Donsker classes

A class of functions is considered a Donsker class if it satisfies Donsker's theorem, a functional generalization of the central limit theorem.

Let ${\displaystyle {\mathcal {F}}}$ be a collection of square integrable functions on a probability space ${\displaystyle ({\mathcal {X}},{\mathcal {A}},P)}$. The empirical process ${\displaystyle \mathbb {G} _{n}}$ is the stochastic process on the set ${\displaystyle {\mathcal {F}}}$ defined by $${\displaystyle \mathbb {G} _{n}(f)={\sqrt {n}}(\mathbb {P} _{n}-P)(f)}$$ where ${\displaystyle \mathbb {P} _{n}}$ is the empirical measure based on an iid sample ${\displaystyle X_{1},\dots ,X_{n}}$ from ${\displaystyle P}$.

The class of measurable functions ${\displaystyle {\mathcal {F}}}$ is called a Donsker class if the empirical process ${\displaystyle (\mathbb {G} _{n})_{n=1}^{\infty }}$ converges in distribution to a tight Borel measurable element in the space ${\displaystyle \ell ^{\infty }({\mathcal {F}})}$.

By the central limit theorem, for every finite set of functions ${\displaystyle f_{1},f_{2},\dots ,f_{k}\in {\mathcal {F}}}$, the random vector ${\displaystyle (\mathbb {G} _{n}(f_{1}),\mathbb {G} _{n}(f_{2}),\dots ,\mathbb {G} _{n}(f_{k}))}$ converges in distribution to a multivariate normal vector as ${\displaystyle n\rightarrow \infty }$. Thus the class ${\displaystyle {\mathcal {F}}}$ is Donsker if and only if the sequence ${\displaystyle (\mathbb {G} _{n})_{n=1}^{\infty }}$ is asymptotically tight in ${\displaystyle \ell ^{\infty }({\mathcal {F}})}$ ¹

Classes of functions which have finite Dudley's entropy integral are Donsker classes. This includes empirical distribution functions formed from the class of functions defined by ${\displaystyle \mathbb {I} _{(-\infty ,t]}}$ as well as parametric classes over bounded parameter spaces. More generally any VC class is also Donsker class.²

Classes of functions formed by taking infima or suprema of functions in a Donsker class also form a Donsker class.²

Donsker's theorem states that the empirical distribution function, when properly normalized, converges weakly to a Brownian bridge—a continuous Gaussian process. This is significant as it assures that results analogous to the central limit theorem hold for empirical processes, thereby enabling asymptotic inference for a wide range of statistical applications.³

The concept of the Donsker class is influential in the field of asymptotic statistics. Knowing whether a function class is a Donsker class helps in understanding the limiting distribution of empirical processes, which in turn facilitates the construction of confidence bands for function estimators and hypothesis testing.³