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 $\mathcal$ be a collection of square integrable functions on a probability space $(\mathcal, \mathcal, P)$. The empirical process $\mathbb_n$ is the stochastic process on the set $\mathcal$ defined by
$\mathbb_n(f) = \sqrt(\mathbb_n - P)(f)$
where $\mathbb_n$ is the empirical measure based on an iid sample $X_1, \dots, X_n$ from $P$.

The class of measurable functions $\mathcal$ is called a Donsker class if the empirical process $(\mathbb_n)_^$ converges in distribution to a tight Borel measurable element in the space $\ell^(\mathcal)$.

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

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 $\mathbb I_$ as well as parametric classes over bounded parameter spaces. More generally any VC class is also Donsker class.Vaart AW van der. Asymptotic Statistics. Cambridge University Press; 1998.

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.van der Vaart, A. W., & Wellner, J. A. (1996). Weak Convergence and Empirical Processes. In Springer Series in Statistics. Springer New York. https://doi.org/10.1007/978-1-4757-2545-2

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)

References

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Probability theory
Central limit theorem