Proof 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 $1-O(N^{-1/2})$

Applicability of the record-frozen closeness-set lemma. Fix $N\ge1$. The data of the $N$-th solution are an instance of the setting of the record-frozen closeness-set lemma: the natural numbers $N\ge1$, $l\ge2$, $\tilde{l}\ge1$, $m\ge1$ are those of the common data; $(\beta_{0},\beta_{1})$ is an affine-controlled transition-rate family on $l$ states with control set $\mathcal{A}$ and Lipschitz constant $\Lambda$, with $\mathcal{A}$ nonempty, convex and compact, and $R=\sup_{\alpha\in\mathcal{A}}|\alpha|$ is its control bound; $\beta$ is its transition-rate family, $\tilde{\beta}$ the observation-rate family, $T$ the horizon, $(\Omega,\mathcal{F},P)$ the driving system, $h^{(N)}$ an $\mathcal{A}$-valued observation-driven control policy with horizon $T$, control dimension $m$ and $\tilde{l}$ channels, and the $N$-th solution is a solution on $[0,T]$ for these data with regular event $\Omega_{0}$ and observation record $W$; the family $(\tau_{j},\upsilon_{j})_{j\ge1}$ and the realized control $\hat{\alpha}$ are those fixed in the Data, formed from claims 1 and 2 of the realized-control lemma with the same point $a_{0}\in\mathcal{A}$ and the same dense sequence fixed in the setting of that lemma as in the control- and flow-closeness lemma (neither enters the conclusions used here, which concern points of $\Omega_{0}$). As comparison data take $S^{*}=S$ and $K^{*}=1$: the components of $S$ are continuous on $[0,T]$ by condition 1 of the definition of a mean-field trajectory pair, and $|S_{t}|\le1$ because $S_{t}$ lies in the probability simplex, whose points have coordinates in $[0,1]$ summing to $1$, so that $(S^{\gamma}_{t})^{2}\le S^{\gamma}_{t}$ for each $\gamma$ and $|S_{t}|^{2}=\sum_{\gamma}(S^{\gamma}_{t})^{2}\le\sum_{\gamma}S^{\gamma}_{t}=1$, whence $|S_{t}|\le1$ as $|S_{t}|\ge0$; and the map $A:[0,T]\to\mathcal{A}$ has $\mathcal{B}_{[0,T]}$-measurable components as recorded in the Data. As path data take $E=\{0_{\mathbb{R}^{l}}\}$, the origin of $\mathbb{R}^{l}$ (a nonempty finite subset of $\mathbb{R}^{l}$); the path data enter only claims 3 and 4 of that lemma, which are not used here. Hence claims 1 and 2 of the record-frozen closeness-set lemma hold for the $N$-th solution, with the control discrepancy $\mathsf{d}$ of the Data.

Applicability of the envelope lemma. The setting of the pre-stopping envelope lemma consists of the setting of the extended good-set stopping-time lemma, the setting of the pathwise tracking lemma for the same data, policy, solution and realized control (only claims 1 and 2 of which are used there, its hypothesis (LipC) not being assumed and its population cost data being arbitrary), a mean-field trajectory pair $(S,A)$ whose control is the map $A$ of the extended good-set setting and whose initial state is $S_{0}=x_{0}$, the stipulation $S^{*}=S$, and a barrier $\varepsilon_{Y}>0$. All of this is available for the $N$-th solution: the extended good-set setting determined by the present data (the same $\hat{\alpha}$, $x_{0}$, $\Phi$, $Y$, $A$, $\mathcal{E}$ and $S^{*}=S$ as in the control- and flow-closeness lemma, none of which involves the four thresholds) is instantiated here with $\varepsilon_{Y}=c_{\mathcal{E}}=\delta=\theta_{\mathrm{out}}=1$; the objects and conclusions of the envelope lemma used below (its claims 2, 4 and 6) involve the thresholds only through the shift $\overline{\mathfrak{b}}=\varepsilon_{Y}+Q$, which is undone below, the parts of its claim 2 that retain $\varepsilon_{Y}$ (the bounds on the second and fourth moments of $\overline{\mathfrak{b}}$) not being used; the setting of the tracking lemma is part of the standing hypotheses of the lower bound theorem, which even assumes its hypothesis (LipC); $(S,A)$ is a mean-field trajectory pair for $\beta$ with horizon $T$ by the definition of a stationary mean-field triple, its control is the map $A$ of the setting, and $S_{0}=x_{0}$ by assumption; and we take $\varepsilon_{Y}=1$. The number $\kappa_{0}=1+\mathbb{E}[|\mathfrak{s}_{0}|^{4}]$ of the envelope lemma is the number so named in the lower bound theorem, the state fluctuation $\mathfrak{s}_{t}=\sqrt{N}(\Sigma_{t}-S_{t})$ being formed from the same trajectory pair. By the definitions in the envelope lemma, $Q=\overline{\mathfrak{b}}-\varepsilon_{Y}=\overline{M}+\Lambda_{b}e^{\Lambda_{b}T}I+e^{\Lambda_{b}T}N^{-1/2}|\mathfrak{s}_{0}|$, which does not involve $\varepsilon_{Y}$, and $c_{Q}$ is the constant displayed in the statement.

Proof of claim 1. Fix $N\ge1$. By claim 2 of the envelope lemma, $\overline{\mathfrak{b}}$ is a random variable, hence so is $Q=\overline{\mathfrak{b}}-1$ (claims 1 and 2 of the arithmetic lemma for measurable functions, the constant function $-1$ being measurable), and $\mathbb{E}[Q^{4}]\le c_{Q}\kappa_{0}N^{-2}$. Hypothesis (I$'$) gives $\kappa_{0}\le\kappa^{\sharp}$, hence $\mathbb{E}[Q^{4}]\le c_{Q}\kappa^{\sharp}N^{-2}$, because $N^{-2}>0$ and $c_{Q}\ge0$, being a sum of products of the nonnegative numbers $c_{M}$, $\kappa_{T}$, $\Lambda_{b}^{4}$, $T^{4}$ and values of the exponential function. For real $\theta>0$, claim 4 of the envelope lemma gives $P(\overline{\mathfrak{b}}\ge\varepsilon_{Y}+\theta)\le c_{Q}\kappa_{0}\theta^{-4}N^{-2}\le c_{Q}\kappa^{\sharp}\theta^{-4}N^{-2}$, and $\{\overline{\mathfrak{b}}\ge\varepsilon_{Y}+\theta\}=\{Q\ge\theta\}$ since $Q=\overline{\mathfrak{b}}-\varepsilon_{Y}$. Finally, claim 6 of the envelope lemma states that $|\mathfrak{s}_{t}(\omega)|\le\sqrt{N}\,(Y_{t}(\omega)+Q(\omega))$ for every $\omega\in\Omega_{0}$ and $t\in[0,T]$, where $Y_{t}(\omega)=|\Phi_{t}(\omega)-S^{*}_{t}|=|\Phi_{t}(\omega)-S_{t}|$ is the deviation of the extended good-set setting with $S^{*}=S$. Since $|\mathfrak{s}_{t}(\omega)|=\sqrt{N}\,|\Sigma_{t}(\omega)-S_{t}|$ and $\sqrt{N}>0$, dividing by $\sqrt{N}$ gives $|\Sigma_{t}(\omega)-S_{t}|\le|\Phi_{t}(\omega)-S_{t}|+Q(\omega)$. (The same bound follows from the noise-majorant lemma, whose setting is that of the envelope lemma and which gives $|\Sigma_{t}(\omega)-\Phi_{t}(\omega)|\le Q(\omega)$ on $\Omega_{0}$, together with the triangle inequality of claim 6 of the elementary properties of the Euclidean norm.)

Proof of claim 2. Existence of $N_{2}$. By clause 1 of the Archimedean property there is a natural number $N'\ge C_{\mathrm{esc}}^{2}$, and the larger of $N'$ and $N_{1}$ is a natural number $N_{2}$ with $N_{2}\ge N_{1}$ and $N_{2}\ge C_{\mathrm{esc}}^{2}$.

Applicability of claim 4 of the control- and flow-closeness lemma. Fix $N\ge N_{2}$ and put $\varepsilon_{\flat}=N^{-1/2}$ and $N_{\flat}=N$. Then $0<\varepsilon_{\flat}\le1$ because $N\ge1$ gives $\sqrt{N}\ge1$; $N_{\flat}=N\ge N_{2}\ge N_{1}$; and $\varepsilon_{\flat}N_{\flat}=N/\sqrt{N}=\sqrt{N}\ge\sqrt{C_{\mathrm{esc}}^{2}}=|C_{\mathrm{esc}}|\ge C_{\mathrm{esc}}$, because $N\ge C_{\mathrm{esc}}^{2}$ and the nonnegative square root is nondecreasing (if $0\le a\le b$ then $\sqrt{a}\le\sqrt{b}$, since $\sqrt{a}>\sqrt{b}$ would give $a>b$), and $\sqrt{C_{\mathrm{esc}}^{2}}=|C_{\mathrm{esc}}|$ as $|C_{\mathrm{esc}}|$ is nonnegative with square $C_{\mathrm{esc}}^{2}$. Hence claim 4 of that lemma applies to the $N$-th solution with these $\varepsilon_{\flat}$ and $N_{\flat}$ (and $N\ge N_{\flat}$), and furnishes the event $\Omega^{\flat}_{N}=\Omega^{\flat}_{N}(N^{-1/2})$, which belongs to $\mathcal{G}_{T}$ and hence to $\mathcal{F}$, with $P(\Omega\setminus\Omega^{\flat}_{N})\le\varepsilon_{\flat}=N^{-1/2}$ and such that, for every $\omega\in\Omega^{\flat}_{N}$ and every $t\in[0,T]$,

using $\varepsilon_{\flat}N=\sqrt{N}$ and $1/\sqrt{N}=N^{-1/2}$. Now $(\sqrt{N})^{1/2}=N^{1/4}$ by the definition of $N^{1/4}$; the nonnegative square root of a product of nonnegative reals is the product of their square roots (both sides being nonnegative with the same square); and $(N^{-1/2})^{1/2}=N^{-1/4}$, since $N^{-1/4}=1/N^{1/4}$ is nonnegative with square $1/(N^{1/4})^{2}=1/\sqrt{N}=N^{-1/2}$. So the right-hand sides are $(TC_{\mathrm{ctl}})^{1/2}N^{-1/4}=\varepsilon_{\mathrm{ctl}}(N)$ and $C_{\mathrm{flw}}N^{-1/4}$ respectively.

$\Omega^{\mathrm{cl}}_{N}$ is an event. $\Omega_{0}\in\mathcal{F}$ by the definition of a solution; $\Omega^{\flat}_{N}\in\mathcal{F}$ as just recalled; and $\{Q<N^{-1/4}\}\in\mathcal{F}$ because $Q$ is a random variable by claim 1. A $\sigma$-algebra is closed under finite intersections, so $\Omega^{\mathrm{cl}}_{N}\in\mathcal{F}$.

The probability bound. A point of $\Omega$ outside $\Omega^{\mathrm{cl}}_{N}$ lies outside $\Omega_{0}$, or outside $\Omega^{\flat}_{N}$, or in $\{Q\ge N^{-1/4}\}$; so $\Omega\setminus\Omega^{\mathrm{cl}}_{N}\subseteq(\Omega\setminus\Omega_{0})\cup(\Omega\setminus\Omega^{\flat}_{N})\cup\{Q\ge N^{-1/4}\}$. Now $P(\Omega_{0})=1$ by the definition of a solution, so $P(\Omega\setminus\Omega_{0})=1-P(\Omega_{0})=0$ by claim 3 of the basic properties of a measure; $P(\Omega\setminus\Omega^{\flat}_{N})\le N^{-1/2}$; and, by claim 1 with $\theta=N^{-1/4}$,

using $(N^{-1/4})^{-4}=1/(N^{-1/4})^{4}=(N^{1/4})^{4}=N$. By the monotonicity and countable subadditivity of the probability (claims 2 and 4 of the basic properties of a measure, the latter applied to the three sets padded by empty sets), $P(\Omega\setminus\Omega^{\mathrm{cl}}_{N})\le0+N^{-1/2}+c_{Q}\kappa^{\sharp}N^{-1}$.

The three bounds. Let $\omega\in\Omega^{\mathrm{cl}}_{N}$ and $t\in[0,T]$. Since $\omega\in\Omega_{0}$, claim 1 gives $|\Sigma_{t}(\omega)-S_{t}|\le|\Phi_{t}(\omega)-S_{t}|+Q(\omega)$; since $\omega\in\Omega^{\flat}_{N}$, $|\Phi_{t}(\omega)-S_{t}|\le C_{\mathrm{flw}}N^{-1/4}$; and $Q(\omega)<N^{-1/4}$ by the definition of $\Omega^{\mathrm{cl}}_{N}$. Adding, $|\Sigma_{t}(\omega)-S_{t}|\le(C_{\mathrm{flw}}+1)N^{-1/4}=\varepsilon_{S}(N)$, the first bound. The second bound, $\int_{[0,t]}|\hat{\alpha}(u,\omega)-A_{u}|\,du\le\varepsilon_{\mathrm{ctl}}(N)$, was recalled above for every $\omega\in\Omega^{\flat}_{N}$. For the third, $\omega\in\Omega_{0}$, so claim 2 of the record-frozen closeness-set lemma, applicable to the $N$-th solution as verified at the start of the proof, gives $\mathsf{d}(W(\omega))=\int_{[0,T]}|\hat{\alpha}(u,\omega)-A_{u}|\,du$, which is the second bound at $t=T$; hence $\mathsf{d}(W(\omega))\le\varepsilon_{\mathrm{ctl}}(N)$. This completes the proof.