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Large deviations, moderate deviations and LIL for empirical processes. (English) Zbl 0793.60032

Summary: Let \((X_ n)_{n\geq 1}\) be a sequence of i.i.d. r.v.’s with values in a measurable space \((E,{\mathcal E})\) of law \(\mu\), and consider the empirical process \(L_ n(f)=(1/n)\sum^ n_{k=1}f(X_ k)\) with \(f\) varying in a class of bounded functions \({\mathcal F}\). Using a recent isoperimetric inequality of Talagrand, we obtain necessary and sufficient conditions for the large deviation estimations, the moderate deviation estimations and the LIL of \(L_ n(\cdot)\) in the Banach space of bounded functionals \(\ell_ \infty({\mathcal F})\). The extension to the unbounded functionals is also discussed.

MSC:

60F10 Large deviations
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60G50 Sums of independent random variables; random walks
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