Functional central limit theorem by bracketing entropy

If $\mathrm{\parallel }F{\mathrm{\parallel }}_{P,2}=0$, every member of a countable pointwise-dense subclass is zero $P$-almost surely. By [F15], countable subadditivity gives a probability-one event on which all those functions vanish at every observation. Pointwise approximation then makes every $f\in \mathcal{F}$ vanish at the observations; dominated convergence gives $Pf=0$ for each $f$. Thus ${\mathbb{G}}_n$ is the zero process in outer probability and the conclusion holds with the zero Gaussian bridge.

Assume $\mathrm{\parallel }F{\mathrm{\parallel }}_{P,2}>0$. The countable pointwise-dense subclass has a common measurable conull set on which its members are bounded by $F$; pointwise approximation extends that bound to every $f\in \mathcal{F}$ on the same set. By [F15], all observations lie in it on one measurable event of probability one. On that event, ${\sup }_{f\in \mathcal{F}}\mathrm{\mid }{\mathbb{G}}_n(f)\mathrm{\mid }\le \sqrt{n}(P_{n}F+PF)<\mathrm{\infty }$ for every $n$, by [L3]. Set the entire empirical path to zero off that event, as required by the Donsker definition. Each finite $L^2(P)$ bracket of width $r$ has ${\rho }_P$-diameter at most $r$, because ${\rho }_P(f,g)\le \mathrm{\parallel }f-g{\mathrm{\parallel }}_{P,2}\le \mathrm{\parallel }u-l{\mathrm{\parallel }}_{P,2}$. The entropy integral being finite forces $N_{[]}(r,\mathcal{F},L^2(P))<\mathrm{\infty }$ for every $r>0$: if it were infinite at one positive scale, monotonicity would make the integral infinite on an interval below that scale. Therefore the quotient of $\mathcal{F}$ by zero ${\rho }_P$-distance is totally bounded. The countable pointwise-dense subclass is dense in this quotient, since $\mathrm{\mid }{f}_k-f{\mathrm{\mid }}^2\le 4F^2$ and [F7] gives $\mathrm{\parallel }{f}_k-f{\mathrm{\parallel }}_{P,2}\to 0$.

Enumerate the countable dense subclass as ${\mathcal{F}}_0=\{f_j:j\in J\}$. For any finite list $f_{j_1},\dots ,f_{j_m}$, the covariance matrix ${\mathrm{\Sigma }}_{ab}=K(f_{j_a},f_{j_b})$ is positive semidefinite, since $c^T\mathrm{\Sigma }c=P[{\sum }_a{c}_a(f_{j_a}-P{f}_{j_a}){]}^2\ge 0$. [F3] supplies the centered Gaussian law $N_m(0,\mathrm{\Sigma })$. For any sublist, both the corresponding marginal and the Gaussian law constructed from its covariance have, in every direction, the same centered normal projection; [F14] makes the laws equal. Thus these finite-dimensional laws are consistent.

The coordinate space $\mathbb{R}$ with its Borel sigma-algebra is standard Borel by [F13], so [F4] extends the consistent Gaussian laws to a centered Gaussian process ${\mathbb{G}}_0$ indexed by ${\mathcal{F}}_0$, with covariance $K$. Each increment ${\mathbb{G}}_0(f)-{\mathbb{G}}_0(g)$ is centered normal with variance ${\rho }_P(f,g{)}^2$.

For any centered normal $Y$ with variance ${\sigma }^2$, if $\sigma =0$ then [F16] makes $Y=0$ almost surely and $\mathbb{E}{e}^{\lambda Y}=1$. If $\sigma >0$, [F16]–[F17] give $\mathbb{E}{e}^{\lambda Y}={\int }_{\mathbb{R}}{e}^{\lambda \sigma x}\varphi (x)dx=e^{{\lambda }^2{\sigma }^2\mathrm{/}2}{\int }_{\mathbb{R}}\varphi (x-\lambda \sigma )dx,$ by completing the square. For $a=\lambda \sigma$ and positive integer $R>\mathrm{\mid }a\mathrm{\mid }$, apply [F18] to the translation $x\mapsto x-a$, whose derivative is the integrable constant one, and the continuous integrand $\varphi$. Both compact integrands are continuous and bounded, so the Riemann–Lebesgue bridge in [F18], with Countable Choice from [A1], gives ${\int }_{[-R,R]}\varphi (x-a)dx={\int }_{[-R-a,R-a]}\varphi (u)du$. These intervals increase to $\mathbb{R}$ as $R\to \mathrm{\infty }$; [F19] on both sides and the normalization in [F16] give ${\int }_{\mathbb{R}}\varphi (x-a)dx=1$. Thus every centered Gaussian increment with variance ${\sigma }^2$ has MGF proxy ${\sigma }^2$, as needed for chaining.

For $\delta >0$, index the Gaussian increment process by pairs $(f,g)\in {\mathcal{F}}_0^2$ with ${\rho }_P(f,g)\le \delta$, and write $Y_{f,g}={\mathbb{G}}_0(f)-{\mathbb{G}}_0(g)$. Its canonical pseudometric is $d_Y((f,g),(f^{\mathrm{\prime }},g^{\mathrm{\prime }}))=\{E[(Y_{f,g}-Y_{f^{\mathrm{\prime }},g^{\mathrm{\prime }}}{)}^2]{\}}^{1\mathrm{/}2}$. The triangle inequality in $L^2(P)$ gives $d_Y\le {\rho }_P(f,f^{\mathrm{\prime }})+{\rho }_P(g,g^{\mathrm{\prime }})$, while every $Y_{f,g}$ has standard deviation at most $\delta$, so the $d_Y$-diameter of this pair set is at most $2\delta$. By step 4.1 its increments are centered sub-Gaussian with their $d_Y$-scale. A product of two $ϵ\mathrm{/}4$-nets of ${\mathcal{F}}_0$ covers the pair set within $d_Y$-distance $ϵ\mathrm{/}2$. Its centers might lie outside the restricted pair set; from each product ball that meets the pair set, choose one pair inside it. The triangle inequality then gives an $ϵ$-net with centers in the pair set and cardinality at most $N(ϵ\mathrm{/}4,{\mathcal{F}}_0,{\rho }_P{)}^2$. Applying [F5] with the diagonal base point $(f_{\ast },f_{\ast })$ yields $E{\sup }_{f,g\in {\mathcal{F}}_0:{\rho }_P(f,g)\le \delta }\mathrm{\mid }{\mathbb{G}}_0(f)-{\mathbb{G}}_0(g)\mathrm{\mid }\le C{\int }_0^{C^{\mathrm{\prime }}\delta }\sqrt{\log N(ϵ,{\mathcal{F}}_0,{\rho }_P)}dϵ,$ which tends to zero as $\delta \downarrow 0$ by the bracketing cover bound in step 1.2.

Choose rational ${\delta }_k\downarrow 0$ so that the expectation in step 5.1 is at most $2^{-3k}$. By [F11], the probability that the corresponding modulus exceeds $2^{-k}$ is at most $2^{-2k}$, and the first Borel–Cantelli lemma makes only finitely many such events occur almost surely. Thus the Gaussian sample paths are uniformly ${\rho }_P$-continuous on ${\mathcal{F}}_0$ almost surely. By [A2], Countable Choice is available for the metric-completion construction in [F8], so the metric quotient embeds densely in a complete metric completion $T$; compactness and the pathwise extension are established below.

Fix $\epsilon >0$ and take a finite $\epsilon \mathrm{/}2$-net in the totally bounded metric quotient from step 1.2. Its image in $T$ is still a finite $\epsilon \mathrm{/}2$-net for the dense embedded quotient. For any $x\in T$, density gives a quotient point $q$ with $d(x,q)<\epsilon \mathrm{/}2$, and some net point $y$ satisfies $d(q,y)\le \epsilon \mathrm{/}2$; hence $d(x,y)<\epsilon$. Thus $T$ is totally bounded, and it is compact by its completeness from [F8] and the complete-plus-totally-bounded theorem in [F8]. To extend a path $G_0$ that is uniformly ${\rho }_P$-continuous on the listed dense subset ${\mathcal{F}}_0$, for each $x\in T$ and $n\ge 1$ let $f_n(x)$ be the first listed point of ${\mathcal{F}}_0$ within $1\mathrm{/}n$ of $x$. The sequence $G_0(f_n(x))$ is Cauchy by uniform continuity and therefore has a real limit by [F13]. Any two sequences from ${\mathcal{F}}_0$ converging to $x$ have pairwise distances tending to zero, so uniform continuity makes their value limits equal. Define ${\bar{G}}_0(x)$ by this common limit. To verify continuity, given $\eta >0$ choose $\tau >0$ so that $d(u,v)<\tau$ for $u,v\in {\mathcal{F}}_0$ implies $\mathrm{\mid }{G}_0(u)-G_0(v)\mathrm{\mid }<\eta \mathrm{/}2$. If $d(x,y)<\tau \mathrm{/}3$, approximating sequences $u_n\to x$ and $v_n\to y$ eventually satisfy $d(u_n,v_n)<\tau$; passing to their value limits gives $\mathrm{\mid }{\bar{G}}_0(x)-{\bar{G}}_0(y)\mathrm{\mid }\le \eta \mathrm{/}2<\eta$. Density makes this extension unique.

Uniform continuity on the countable set ${\mathcal{F}}_0$ is a measurable path event: for each integer $m$, the condition that some positive rational radius makes every coordinate increment below $1\mathrm{/}m$ for pairs in ${\mathcal{F}}_0$ closer than that radius is a countable union and intersection of measurable coordinate events. By step 6.1 this event has probability one. On it use the extension from step 7.1, and define the function to be zero off it. The resulting map is a Borel random element of $C(T,\mathbb{R})$: evaluations at the countable dense set ${\mathcal{F}}_0$ are measurable, and for continuous functions the supremum distance is the supremum over ${\mathcal{F}}_0$, so preimages of open balls are measurable; separability makes every open set a countable union of such balls. By [F9], $C(T,\mathbb{R})$ is complete and separable; [F10] makes it Polish and its Gaussian law tight. For any finite list $f_1,\dots ,f_m\in \mathcal{F}$, choose pointwise approximants $f_{i,k}\in {\mathcal{F}}_0$ for each $i$. By [F7], $\mathrm{\parallel }{f}_{i,k}-f_i{\mathrm{\parallel }}_{P,2}\to 0$. Fix coefficients $c_1,\dots ,c_m$ and put $S_k={\sum }_i{c}_i{\mathbb{G}}_0(f_{i,k})$ and $S={\sum }_i{c}_i{\bar{G}}_0([f_i])$. Step 7.1 gives $S_k\to S$ almost surely. Each $S_k$ is centered normal with variance ${\sigma }_k^2\to {\sigma }^2={\sum }_{i,j}{c}_i{c}_{j}K(f_i,f_j)=P[{\sum }_i{c}_i(f_i-P{f}_i){]}^2\ge 0$. Dominated convergence [F7] applied to $e^{it{S}_k}$, whose modulus is one, gives convergence of characteristic functions. By [F12], the limit is $e^{-t^2{\sigma }^2\mathrm{/}2}$ and determines the centered normal law of variance ${\sigma }^2$. Every linear combination of the limiting vector is therefore centered normal; [F3] identifies that vector as multivariate normal with covariance $(K(f_i,f_j))$. This proves that the pathwise extension has the required Gaussian finite-dimensional laws. Define $R:C(T,\mathbb{R})\to {\mathrm{\ell }}^{\mathrm{\infty }}(\mathcal{F})$ by $R(z)(f)=z([f])$, where $[f]$ is the image of $f$ in the metric quotient and its completion. Since these images are dense in $T$ and $z-w$ is continuous, $\mathrm{\parallel }R(z)-R(w){\mathrm{\parallel }}_{\mathcal{F}}={\sup }_{f\in \mathcal{F}}\mathrm{\mid }z([f])-w([f])\mathrm{\mid }={\sup }_{t\in T}\mathrm{\mid }z(t)-w(t)\mathrm{\mid }$, so $R$ is an isometry and is continuous. Its pushforward of the tight Borel law on $C(T,\mathbb{R})$ is a tight Borel Gaussian bridge on ${\mathrm{\ell }}^{\mathrm{\infty }}(\mathcal{F})$.

Fix any finite $f_1,\dots ,f_m\in \mathcal{F}$. The vectors $(f_1(X_i)-P{f}_1,\dots ,f_m(X_i)-P{f}_m)$ are iid and have finite second moments, bounded in squared norm by $2m(F(X_i{)}^2+(PF{)}^2)$ almost surely. [F2] gives convergence of their normalized sums to $N_m(0,(K(f_a,f_b){)}_{a,b})$, the corresponding finite-dimensional law of the Gaussian bridge from step 8.1. This proves finite-dimensional convergence, with the asymptotic-distribution convention of [F1].

By [F6], the empirical processes are asymptotically equicontinuous in ${\rho }_P$. For each $ϵ>0$, the totally bounded quotient from step 1.2 has a finite $ϵ\mathrm{/}2$-net with centers in the dense subset ${\mathcal{F}}_0$; density then makes it an $ϵ$-net for all of $\mathcal{F}$. List its finitely many centers and assign to each $f$ the first center within distance $ϵ$, obtaining ${\pi }_{ϵ}(f)$. Define ${\mathbb{G}}_{n,ϵ}(f)={\mathbb{G}}_n({\pi }_{ϵ}(f))$ and ${\mathbb{G}}_{ϵ}(f)={\mathbb{G}}_P({\pi }_{ϵ}(f))$. Their ranges lie in a finite-dimensional subspace of ${\mathrm{\ell }}^{\mathrm{\infty }}(\mathcal{F})$; the linear map from the selected coordinate vector to that subspace is continuous in the supremum norm. Thus the finite-dimensional convergence in step 9.1, tested after composition with any bounded continuous functional, gives ${\mathbb{G}}_{n,ϵ}\Rightarrow {\mathbb{G}}_{ϵ}$ for each fixed $ϵ$.

The sup-norm error $\mathrm{\parallel }{\mathbb{G}}_n-{\mathbb{G}}_{n,ϵ}{\mathrm{\parallel }}_{\mathcal{F}}$ is bounded by the empirical-process modulus over pairs at ${\rho }_P$-distance at most $ϵ$; it is therefore bounded by the modulus at radius $2ϵ$ in [F6], which tends to zero in outer probability as $ϵ\downarrow 0$. Uniform continuity of the Gaussian paths gives $\mathrm{\parallel }{\mathbb{G}}_P-{\mathbb{G}}_{ϵ}{\mathrm{\parallel }}_{\mathcal{F}}\to 0$ almost surely. For bounded uniformly continuous $H$, put ${\omega }_H(r)={\sup }_{\mathrm{\parallel }x-y{\mathrm{\parallel }}_{\mathcal{F}}\le r}\mathrm{\mid }H(x)-H(y)\mathrm{\mid }$, which tends to zero as $r\downarrow 0$. If $Z,Z^{\mathrm{\prime }}$ are random elements and $P^{\ast }(\mathrm{\parallel }Z-Z^{\mathrm{\prime }}\mathrm{\parallel }>r)=p$, a measurable event of probability arbitrarily close to $p$ covers the exceptional set; outside it $\mathrm{\mid }H(Z)-H(Z^{\mathrm{\prime }})\mathrm{\mid }\le {\omega }_H(r)$, and everywhere the difference is at most $2\mathrm{\parallel }H{\mathrm{\parallel }}_{\mathrm{\infty }}$. This gives $\mathrm{\mid }{E}^{\ast }H(Z)-E^{\ast }H(Z^{\mathrm{\prime }})\mathrm{\mid }\le {\omega }_H(r)+2\mathrm{\parallel }H{\mathrm{\parallel }}_{\mathrm{\infty }}p$ by measurable majorants. Use the three-term bound $\mathrm{\mid }{E}^{\ast }H({\mathbb{G}}_n)-EH({\mathbb{G}}_P)\mathrm{\mid }\le \mathrm{\mid }{E}^{\ast }H({\mathbb{G}}_n)-E^{\ast }H({\mathbb{G}}_{n,ϵ})\mathrm{\mid }+\mathrm{\mid }EH({\mathbb{G}}_{n,ϵ})-EH({\mathbb{G}}_{ϵ})\mathrm{\mid }+\mathrm{\mid }EH({\mathbb{G}}_{ϵ})-EH({\mathbb{G}}_P)\mathrm{\mid }\mathrm{.}$ For fixed $ϵ$ the middle term tends to zero by step 10.1; then letting $ϵ\downarrow 0$ makes the other two terms vanish by the outer-probability approximation bound and ${\omega }_H(r)\to 0$. This is the outer weak convergence in Donsker classes and asymptotic equicontinuity, with the tight Gaussian bridge from step 8.1. AC is used in [F2], [F4], [F10], and the countable-choice completion in [F8]; no finite-dimensional limit is used as a substitute for the process-level equicontinuity step.