Where does "Uniform" in "Uniform central limit theorem" come from? We may all know about the CLT. Today I have seen two articles where the use a new term (to me), that is "Uniform central limit theorem".

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*A uniform central limit theorem and efficiency for deconvolution estimators


*Empirical Processes in M-Estimation
My question is that where the word "uniform" come from? Does it come from "uniform Donsker class"?
 A: Uniform Central Limit theorems relate to functional extensions of the classic central limit theorem (extending from application to single valued random variables to random functions). They are about the convergence of random functions. For instance think about a the Wiener process as a limit of a random walk. With a random walk we do not deal with the convergence of a single random value, but with the convergence of an entire function (the entire walk).
Uniform relates to uniform convergence.
See also

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*The Glivenko Cantelli theorem, which is about a uniform law of large numbers (more specifically the case of convergence of the empirical distribution to the true distribution function). The theorem proves that this convergence is not just pointwise, but also uniform.


*Donsker's theorem, which states that a scaled random walk converges uniformly in distribution to a normal distribution.


*There is a whole book written about this topic.
Dudley, Richard M. Uniform central limit theorems. Vol. 142. Cambridge university press, 2014.
