Mean Deviation Bound for the Aggregate Fluctuation Covariance along a Mean-Field Trajectory Pair

Adopt the setting of the \reftext{def:n-agent-fluctuation-processes-2026a}{fluctuation processes of the controlled $N$-agent dynamics}: a \reftext{def:transition-rate-family-2026a}{transition-rate family} $\beta$ with rate bound $B$ on $l$ states with control dimension $m$, an \reftext{def:observation-rate-family-2026a}{observation-rate family} $\tilde{\beta}$, a horizon $T>0$, an \reftext{def:n-agent-driving-system-2026a}{$N$-agent driving system} $(\Omega,\mathcal{F},P)$, an \reftext{def:observation-driven-control-policy-2026a}{observation-driven control policy} $h$, a \reftext{def:n-agent-controlled-dynamics-2026a}{solution} on $[0,T]$ with regular event $\Omega_0$, empirical state measure $\Sigma_t$, and control $\alpha_t$, a \reftext{def:mean-field-trajectory-pair-2026a}{mean-field trajectory pair} $(S,A)$ for $\beta$ with horizon $T$, and the fluctuation processes $\mathfrak{s}_t=\sqrt{N}(\Sigma_t-S_t)$ and $\mathfrak{a}_t=\sqrt{N}(\alpha_t-A_t)$. Assume that $\beta$ admits a \reftext{def:c2-transition-rate-extension-2026a}{twice continuously differentiable extension} $(U,\bar{\beta})$ with derivative bound $K$. Let $\Theta$ be the \reftext{def:aggregate-fluctuation-covariance-2026a}{aggregate fluctuation covariance} of $\beta$, write $\mathbb{E}$ for the \reftext{def:expectation-variance-2026a}{expectation} and $|\cdot|$ for the Euclidean norm (\reftext{def:euclidean-distance-rn-2026a}{Euclidean distance} to the origin), set $z_s=(\mathfrak{s}_s,\mathfrak{a}_s)\in\mathbb{R}^{l+m}$ under the coordinate identification of the extension definition, and set

Products with an infinite factor are read with the convention $0\cdot\infty=0$ (if $c_\Theta=0$, the rate bound forces $\beta$, and hence $\Theta$ and every left-hand side below, to vanish identically). Then, for all $\gamma,\delta\in\{1,\dots,l\}$:

\textbf{(a) (Pointwise deviation bound.)} At every $\omega\in\Omega$ and every $s\in[0,T]$,

\textbf{(b) (Mean deviation bound.)} For every $s\in[0,T]$, with the expectation $\mathbb{E}[\Theta^{\gamma\delta}(\Sigma_s,\alpha_s)]$ finite, and bounded and \reftext{def:measurable-function-2026a}{measurable} as a function of $s$, by clause (a) of the \reftext{lem:fluctuation-weighted-second-moment-2026a}{weighted second-moment evolution lemma} (applied with the constant matrix family $Z_t=0$, whose densities $\dot{z}^{\gamma\delta}\equiv0$ are continuous), and with the right-hand side valued in $[0,\infty]$,

\textbf{(c) (Integrated deviation bound.)} The function $s\mapsto\Theta^{\gamma\delta}(S_s,A_s)$ is \reftext{def:continuity-closed-interval-c54-2026b}{continuous} on $[0,T]$ — by \reftext{thm:composition-continuous-euclidean-2026a}{continuity of compositions} and of \reftext{thm:sum-product-continuous-real-2026a}{sums and products of continuous functions} along the continuous trajectory pair, each $\beta(\sigma,\sigma',\cdot,\cdot)$, over ordered pairs $\sigma\neq\sigma'$, agreeing on the \reftext{def:probability-simplex-2026a}{probability simplex} product $\Delta^l\times\mathbb{R}^m$ with the continuous extension $\bar{\beta}(\sigma,\sigma',\cdot,\cdot)$ — and hence \reftext{lem:continuous-compact-interval-bounded-2026a}{bounded}, so the function $s\mapsto\big|\mathbb{E}[\Theta^{\gamma\delta}(\Sigma_s,\alpha_s)]-\Theta^{\gamma\delta}(S_s,A_s)\big|$ is measurable and bounded and its \reftext{lem:interval-lebesgue-toolkit-2026a}{Lebesgue integral} over $[0,T]$ is defined and finite. With

(the script $\mathcal{S}$ is distinct from the trajectory $S$), both well defined in $[0,\infty]$ with $\mathcal{S}\le4\,N\,T$ by clause (a) of the \reftext{lem:fluctuation-state-moment-bound-2026a}{a priori second-moment bound} and \reftext{thm:linearity-monotonicity-integral-2026a}{monotonicity of the integral}, one has, in $[0,\infty]$,