Empirical, centered, and Rademacher processes

Let $(\mathcal{X},\mathcal{A},P)$ be a probability space, let $n\ge 1$ be an integer, and let $Z_1,\dots ,Z_n$ be IID random elements with law $P$. For a class $\mathcal{F}$ of measurable, integrable real-valued functions on $\mathcal{X}$, write

$Pf:=\int fdP,P_{n}f:=\frac{1}{n}{\sum }_{i=1}^{n}f(Z_i),{\mathbb{G}}_n(f):=\sqrt{n}(P_{n}f-Pf)\mathrm{.}$

The map $f\mapsto P_{n}f-Pf$ is the centered empirical process, and ${\mathbb{G}}_n$ is its $\sqrt{n}$-scaled version. More generally, for independent processes $X_1(t),\dots ,X_n(t)$ whose marginal expectations are finite for each index $t\in T$, the centered process sum is

$S_n(t):={\sum }_{i=1}^n(X_i(t)-\mathbb{E}{X}_i(t)),t\in T\mathrm{.}$

On an auxiliary probability-space extension, let ${\sigma }_1,\dots ,{\sigma }_n$ be independent signs, each taking $+1$ and $-1$ with probability $1\mathrm{/}2$, independent of the observations. The empirical Rademacher process is

$R_n(f):={\sum }_{i=1}^n{\sigma }_{i}f(Z_i),f\in \mathcal{F}\mathrm{.}$

The centered Rademacher process ${\sum }_i{\sigma }_i(f(Z_i)-Pf)$ is a distinct process; no equality between it and $R_n$ is part of this definition.

For a countable class of measurable functions, each displayed supremum of measurable sample paths is measurable. For an uncountable class, ordinary probability or expectation is used only when a specified countable subclass has the same supremum almost surely for the process in question; this is the separable-version convention used here. Otherwise state results using outer probability

$P^{\ast }(A):=\inf \{P(B):A\subseteq B,B\in \mathcal{A}\}$

and, for a nonnegative possibly nonmeasurable random quantity $Y$, outer expectation

${\mathbb{E}}^{\ast }Y:=\inf \{\mathbb{E}U:U\text{ is measurable, nonnegative, and }U\ge Y\}\mathrm{.}$

In particular, measurability of every $f$ separately does not by itself make an uncountable supremum measurable.