Central limit theorems for stochastic processes under random entropy conditions
type: Item
instance of: scholarly article
title: Central limit theorems for stochastic processes under random entropy conditions
zbMATH Open document ID: 0594.60031
DOI: 10.1007/BF00318707
author: Kenneth S. Alexander
publication date: 1987-01-01T00:00:00Z
review text: Necessary and sufficient conditions are found for the weak convergence of the row sums of an infinitesimal row-independent triangular array \((\phi_{nj})\) of stochastic processes, indexed by a set S, to a sample- continuous Gaussian process, when the array satisfies a ''random entropy'' condition, analogous to one used by \textit{E. Giné} and \textit{J. Zinn} [Ann. Probab. 12, 929-989 (1984; Zbl 0553.60037)] for empirical processes. This entropy condition is satisfied when S is a class of sets or functions with the Vapnik-Červonenkis property and each \(\phi_{nj}\) is of the form \(\phi_{nj}(f)=\int fd\nu_{nj}\) for some reasonable random finite signed measure \(\nu_{nj}.\) As a result we obtain necessary and sufficient conditions for the weak convergence of (possibly non-i.i.d.) partial-sum processes, and new sufficient conditions for empirical processes, indexed by Vapnik- Červonenkis classes. Special cases include \textit{Yu. V. Prokhorov}'s [Teor. Veroyatn. Primen. 1, 177-237 (1956; Zbl 0075.290); English translation in Theory Probab. Appl. 1, 188-214 (1956)] central limit theorem for processes on the unit interval, \textit{M. J. Wichura}'s [Ann. Math. Stat. 40, 681-687 (1969; Zbl 0214.177)] sufficient conditions for weak convergence of multiparameter processes, \textit{D. Pollard}'s [J. Aust. Math. Soc., Ser. A 33, 235-248 (1982; Zbl 0504.60023)] central limit theorem for empirical processes, and \textit{G. R. Shorack}'s [Stat. Neerl. 33, 169-189 (1979; Zbl 0436.60009)] theorems on weighted empirical processes.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 60G57
zbMATH DE Number: 3956109
zbMATH Keywords: weak convergence
zbMATH Keywords: empirical processes
zbMATH Keywords: Vapnik-Červonenkis property
zbMATH Keywords: central limit theorem
zbMATH Keywords: central limit theorem for empirical processes
zbMATH Keywords: weighted empirical process
zbMATH Keywords: metric entropy
MaRDI profile type: Publication
cites work: Probability inequalities for empirical processes and a law of the iterated logarithm
cites work: A uniform central limit theorem for set-indexed partial-sum processes with finite variance
cites work: Q3906818
cites work: Functional law of the iterated logarithm and uniform central limit theorem for partial-sum processes indexed by sets
cites work: A strong law of large numbers for partial-sum processes indexed by sets
cites work: Q5804555
cites work: Justification and Extension of Doob's Heuristic Approach to the Kolmogorov- Smirnov Theorems
cites work: The sizes of compact subsets of Hilbert space and continuity of Gaussian processes
cites work: Sample functions of the Gaussian process
cites work: Metric entropy of some classes of sets with differentiable boundaries
cites work: Central limit theorems for empirical measures
cites work: Q3968229
cites work: Q4035374
cites work: Some limit theorems for empirical processes (with discussion)
cites work: Variance of set-indexed sums of mixing random variables and weak convergence of set-indexed processes
cites work: Q4769223
cites work: Probability Inequalities for Sums of Bounded Random Variables
cites work: Central limit theorems for C(S)-valued random variables
cites work: Q4741477
cites work: Q3038283
cites work: Random Fourier Series with Applications to Harmonic Analysis. (AM-101)
cites work: Q3700501
cites work: Q4741476
cites work: Convergence of stochastic processes
cites work: Q3962244
cites work: Q5513104
cites work: The weighted empirical process of row independent random variables with arbitrary distribution functions
cites work: Q4084474
cites work: Inequalities with Applications to the Weak Convergence of Random Processes with Multi-Dimensional Time Parameters
cites work: Q3249860
full work available at URL: https://doi.org/10.1007/bf00318707
OpenAlex ID: W2019603730
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
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description: scientific article; zbMATH DE number 3956109