Glivenko-Cantelli class

In empirical process theory, a Glivenko-Cantelli class $\mathcal{F}$ whose associated empirical process satisfies an LLN. That is, if

$${\Delta }_n(\mathcal{F})\to 0,\text{for }{\Delta }_n(\mathcal{F})=\underset{f\in \mathcal{F}}{\sup }\left|\frac{1}{n}\sum _{i\le n}f(X_i)-\mathbb{E}f(X)\right|,$$

where the converge is either in probability or almost surely. Under various regularity conditions, $\mathcal{F}$ is a GC class iff it is a VC class (meaning it has finite VC dimension).

The most famous of example of a GC class is the class of indicator functions $\mathcal{F}=\{f(x)=1(-\infty ,x):x\in \mathbb{R}\}$. This gives that the uniform convergence of the cdf (see cdf concentration).

If $\mathcal{F}$ is sufficiently rich it can fail to be a GC class. Suppose that $X_1,\dots ,X_n\sim P$ where $P$ is a continuous distribution over $[0,1]$ and let $\mathcal{F}$ be the indicators $1_A(\cdot )$ for all finite subsets $A$ of $[0,1]$. Then $\mathbb{E}f(X)$ = $\mathbb{E}{1}_A(X)=P(A)=0$ for all such $A$ by continuity of $P$. Meanwhile, for the set $A=\{X_1,\dots ,X_{\}}$ we have $1_A(X_i)=0$ for all $i$, so ${\Delta }_n((\mathcal{F})=1$. Hence $\mathcal{F}$ is not GC.