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.

Definition

Let $ℱ$ be a collection of square integrable functions on a probability space $(𝒳,𝒜,P)$. The empirical process ${𝔾}_n$ is the stochastic process on the set $ℱ$ defined by $${𝔾}_n(f)=\sqrt{n}({\mathbb{P}}_n-P)(f)$$ where ${\mathbb{P}}_n$ is the empirical measure based on an iid sample $X_1,\dots ,X_n$ from $P$.

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

By the central limit theorem, for every finite set of functions $f_1,f_2,\dots ,f_k\in ℱ$, the random vector $({𝔾}_n(f_1),{𝔾}_n(f_2),\dots ,{𝔾}_n(f_k))$ converges in distribution to a multivariate normal vector as $n\to \mathrm{\infty }$. Thus the class $ℱ$ is Donsker if and only if the sequence $({𝔾}_n{)}_{n=1}^{\mathrm{\infty }}$ is asymptotically tight in ${\ell }^{\mathrm{\infty }}(ℱ)$

Examples and Sufficient Conditions

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 ${𝕀}_{(-\mathrm{\infty },t]}$ as well as parametric classes over bounded parameter spaces. More generally any VC class is also Donsker class.

Properties

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

Donsker's Theorem

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.

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.

See also

• Empirical process
• Central limit theorem
• Brownian bridge
• Glivenko–Cantelli theorem
• Vapnik–Chervonenkis theory
• Weak convergence (probability)