Donsker classes and asymptotic equicontinuity

For a class $\mathcal{H}$ of measurable real functions under a probability measure $Q$, an envelope $H$ means a nonnegative measurable function with $\mathrm{\mid }h\mathrm{\mid }\le H$ $Q$-almost surely for every $h\in \mathcal{H}$. Cauchy–Schwarz Cauchy-Schwarz inequality for $L^2$ gives $QH\le \mathrm{\parallel }H{\mathrm{\parallel }}_{Q,2}<\mathrm{\infty }$ when $H\in L^2(Q)$.

Let $(S,\mathcal{A},P)$ be a probability space, let $X_1,X_2,\dots$ be iid with law $P$, and let $\mathcal{F}$ be a nonempty class of measurable real-valued functions with a nonnegative measurable envelope $F$ satisfying $P{F}^2<\mathrm{\infty }$. A class is pointwise measurable if it has a countable subclass ${\mathcal{F}}_0$ such that for each $f\in \mathcal{F}$ there is a sequence in ${\mathcal{F}}_0$ converging pointwise to $f$. Write ${\mathbb{G}}_n(f)=\sqrt{n}(P_{n}f-Pf)$ using Empirical, centered, and Rademacher processes, and put

${\rho }_P(f,g)=\{P[(f-g)-P(f-g){]}^2{\}}^{1\mathrm{/}2}\mathrm{.}$

This is a pseudometric: distinct functions can have zero distance when their centered difference vanishes $P$-almost surely. The empirical process is asymptotically equicontinuous in probability with respect to ${\rho }_P$ if for every $\eta >0$,

${\lim }_{\delta \downarrow 0}{\mathrm{lim\hspace{0.17em}sup}}_{n\to \mathrm{\infty }}{P}^{\ast }({\sup }_{f,g\in \mathcal{F}:{\rho }_P(f,g)<\delta }\mathrm{\mid }{\mathbb{G}}_n(f)-{\mathbb{G}}_n(g)\mathrm{\mid }>\eta )=0.$

Here $P^{\ast }(A)=\inf \{P(B):A\subseteq B,B\in \mathcal{A}\}$ is outer probability; bounded nonmeasurable functionals are handled by the corresponding outer expectation $E^{\ast }Y=\inf \{E{Y}^{\mathrm{\prime }}:Y^{\mathrm{\prime }}\text{ measurable and }{Y}^{\mathrm{\prime }}\ge Y\}$. For a pointwise-measurable class these conventions agree with ordinary probability whenever the displayed supremum is measurable.

Write ${\mathrm{\ell }}^{\mathrm{\infty }}(\mathcal{F})$ for the bounded real-valued functions on the index set $\mathcal{F}$, with norm $\mathrm{\parallel }z{\mathrm{\parallel }}_{\mathcal{F}}={\sup }_{f\in \mathcal{F}}\mathrm{\mid }z(f)\mathrm{\mid }$. The class is $P$-Donsker if, for each $n$, ${\mathbb{G}}_n$ has bounded paths on a measurable event of probability one and, after setting the entire path to zero off that event, the resulting ${\mathrm{\ell }}^{\mathrm{\infty }}(\mathcal{F})$-valued process converges weakly in the outer-expectation sense to a tight Borel law of a centered Gaussian process ${\mathbb{G}}_P$ with covariance

$\mathrm{Cov}({\mathbb{G}}_P(f),{\mathbb{G}}_P(g))=P[(f-Pf)(g-Pg)]\mathrm{.}$

Here outer weak convergence means $E^{\ast }H({\mathbb{G}}_n)\to EH({\mathbb{G}}_P)$ for every bounded uniformly continuous real function $H$ on ${\mathrm{\ell }}^{\mathrm{\infty }}(\mathcal{F})$; the outer expectation is the infimum of expectations of measurable majorants. The law is tight in the sense of Tight family of probability measures. Finite-dimensional convergence means weak convergence of the vector of process values at every finite list of indices, as in Convergence in distribution of random elements.