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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

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.

Suggested Citation

  • Ziegler, Klaus, 1997. "Functional Central Limit Theorems for Triangular Arrays of Function-Indexed Processes under Uniformly Integrable Entropy Conditions," Journal of Multivariate Analysis, Elsevier, vol. 62(2), pages 233-272, August.
    • Handle: RePEc:eee:jmvana:v:62:y:1997:i:2:p:233-272
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    References listed on IDEAS

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    1. Alexander, Kenneth S., 1987. "The central limit theorem for weighted empirical processes indexed by sets," Journal of Multivariate Analysis, Elsevier, vol. 22(2), pages 313-339, August.
    2. Ossiander, Mina & Pyke, Ronald, 1985. "Lévy's Brownian motion as a set-indexed process and a related central limit theorem," Stochastic Processes and their Applications, Elsevier, vol. 21(1), pages 133-145, December.
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    Cited by:

    1. Quansheng Liu & Emmanuel Rio & Alain Rouault, 2003. "Limit Theorems for Multiplicative Processes," Journal of Theoretical Probability, Springer, vol. 16(4), pages 971-1014, October.
    2. 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.
    3. 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.
    4. Daniel Gaigall, 2020. "Testing marginal homogeneity of a continuous bivariate distribution with possibly incomplete paired data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(4), pages 437-465, May.
    5. 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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