The Annals of Probability

Rates of Convergence for Empirical Processes of Stationary Mixing Sequences

Bin Yu

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Abstract

Classical empirical process theory for Vapnik-Cervonenkis classes deals mainly with sequences of independent variables. This paper extends the theory to stationary sequences of dependent variables. It establishes rates of convergence for $\beta$-mixing and $\phi$-mixing empirical processes indexed by classes of functions. The method of proof depends on a coupling of the dependent sequence with sequences of independent blocks, to which the classical theory can be applied. A uniform $O(n^{-s/(1+s)})$ rate of convergence over V-C classes is established for sequences whose mixing coefficients decay slightly faster than $O(n^{-s})$.

Article information

Source
Ann. Probab., Volume 22, Number 1 (1994), 94-116.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176988849

Digital Object Identifier
doi:10.1214/aop/1176988849

Mathematical Reviews number (MathSciNet)
MR1258867

Zentralblatt MATH identifier
0802.60024

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60F17: Functional limit theorems; invariance principles 60G10: Stationary processes

Keywords
Empirical process $\beta$-mixing rate of convergence random metric entropy V-C class blocking

Citation

Yu, Bin. Rates of Convergence for Empirical Processes of Stationary Mixing Sequences. Ann. Probab. 22 (1994), no. 1, 94--116. doi:10.1214/aop/1176988849. https://projecteuclid.org/euclid.aop/1176988849


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