Compact iid empirical convergence and propagation of chaos

For any fixed $\nu \in {\mathcal{P}}_1(Z)$, the map $z\mapsto {\mathcal{W}}_1(N^{-1}{\sum }_i{\delta }_{z_i},\nu )$ is continuous. Indeed, split a coupling's mass at each repeated atom equally among its labels, and then move label $i$ from $z_i$ to $z_i^{\mathrm{\prime }}$ while keeping its target. The cost increases by at most $N^{-1}{\sum }_i{d}_Z(z_i,z_i^{\mathrm{\prime }})$. Taking infima and reversing the roles gives the same bound for the absolute difference of the two Wasserstein values. Thus the initial Wasserstein error is a measurable random variable, bounded by $D_K$ on $K^N$. For a finite Borel partition $\mathcal{P}=\{C_1,\dots ,C_M\}$ of $K$ with cell diameters at most $\delta$, put $p_j={\mu }_0(C_j)$ and $P_{N,j}=N^{-1}{\sum }_{i=1}^N{\mathbf{1}}_{C_j}(Z_i(0))$. A coupling matches the common mass $\min (p_j,P_{N,j})$ inside each cell at cost at most $\delta$; the leftover mass is $r=\frac{1}{2}{\sum }_j\mathrm{\mid }{P}_{N,j}-p_j\mathrm{\mid }$ and can be coupled across $K$ at cost at most $D_{K}r$. Product measures in [F6] construct these within-cell and residual couplings (if $r=0$, no residual coupling is needed). Hence ${\mathcal{W}}_1({\mu }_N(0),{\mu }_0)\le \delta +(D_K\mathrm{/}2){\sum }_j\mathrm{\mid }{P}_{N,j}-p_j\mathrm{\mid }$. Independence [F1] gives $\mathbb{E}\mathrm{\mid }{P}_{N,j}-p_j{\mathrm{\mid }}^2=p_j(1-p_j)\mathrm{/}N$, so [F7] gives $\mathbb{E}\mathrm{\mid }{P}_{N,j}-p_j\mathrm{\mid }\le \sqrt{p_j\mathrm{/}N}$; therefore the sum of expected discrepancies is at most $\sqrt{M\mathrm{/}N}$. Partition a bounded velocity box containing $K$ and the position torus into rectangular cells of side at most a fixed multiple of $\delta$; intersect with $K$ and disjointize the cover. There are at most $C_K{\delta }^{-2d}$ cells, each with $d_Z$-diameter at most $\delta$. This proves the expected initial bound in the Statement.

For each ${\mu }_0$, take every coordinate space to be $(Z,\mathcal{B}(Z))$ and set every history-dependent kernel to $K_n(z_0,\dots ,z_n,A)={\mu }_0(A)$. For each fixed $A$ this is constant, hence measurable, and it is a probability measure in $A$. The Ionescu--Tulcea theorem [F2], under full Choice, supplies a probability on the countable product whose finite-prefix laws are the iterated products of ${\mu }_0$. Reindexing coordinates from 0 to 1 gives the sequence in the Statement; its common marginals and product finite-prefix laws make its coordinates mutually independent and identically distributed by [F1].

For a centered Bernoulli variable $Y={\mathbf{1}}_{C_j}(Z_i)-p_j$, $\mathbb{E}{Y}^2=p_j(1-p_j)$ and $\mathbb{E}{Y}^4\le p_j$. Expanding the fourth power of the sum of independent centered variables leaves only terms with four equal indices or two equal pairs, so $\mathbb{E}\mathrm{\mid }{P}_{N,j}-p_j{\mathrm{\mid }}^4=N^{-4}(N\mathbb{E}{Y}^4+3N(N-1)(\mathbb{E}{Y}^2{)}^2)\le 4N^{-2}\mathrm{.}$ For every rational $\epsilon >0$, the elementary bound $\mathbb{P}(\mathrm{\mid }{P}_{N,j}-p_j\mathrm{\mid }>\epsilon )\le {\epsilon }^{-4}\mathbb{E}\mathrm{\mid }{P}_{N,j}-p_j{\mathrm{\mid }}^4$ makes ${\sum }_N\mathbb{P}(\mathrm{\mid }{P}_{N,j}-p_j\mathrm{\mid }>\epsilon )<\mathrm{\infty }$. By countable subadditivity, the probability of such deviations occurring for some $N\ge n$ is bounded by the tail of that convergent series, which tends to zero; hence for this rational $\epsilon$ the deviations occur only finitely often almost surely. Intersect over rational $\epsilon >0$, over the finitely many cells of each of a countable sequence of partitions with mesh ${\delta }_{\mathrm{\ell }}\downarrow 0$. The coupling bound in step 1.1 then gives ${\mathrm{lim\hspace{0.17em}sup}}_N{\mathcal{W}}_1({\mu }_N(0),{\mu }_0)\le {\delta }_{\mathrm{\ell }}$ for every $\mathrm{\ell }$, so the initial Wasserstein distance tends to zero almost surely.

Step 1.1 and $M_{\delta }\le C_K{\delta }^{-2d}$ give $\mathbb{E}{\mathcal{W}}_1({\mu }_N(0),{\mu }_0)\le \delta +(D_K\sqrt{C_K}\mathrm{/}2){\delta }^{-d}{N}^{-1\mathrm{/}2}$. Taking $\delta =N^{-1\mathrm{/}(2d+2)}$ bounds both terms by constants times that power of $N$. Step 2.1 gives almost-sure convergence. To identify the trajectories used by [F4] and [F5], let ${\mathrm{\Xi }}_N:Z^N\to (T^{\ast }{\mathbb{T}}_L^d{)}^N$ be ${\mathrm{\Xi }}_N((x_i,v_i{)}_{i=1}^N)=((L{x}_i<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>L,P_0{v}_i{)}_{i=1}^N)$ and set ${\tau }_0=mL\mathrm{/}{P}_0$, $k(x)=(mL\mathrm{/}{P}_0^2)K(Lx)$ as in [F3]. For the first $N$ iid initial coordinates, define $(Z_i(s){)}_{i=1}^N:={\mathrm{\Xi }}_N^{-1}\text{\hspace{0.17em}⁣}\left({\mathrm{\Phi }}_{{\tau }_{0}s}^N\right({\mathrm{\Xi }}_N((Z_i(0){)}_{i=1}^N)))={\left((q_i({\tau }_{0}s)\mathrm{/}L,p_i({\tau }_{0}s)\mathrm{/}{P}_0)\right)}_{i=1}^N$, where ${\mathrm{\Phi }}_t^N$ is the physical flow from [F4]. This is its conjugate in the dimensionless coordinates $Z={\mathbb{T}}^d\times {\mathbb{R}}^d$ and time $s$; differentiating and using the model equations gives ${\dot{X}}_i=V_i$ and ${\dot{V}}_i=N^{-1}{\sum }_{j\mathrm{\ne }i}k(X_i-X_j)$. Since $k(0)=0$, ${\mu }_N(s)=N^{-1}{\sum }_i{\delta }_{Z_i(s)}$ is the empirical characteristic evolution to which [F5] applies. The coordinate scaling commutes with particle permutations, so the exchangeability used below follows from [F4] and the iid initial law. On $[-T,T]$ the particle trajectories depend continuously on time and their initial vector by [F4]. The characteristics carrying ${\mu }_0$ have uniformly bounded velocities there because $K$ is compact and $k$ is bounded. For two times $s,t$, transport the target of any empirical-to-${\mu }_s$ coupling by the characteristic map from $s$ to $t$, and move its labelled source atoms from $Z_i(s)$ to $Z_i(t)$. The cost changes by at most $N^{-1}{\sum }_i{d}_Z(Z_i(s),Z_i(t))+\int d_Z({\chi }_{s}z,{\chi }_{t}z)d{\mu }_0(z)$, which tends to zero as $t\to s$; reversing times proves continuity of the error. At each fixed time its dependence on the initial vector is continuous by the labelled-coupling bound above and [F4]. Its supremum over $[-T,T]$ therefore equals the supremum over a countable dense set of times and is measurable. Applying [F5] at every $\mathrm{\mid }s\mathrm{\mid }\le T$ yields the supremum bound with factor $e^{CT}$, both in expectation and almost surely.

The joint law of the finite particle vector is permutation invariant by [F4] and iid symmetry [F1]. Given the finite-particle configuration at time $s$, draw $k$ indices independently and uniformly from $\{1,\dots ,N\}$. The resulting tuple has conditional law ${\mu }_N(s{)}^{\otimes k}$. The chance of an index collision is at most $\left(\genfrac{}{}{0px}{}{k}{2}\right)\mathrm{/}N$ by a union bound. After averaging over the exchangeable particle vector, conditioning the index sampling on no collision makes the tuple's law equal to that of the first $k$ particles. Thus the expectation of $\mathrm{\Psi }$ under the true $k$-law and under ${\mu }_N(s{)}^{\otimes k}$ differs by at most $2\mathrm{\parallel }\mathrm{\Psi }{\mathrm{\parallel }}_{\mathrm{\infty }}\left(\genfrac{}{}{0px}{}{k}{2}\right)\mathrm{/}N$. For any coupling $\pi$ of ${\mu }_N(s)$ and ${\mu }_s$, its $k$-fold product from [F6] couples their product laws, and the sum-metric Lipschitz bound gives a test-integral difference at most $k\mathrm{Lip}(\mathrm{\Psi })\int d_{Z}d\pi$. Taking the infimum over $\pi$, then expectation, and using [F5] gives the second term in the displayed bound. The estimate tends to zero uniformly for $\mathrm{\mid }s\mathrm{\mid }\le T$, establishing fixed-$k$ propagation of chaos in the stated test class.