Donsker classes explained

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

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

The class of measurable functions

l{F} is called a Donsker class if the empirical process ( G _{n)} ^{inf ty} ^{n =1} converges in distribution to a tight Borel measurable element in the space \ell ^{infty} ( l{F}) .

By the central limit theorem, for every finite set of functions

f _{1,} f _{2,} ... , f _{k} \in l{F} , the random vector ( G _{n(f} _{1),} G _{n(f} _{2),} ... , G _{n(f} _{k))} converges in distribution to a multivariate normal vector as n → inf ty . Thus the class l{F} is Donsker if and only if the sequence ( G _{n)} ^{inf ty} ^{n =1} is asymptotically tight in \ell ^{infty} ( l{F}) ^[1]

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 _{(-infty,} as well as parametric classes over bounded parameter spaces. More generally any VC class is also Donsker class.^[2]

Properties

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

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.^[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]

See also

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

Notes and References

1. Book: van der Vaart . A. W. . Wellner . Jon A. . Weak Convergence and Empirical Processes . 2023 . 139 . en.
2. Vaart AW van der. Asymptotic Statistics. Cambridge University Press; 1998.
3. 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