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Functional Central Limit Theorems for Triangular Arrays of Function-Indexed Processes under Uniformly Integrable Entropy Conditions

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  • Ziegler, Klaus
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    Abstract

    Functional central limit theorems for triangular arrays of rowwise independent stochastic processes are established by a method replacing tail probabilities by expectations throughout. The main tool is a maximal inequality based on a preliminary version proved by P. Gaenssler and Th. Schlumprecht. Its essential refinement used here is achieved by an additional inequality due to M. Ledoux and M. Talagrand. The entropy condition emerging in our theorems was introduced by K. S. Alexander, whose functional central limit theorem for so-calledmeasure-like processeswill be also regained. Applications concern, in particular, so-calledrandom measure processeswhich include function-indexed empirical processes and partial-sum processes (with random or fixed locations). In this context, we obtain generalizations of results due to K. S. Alexander, M. A. Arcones, P. Gaenssler, and K. Ziegler. Further examples include nonparametric regression and intensity estimation for spatial Poisson processes.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 62 (1997)
    Issue (Month): 2 (August)
    Pages: 233-272

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    Handle: RePEc:eee:jmvana:v:62:y:1997:i:2:p:233-272

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    Keywords: functional central limit theorem asymptotic equicontinuity symmetrization metric entropy VC graph class maximal inequality empirical processes partial sum processes random measure processes sequential empirical process smoothing by convolution nonparametric regression intensity estimation;

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    Cited by:
    1. Rackauskas, Alfredas & Suquet, Charles & Zemlys, Vaidotas, 2007. "A Hölderian functional central limit theorem for a multi-indexed summation process," Stochastic Processes and their Applications, Elsevier, vol. 117(8), pages 1137-1164, August.
    2. Kosorok, Michael R., 2003. "Bootstraps of sums of independent but not identically distributed stochastic processes," Journal of Multivariate Analysis, Elsevier, vol. 84(2), pages 299-318, February.
    3. Bae, Jongsig & Hwang, Changha & Jun, Doobae, 2012. "The uniform central limit theorem for the tent map," Statistics & Probability Letters, Elsevier, vol. 82(5), pages 1021-1027.

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