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.

A class of functions is considered a Donsker class if it satisfies Donsker's theorem, a functional generalization of the central limit theorem. \mathbb{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 \mathcal{F} is called a Donsker class if the empirical process (\mathbb{G}_n)_{n = 1}^{\infty} converges in distribution to a tight Borel measurable element in the space \ell^{\infty}(\mathcal{F}) . By the central limit theorem, for every finite set of functions f_1, f_2, \dots, f_k \in \mathcal{F} , the random vector (\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 n \rightarrow \infty . Thus the class \mathcal{F} is Donsker if and only if the sequence (\mathbb{G}_n)_{n = 1}^{\infty} is asymptotically tight in \ell^{\infty}(\mathcal{F})

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