Restricted First Moments and Tail Bounds for the Martingale Part of the Empirical State Measure

Adopt the setting of the moment bounds for the aggregate compensated counters with $N$ agents, $l$ states, $\tilde{l}$ observation channels, and control dimension $m$: a transition-rate family $\beta$ with control set $\mathcal{A}$, a nonempty subset of Euclidean space $\mathbb{R}^m$, and rate bound $B$, an observation-rate family $\tilde{\beta}$, a horizon $T>0$, an $N$-agent driving system $(\Omega,\mathcal{F},P)$, an observation-driven control policy $h$ which is $\mathcal{A}$-valued, and a solution on $[0,T]$ with regular event $\Omega_0$, empirical state measure $\Sigma_t$, and control $\alpha_t$. Let $b$ be the aggregate state drift of $\beta$ and let $M_t=(M^\gamma_t)_{\gamma\in\{1,\dots,l\}}$ be the martingale part of the martingale decomposition of the empirical state measure, so that $M^\gamma_t=\Sigma^\gamma_t-\Sigma^\gamma_0-\int_{[0,t]}\mathbf{1}_{\Omega_0}\,b^\gamma(\Sigma_s,\alpha_s)\,ds$ at every point of $\Omega$, where $\mathbf{1}_{D}$ denotes the function equal to $1$ on a set $D$ and $0$ off it. Write $|\cdot|$ for the Euclidean norm (Euclidean distance to the origin), $\mathbb{E}$ for the expectation, $\int_{[0,t]}\cdot\,ds$ for the Lebesgue integral over the compact interval $[0,t]$, and let

$$c_M=6\,l^2\big(2(l-1)\big)^4,\qquad \kappa_T=BT+(BT)^2,\qquad K_M=2+2(l-1)BT .$$

For a nonnegative real $x$, $x^{1/4}$ denotes the nonnegative fourth root, the nonnegative square root applied twice, $x^{3/4}=(x^{1/4})^3$, and for $x>0$, $x^{-1/2}$ is the reciprocal of the nonnegative square root; by the uniqueness of nonnegative square roots, the nonnegative square root, and hence also $x\mapsto x^{1/4}$, is multiplicative and nondecreasing on $[0,\infty)$.

(a) (A Hölder-type inequality.) Let $X$ be a random variable on $(\Omega,\mathcal{F},P)$ with $X\ge0$ pointwise and $\mathbb{E}[X^4]<\infty$, and let $D\in\mathcal{F}$ be an event (the letter $F$ is avoided for events, being the mean-field cost in companion results). Then $\mathbf{1}_DX$ is integrable and

$$\mathbb{E}\big[\mathbf{1}_D\,X\big]\ \le\ P(D)^{3/4}\,\mathbb{E}\big[X^4\big]^{1/4}.$$

(b) (Measurability and fourth moments.) For every $\gamma$ the map $(t,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)M^\gamma_t(\omega)$ on $[0,T]\times\Omega$ is measurable with respect to the product $\sigma$-algebra of the trace Borel $\sigma$-algebra on $[0,T]$ and $\mathcal{F}$, and $|\mathbf{1}_{\Omega_0}M^\gamma_t|\le K_M$ everywhere. For every $\omega\in\Omega$ the path $t\mapsto\mathbf{1}_{\Omega_0}(\omega)|M_t(\omega)|$ is measurable and bounded on $[0,T]$, and

$$I(\omega)=\int_{[0,T]}\mathbf{1}_{\Omega_0}(\omega)\,|M_t(\omega)|\,dt$$

defines a random variable with $0\le I\le\sqrt{l}\,K_MT$; it equals $\int_{[0,T]}|M_t(\omega)|\,dt$ for $\omega\in\Omega_0$ and $0$ for $\omega\notin\Omega_0$ (for $\omega\in\Omega_0$ it is the quantity $\mathcal{M}_T(\omega)$ of Pointwise-in-Time Tracking of the Mean-Field Flow and Cost along the Realized Control of the Controlled N-Agent Dynamics, when the setting of that lemma is in force). Moreover, for every $t\in[0,T]$,

$$\mathbb{E}\big[|M_t|^4\big]\le c_M\,\kappa_T\,N^{-2}\qquad\text{and}\qquad \mathbb{E}\big[I^4\big]\le T^4\,c_M\,\kappa_T\,N^{-2}.$$

(c) (Restricted first moments.) For every event $D\in\mathcal{F}$ and every $t\in[0,T]$,

$$\mathbb{E}\big[\mathbf{1}_D\,|M_t|\big]\le(c_M\kappa_T)^{1/4}\,N^{-1/2}\,P(D)^{3/4}\qquad\text{and}\qquad \mathbb{E}\big[\mathbf{1}_D\,I\big]\le T\,(c_M\kappa_T)^{1/4}\,N^{-1/2}\,P(D)^{3/4}.$$

(d) (Tail bounds.) For every real $\epsilon>0$ and every $t\in[0,T]$,

$$P\big(|M_t|\ge\epsilon\big)\le c_M\,\kappa_T\,\epsilon^{-4}N^{-2}\qquad\text{and}\qquad P\big(I\ge\epsilon\big)\le T^4\,c_M\,\kappa_T\,\epsilon^{-4}N^{-2}.$$