, Volume 7, Issue 1, pp 47-71

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

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Abstract

This 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 inL p 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 β k =O(k r ) for somer>1. These conditions are almost minimal.

Research supported by NSF Grant No. DMS-8505550 during the authors' visit to the MSRI. Research also supported by ARO Grant DAAL03-91-G-007.