Central limit theorems for empirical andU-processes of stationary mixing sequences

AbstractThis paper gives sufficient conditions for the weak convergence to Gaussian processes of empirical processes andU-processes from stationary β mixing sequences indexed byV-C subgraph classes of functions. If the envelope function of theV-C subgraph class is inLp for some 2<p<∞, we obtain a uniform central limit theorem for the empirical process under the β mixing condition $$k^{p/(p - 2)} (\log k)^{2(p - 1)/(p - 2)} \beta _k \to 0 as k \to \infty $$ In the case that the functions in theV-C subgraph class are uniformly bounded, we obtain uniform central limit theorems for the empirical process and theU-process, provided that the decay rate of the β mixing coefficient satisfies βk=O(k−r) for somer>1. These conditions are almost minimal.

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