## Formalized Name Bracketing Donsker Theorem ## Formalized Statement Let $(S,\mathcal S)$ be a measurable space, let $P$ be a probability measure on $(S,\mathcal S)$, and let $\mathcal F$ be a class of measurable functions $f:S\to\mathbb R$. Suppose that $\mathcal F$ has a measurable envelope $F:S\to[0,\infty]$ satisfying $|f(x)|\le F(x)$ for $P$-a.e. $x\in S$ and every $f\in\mathcal F$, and suppose \begin{align*} \int_S F(x)^2\,dP(x)<\infty. \end{align*} For $\varepsilon>0$, let $N_{[]} (\varepsilon,\mathcal F,L^2(P))$ be the least cardinality of a finite collection of brackets $[l,u]$ covering $\mathcal F$, where $l,u:S\to\mathbb R$ are finite-valued measurable functions in $L^2(P)$, where $l\le u$ $P$-a.e., and where \begin{align*} \left(\int_S (u(x)-l(x))^2\,dP(x)\right)^{1/2}<\varepsilon. \end{align*} Set \begin{align*} H_{[]} (\varepsilon,\mathcal F,L^2(P)):=\log N_{[]} (\varepsilon,\mathcal F,L^2(P)). \end{align*} Assume \begin{align*} \int_{(0,\infty)}\sqrt{H_{[]} (\varepsilon,\mathcal F,L^2(P))}\,d\mathcal L^1(\varepsilon)<\infty. \end{align*} Let $(\Omega,\mathcal A,\mathbb P)$ carry i.i.d. $S$-valued random variables $X_1,X_2,\dots$ with common distribution $P$. For every measurable $h:S\to\mathbb R$ with $h\in L^2(P)$, define \begin{align*} Ph:=\int_S h(x)\,dP(x) \end{align*} and \begin{align*} G_n(h):=n^{-1/2}\sum_{i=1}^{n}\{h(X_i)-Ph\}. \end{align*} Then $\mathcal F$ is $P$-Donsker: the maps $G_n|_{\mathcal F}$ converge in outer distribution in $\ell^\infty(\mathcal F)$ to the tight $P$-Brownian bridge $G_P$, the centred Gaussian process indexed by $\mathcal F$ with covariance \begin{align*} \operatorname{Cov}(G_P(f),G_P(g))=P(fg)-Pf\,Pg \end{align*} for all $f,g\in\mathcal F$. All suprema and weak-convergence assertions are interpreted with outer probability and outer expectation when measurability is not known. ## Proof [proofplan] We prove the theorem by reducing it to the standard bracketing Donsker theorem of Ossiander. The entropy hypothesis first gives finite bracketing covers at all sufficiently small scales and total boundedness in the intrinsic $L^2(P)$ semimetric. We then state the exact external empirical-process theorem used, verify its envelope, measurability, and entropy hypotheses in the present notation, and identify the limiting Gaussian covariance by finite-dimensional central limit theory. [/proofplan] [step:Record the intrinsic semimetric and the effect of brackets] Define the semimetric $d_P:\mathcal F\times\mathcal F\to[0,\infty)$ by \begin{align*} d_P(f,g):=\left(\int_S (f(x)-g(x))^2\,dP(x)\right)^{1/2}. \end{align*} Since $|f|\le F$ $P$-a.e. for every $f\in\mathcal F$ and $F\in L^2(P)$, each $f\in\mathcal F$ belongs to $L^2(P)$. Hence $d_P(f,g)<\infty$ for all $f,g\in\mathcal F$. If $[l,u]$ is a bracket appearing in the definition of $N_{[]} (\varepsilon,\mathcal F,L^2(P))$ and if $f,g\in\mathcal F\cap[l,u]$, then $|f-g|\le u-l$ $P$-a.e. Therefore \begin{align*} d_P(f,g)\le \left(\int_S (u(x)-l(x))^2\,dP(x)\right)^{1/2}. \end{align*} Thus every bracket of $L^2(P)$ width less than $\varepsilon$ has $d_P$-diameter less than $\varepsilon$ on $\mathcal F$. The entropy integral is finite. Hence $H_{[]} (\varepsilon,\mathcal F,L^2(P))<\infty$ for $\mathcal L^1$-a.e. sufficiently small $\varepsilon>0$. Since $\varepsilon\mapsto N_{[]} (\varepsilon,\mathcal F,L^2(P))$ is nonincreasing, this implies $N_{[]} (\varepsilon,\mathcal F,L^2(P))<\infty$ for every sufficiently small $\varepsilon>0$. Choosing one element of $\mathcal F$ from each nonempty bracket in such a cover gives a finite $\varepsilon$-net after quotienting by the relation $d_P(f,g)=0$. Therefore the quotient of $(\mathcal F,d_P)$ is totally bounded. [guided] The semimetric relevant to an empirical process indexed by functions is the $L^2(P)$ distance. We define \begin{align*} d_P(f,g):=\left(\int_S (f(x)-g(x))^2\,dP(x)\right)^{1/2} \end{align*} for $f,g\in\mathcal F$. This quantity is finite because the envelope controls every element of the class: $|f|\le F$ $P$-a.e. and $F\in L^2(P)$ imply $f\in L^2(P)$, and hence $f-g\in L^2(P)$. Now take a bracket $[l,u]$ from the definition of the bracketing number. The endpoint functions are finite-valued, measurable, and belong to $L^2(P)$, and the bracket relation means $l\le h\le u$ $P$-a.e. for every $h\in[l,u]$. If $f,g\in\mathcal F\cap[l,u]$, then both functions lie between the same lower and upper endpoints. Hence $|f-g|\le u-l$ $P$-a.e., and therefore \begin{align*} d_P(f,g)\le \left(\int_S (u(x)-l(x))^2\,dP(x)\right)^{1/2}. \end{align*} So a small bracket is also a small set for the semimetric $d_P$. The entropy integral assumption forces finite bracketing covers at arbitrarily small scales. Indeed, if the bracketing number were infinite on a whole interval $(0,a)$, then the integral of $\sqrt{H_{[]}}$ over that interval would be infinite. Monotonicity in the radius then gives finite covers for every sufficiently small radius. Choosing one representative from each nonempty bracket gives a finite net in the quotient space obtained by identifying functions at $d_P$-distance zero. This proves the total boundedness needed for the limiting Brownian bridge and for asymptotic equicontinuity. [/guided] [/step] [step:State the external bracketing Donsker theorem used] We use the following standard external theorem, usually called Ossiander's bracketing central limit theorem or the bracketing Donsker theorem. [claim:External bracketing Donsker theorem] Let $(S,\mathcal S,P)$ be a probability space and let $\mathcal G$ be a class of measurable real-valued functions on $S$ with measurable envelope $G:S\to[0,\infty]$ satisfying $G\in L^2(P)$. Suppose \begin{align*} \int_{(0,\infty)}\sqrt{H_{[]} (\varepsilon,\mathcal G,L^2(P))}\,d\mathcal L^1(\varepsilon)<\infty, \end{align*} where brackets have finite-valued measurable $L^2(P)$ endpoints. Then $\mathcal G$ is $P$-Donsker. More precisely, for i.i.d. observations with law $P$, the empirical processes indexed by $\mathcal G$ converge in outer distribution in $\ell^\infty(\mathcal G)$ to a tight centred Gaussian process $G_P$ with covariance \begin{align*} \operatorname{Cov}(G_P(g_1),G_P(g_2))=P(g_1g_2)-Pg_1\,Pg_2. \end{align*} [/claim] [proof] This is the standard bracketing Donsker theorem in empirical-process theory. Its proof combines the bracketing maximal inequality for the empirical process with the asymptotic equicontinuity criterion for weak convergence in $\ell^\infty(\mathcal G)$. The maximal inequality gives, for the bracketing tail below scale $\delta$, a universal constant multiple of \begin{align*} \int_{(0,\delta)}\sqrt{H_{[]} (\varepsilon,\mathcal G,L^2(P))}\,d\mathcal L^1(\varepsilon) \end{align*} up to the usual negligible finite-sample envelope term. Since this tail integral tends to $0$ as $\delta\downarrow0$, the empirical process is asymptotically equicontinuous in the $L^2(P)$ semimetric. Finite-dimensional convergence follows from the multivariate central limit theorem because $G\in L^2(P)$. The asymptotic equicontinuity plus finite-dimensional convergence theorem for empirical processes then yields outer weak convergence in $\ell^\infty(\mathcal G)$ to the tight Gaussian limit with the stated covariance. [/proof] This external theorem is quoted here with exactly the hypotheses used below: measurable functions, a square-integrable measurable envelope, finite-valued measurable $L^2(P)$ bracket endpoints, finite bracketing entropy integral, and outer probability/expectation for nonmeasurable suprema. [/step] [step:Verify the external theorem for the present class] We apply the external bracketing Donsker theorem with $\mathcal G:=\mathcal F$ and $G:=F$. The class $\mathcal F$ consists of measurable real-valued functions on $(S,\mathcal S)$ by hypothesis. The function $F:S\to[0,\infty]$ is measurable, satisfies $|f|\le F$ $P$-a.e. for every $f\in\mathcal F$, and belongs to $L^2(P)$ because \begin{align*} \int_S F(x)^2\,dP(x)<\infty. \end{align*} Thus $F$ is a square-integrable measurable envelope for $\mathcal F$. The bracketing number in the statement is defined using finite collections of brackets $[l,u]$ whose endpoints $l,u:S\to\mathbb R$ are finite-valued measurable functions in $L^2(P)$ and whose width is measured by the $L^2(P)$ norm of $u-l$. This is exactly the bracketing convention required by the external theorem. Finally, the displayed hypothesis gives \begin{align*} \int_{(0,\infty)}\sqrt{H_{[]} (\varepsilon,\mathcal F,L^2(P))}\,d\mathcal L^1(\varepsilon)<\infty. \end{align*} All hypotheses of the external bracketing Donsker theorem are therefore satisfied. [/step] [step:Identify the finite-dimensional Gaussian covariance] Let $k\in\mathbb N$ and let $f_1,\dots,f_k\in\mathcal F$. Define the random vector $Y_i:\Omega\to\mathbb R^k$ by \begin{align*} Y_i:=\bigl(f_1(X_i)-Pf_1,\dots,f_k(X_i)-Pf_k\bigr). \end{align*} The random vectors $Y_1,Y_2,\dots$ are i.i.d. because $X_1,X_2,\dots$ are i.i.d. Each coordinate has finite second moment since $|f_j|\le F$ $P$-a.e. and $F\in L^2(P)$. Therefore the multivariate central limit theorem applies and gives \begin{align*} (G_n(f_1),\dots,G_n(f_k))\xrightarrow{d}\mathcal N_k(0,\Sigma), \end{align*} where the covariance matrix $\Sigma\in\mathbb R^{k\times k}$ is defined by \begin{align*} \Sigma_{ab}:=P(f_af_b)-Pf_a\,Pf_b. \end{align*} This is exactly the finite-dimensional covariance prescribed for the $P$-Brownian bridge indexed by $\mathcal F$. [/step] [step:Conclude outer weak convergence in $\ell^\infty(\mathcal F)$] By the verification above, the external bracketing Donsker theorem applies to $\mathcal F$. Hence the restrictions $G_n|_{\mathcal F}$ converge in outer distribution in $\ell^\infty(\mathcal F)$ to a tight centred Gaussian process $G_P$ indexed by $\mathcal F$. The covariance of this process is, for all $f,g\in\mathcal F$, \begin{align*} \operatorname{Cov}(G_P(f),G_P(g))=P(fg)-Pf\,Pg. \end{align*} The preceding finite-dimensional calculation identifies this covariance with the covariance of the limiting normal laws of the empirical process. Therefore the limiting process is the tight $P$-Brownian bridge, and $\mathcal F$ is $P$-Donsker. [/step]