Moment Bounds for the Aggregate Compensated Counters of the Controlled N-Agent Dynamics

Adopt the setting of the \reftext{lem:n-agent-compensated-martingales-2026a}{compensated counters of the controlled $N$-agent dynamics} with $N$ agents, $l$ states, $\tilde{l}$ observation channels, and control dimension $m$: a \reftext{def:transition-rate-family-2026a}{transition-rate family} $\beta$ with rate bound $B$, an \reftext{def:observation-rate-family-2026a}{observation-rate family} $\tilde{\beta}$ with rate bound $\tilde{B}$, 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 $\Omega_0$, counters $N^{i,\sigma\gamma}_t$, consumed clock times $A^{i,\sigma\gamma}_t$, and system filtration $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$, and the compensated counters $M^{i,\sigma\gamma}_t=N^{i,\sigma\gamma}_t-A^{i,\sigma\gamma}_t$ of that lemma. Write $\mathbb{E}$ for the \reftext{def:expectation-variance-2026a}{expectation}. For each ordered pair $(\sigma,\gamma)$ of distinct states define the \textbf{aggregate counter}, the \textbf{aggregate consumed clock time}, and the \textbf{aggregate compensated counter}

the fraktur $\mathfrak{A}^{\sigma\gamma}$ is distinct from the control energy $\mathcal{A}$ of the \reftext{lem:fluctuation-state-moment-bound-2026a}{a priori second-moment bound}. Then, for every ordered pair $(\sigma,\gamma)$ of distinct states:

\textbf{(a) (Martingale, counting-path, and compensator properties.)} $(\mathfrak{M}^{\sigma\gamma}_t)_{t\in[0,T]}$ is a \reftext{def:square-integrable-martingale-2026a}{square-integrable martingale} with respect to $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$ with $\mathfrak{M}^{\sigma\gamma}_0=0$; \reftext{def:almost-surely-2026a}{almost surely}, $t\mapsto\mathcal{N}^{\sigma\gamma}_t$ agrees on $[0,T]$ with a \reftext{def:counting-path-2026a}{counting path}; and at every $\omega\in\Omega$ and all $0\le r\le t\le T$,

\textbf{(b) (Second and fourth moments.)} For every $t\in[0,T]$, the powers $(\mathfrak{M}^{\sigma\gamma}_t)^2$ and $(\mathfrak{M}^{\sigma\gamma}_t)^4$ are integrable, and

\textbf{(c) (Aggregate state martingale.)} Let $M_t=(M^\gamma_t)_{\gamma\in\{1,\dots,l\}}$ be as in the \reftext{thm:n-agent-martingale-decomposition-2026a}{martingale decomposition} (the superscript shapes distinguish the per-clock compensated counters $M^{i,\sigma\gamma}$, the aggregate compensated counters $\mathfrak{M}^{\sigma\gamma}$, and the state martingales $M^\gamma$), and write $|\cdot|$ for the Euclidean norm (\reftext{def:euclidean-distance-rn-2026a}{Euclidean distance} to the origin) and $\sqrt{N}$ for the \reftext{thm:nonnegative-real-has-unique-square-root-2026a}{nonnegative square root}. Then, almost surely, for every $\gamma\in\{1,\dots,l\}$ and every $t\in[0,T]$,

the sum running over $\sigma\in\{1,\dots,l\}$ with $\sigma\neq\gamma$; and consequently, for every $t\in[0,T]$,