Journal of Theoretical Probability

, Volume 7, Issue 1, pp 47–71

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

  • M. A. Arcones
  • B. Yu

DOI: 10.1007/BF02213360

Cite this article as:
Arcones, M.A. & Yu, B. J Theor Probab (1994) 7: 47. doi:10.1007/BF02213360


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 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(kr) for somer>1. These conditions are almost minimal.

Key Words

Empirical process U-statistic mixing V-C class absolute regularity β-mixing 

Copyright information

© Plenum Publishing corporation 1994

Authors and Affiliations

  • M. A. Arcones
    • 1
  • B. Yu
    • 2
  1. 1.Department of MathematicsUniversity of UtahSalt Lake City
  2. 2.Statistics DepartmentUniversity of WisconsinMadison

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