The law of the iterated logarithm for empirical processes on Vapnik- Červonenkis classes
The first author, Ann. Probab. 15, 178-203 (1987; Zbl 0624.60032), gave necessary and sufficient conditions for a VC-graph class \({\mathcal F}\) of functions to be P-Donsker (i.e. such that the empirical process based on P and indexed by \({\mathcal F}\) satisfies the CLT). Here the authors give necessary and sufficient conditions for the empirical process indexed by such classes to satisfy the law of iterated logarithm, both compact and bounded. The results are based on the recent remarkable characterization of the LIL for Banach space valued random variables of \textit{M. Ledoux} and \textit{M. Talagrand}, ibid. 16, 1242-1264 (1988; Zbl 0662.60008), which reduced the LIL to moment conditions and to convergence in probability to zero or stochastic boundedness of \(\| S_ n\| /\sqrt{n \log \log n}\). Because of this, entropy chaining proofs can be employed to obtain sharp results. The main part of the proof consists in showing that for VC-graph classes \({\mathcal F}\), if \(\tilde {\mathcal F}\) is the envelope of \({\mathcal F}\) then \[ E F^ 2(X) Log Log F(X)<\infty \quad implies\quad \| \nu_ n/\sqrt{Log Log n}\|_{{\mathcal F}}\to 0\quad in\quad probability, \] an entropy proof. (Here \(\nu_ n=n^{- 1/2}\sum^{n}_{i=1}(\delta_{X_ i}-P)\), \(X_ i\) i.i.d. (P), and \(Log(x)=\log (xve).)\) The main result is: Let \({\mathcal F}\) be a countably determined VC-graph class of functions on (T,\({\mathcal B},P)\) with envelope \(\tilde {\mathcal F}\) and let \({\mathcal F}_ P=\sup \{| f-Pf|:\quad f\in {\mathcal F}).\) Then the conditions \(\sup_{f\in F}Var(F(x))<\infty\) (\(\{\) f-Pf: \(f\in {\mathcal F}\}\) is totally bounded in \(L_ 2(P))\) and \(E F^ 2(X)/Log Log F(X)<\infty\) imply that the empirical process based on P and indexed by \({\mathcal F}\) satisfies the bounded law of iterated logarithm (the compact LIL), and conversely, if \({\mathcal F}\) satisfies the bounded (compact) LIL then these two conditions hold, but the second for \(F_ P\) instead of \(\tilde {\mathcal F}\). The cluster set is always the unit ball of the rkhs. These results are applied to the LIL for weighted empirical processes.
scientific article; zbMATH DE number 4128156