same measurable space $(\mathbf{R}_{s},\mathcal{R}_{s})$. Claim 5 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 therefore applies with $(\Omega^{\mathrm{ag}},\mathcal{F}^{\mathrm{ag}},P^{\mathrm{ag}})$, the measurable space $(\mathbf{R}_{s},\mathcal{R}_{s})$, the map $W^{(s)}$ in the role of $\mathsf{D}$, the square-integrable random variable $c\cdot X'_{s}$ (square-integrable by (a)) in the role of $X$ and $\mathcal{G}_{s}$ in the role of $\mathcal{G}$ on one side, and $(\Omega',\mathcal{F}',P')$, $\mathsf{D}'$, $X^{\dagger}$ in the role of $X'$ and $\mathcal{G}'$ on the other; its hypothesis on the image measures is the law identity of (a). It gives

$$\mathbb{E}\Bigl[\bigl(c\cdot X'_{s}-\mathbb{E}[c\cdot X'_{s}\mid\mathcal{G}_{s}]\bigr)^{2}\Bigr]=\mathbb{E}'\Bigl[\bigl(X^{\dagger}-\mathbb{E}'[X^{\dagger}\mid\mathcal{G}']\bigr)^{2}\Bigr]. \tag{2.3}$$

The certificate on the copy. By (C1), $\mathcal{G}'=\{\mathsf{D}'^{-1}(E):E\in\mathcal{Y}\}$. Apply Localized Filtering Lower Bound from a Mixture-Weight van Trees Certificate on the probability space $(\Omega',\mathcal{F}',P')$ with the sub-$\sigma$-algebra $\mathcal{G}'$, with $l$ there equal to $1$, the tuple $(X^{\dagger})$ in the role of $X$, the vector $(1)\in\mathbb{R}^{1}$ in the role of its vector $c$, the event $\mathcal{H}=\Omega'$ (which lies in $\mathcal{G}'$, being $\mathsf{D}'^{-1}(\mathbf{R}_{s})$, and has $\mathbf{1}_{\mathcal{H}}\equiv1$), the tolerance $\epsilon$ and the value $\varkappa$, and with the certificate data $d$, $n=d$, $(\mathsf{Y},\mathcal{Y})$, $\varrho_{0}$, $\mathsf{D}'$ in the role of $\mathsf{D}$ (measurable with respect to $\mathcal{F}'$ and $\mathcal{Y}$ by (VT$'$)), $\vartheta_{1},\dots,\vartheta_{d}$ in the role of $\Theta_{1},\dots,\Theta_{d}$, the function $p$, the vectors $\alpha$, $z$, $a_{1},\dots,a_{d}$ and the map $G$. Its condition (C1) asks that every $\mathcal{G}'$-measurable square-integrable random variable be almost surely equal to $g(\mathsf{D}')$ for some $\mathcal{Y}$-measurable $g$ with $g(\mathsf{D}')$ square-integrable; this is given by (C1) of (VT$'$), which even provides equality at every point. Its condition (C2$'$) is (C2$'$) of (VT$'$), and the mixture-weight information $\mathcal{I}_{z}$ of the two conditions is the same number. Its condition (C3) reads $\lVert\alpha\cdot\vartheta+G(\mathsf{D}')-\mathbf{1}_{\Omega'}\,X^{\dagger}\rVert_{2}\le\epsilon$, which is (C3) of (VT$'$). For its condition (C4$'$): by (C4$'$) of (VT$'$) and the identities $c\cdot\psi_{\lambda}(s)=\kappa$ and $\mathcal{A}_{s}(\lambda)=\kappa$ established above,

$$\alpha\cdot z\ \ge\ \kappa-\epsilon\ >\ 0,\qquad \mathcal{I}_{z}\ \le\ \kappa+\epsilon,\qquad \max_{1\le j\le d}|\alpha\cdot a_{j}|\ \le\ \epsilon ;$$

the nonnegative square root being nondecreasing, $\sqrt{\mathcal{I}_{z}}\le\sqrt{\kappa+\epsilon}$; the factor $\max_{j}|\alpha\cdot a_{j}|+\sqrt{\varkappa}$ is nonnegative and at most $\epsilon+(q-\epsilon)=q$, and $\sqrt{\kappa+\epsilon}\ge0$; hence

$$\sqrt{\mathcal{I}_{z}}\,\Bigl(\max_{1\le j\le d}|\alpha\cdot a_{j}|+\sqrt{\varkappa}\Bigr)\ \le\ \sqrt{\kappa+\epsilon}\,\bigl(\epsilon+(q-\epsilon)\bigr)\ =\ \sqrt{\kappa+\epsilon}\;q\ =\ \kappa-\epsilon\ \le\ \alpha\cdot z ,$$

which together with $\alpha\cdot z>0$ is condition (C4$'$) of the lemma with this $\varkappa$. Moreover $\sqrt{\varkappa}\ge\epsilon$ was shown above.

Claim 2 of Localized Filtering Lower Bound from a Mixture-Weight van Trees Certificate is therefore available. Its claim 1, with $\mathbf{1}_{\Omega'}\equiv1$, identifies the quantity bounded there with $\mathbb{E}'[(X^{\dagger}-M')^{2}]$ for a conditional expectation $M'$ of $X^{\dagger}$ given $\mathcal{G}'$, and the lemma records that this quantity does not depend on the choice of $M'$; so its claim 2 gives

$$\mathbb{E}'\Bigl[\bigl(X^{\dagger}-\mathbb{E}'[X^{\dagger}\mid\mathcal{G}']\bigr)^{2}\Bigr]\ \ge\ \bigl(\sqrt{\varkappa}-\epsilon\bigr)^{2}=(q-2\epsilon)^{2}=q^{2}-4\epsilon q+4\epsilon^{2}\ \ge\ q^{2}-4\epsilon q\ \ge\ \kappa-3\epsilon-4\epsilon\sqrt{\kappa}\ \ge\ \kappa-\eta,$$

the third-to-last step by $\epsilon^{2}\ge0$, the second-to-last by the first two elementary estimates ($q^{2}\ge\kappa-3\epsilon$ and $q\le\sqrt{\kappa}$), and the last by the choice $\epsilon\le\eta/(3+4\sqrt{\kappa})$. Combining this with (2.3), (2.2) and (2.1) yields the bound of claim 2 for every $N\ge N_{3}$. 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

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

(in this claim the letter $p$ denotes these probabilities; the certificate density of (VT$'$) is not used here). 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 $\Omega^{\mathrm{ag}}\setminus\mathcal{T}_{k}(s)$ 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 over Adjacent Intervals, Subintervals and Oriented Integrals, and Integrals of Restrictions), 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.