A class of functions is considered a Donsker class if it satisfies Donsker's theorem, a functional generalization of the central limit theorem.
be a collection of square integrable functions on a probability space
The empirical process
is the stochastic process on the set
is the empirical measure based on an iid sample
The class of measurable functions
is called a Donsker class if the empirical process
converges in distribution to a tight Borel measurable element in the space
By the central limit theorem, for every finite set of functions
, the random vector
converges in distribution to a multivariate normal vector as
is Donsker if and only if the sequence
is asymptotically tight in
[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
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.