Rates of convergence for classes of functions: The non-i.i.d. case
type: Item
instance of: scholarly article
title: Rates of convergence for classes of functions: The non-i.i.d. case
zbMATH Open document ID: 0604.60029
DOI: 10.1016/0047-259X(86)90076-X
published in: Journal of Multivariate Analysis
publication date: 1986-01-01T00:00:00Z
review text: Let \(\{X_ i:\) \(i\geq 1\}\) be a sequence of \(\phi\)-mixing random variables taking values in a sample space (X,A). Let \(G_ n\), \(n\geq 1\), be classes of real-valued measurable functions, \(g: X\to {\mathbb{R}}\). For such \(g\in G_ n\), define \[ S_ n(g)=\sum^{n}_{i=1}\{g(X_ i)- Eg(X_ i)\}. \] This paper establishes bounds on the fluctuations of \(S_ n(g)\) over \(g\in G_ n\) and \(n\geq 1\). Almost surely, with respect to the outer probability measure, \[ \overline{\lim}_{n\to \infty}\sup_{g\in G_ n}| H^{-1}(nV)^{-1/2}S_ n(g)| \leq U<+\infty. \] Here V is the supremum of \(Eg^ 2(X_ i)\) over \(g\in G_ n\) and \(i\leq n\). \(H^ 2\) is the entropy of the class \(G_ n\). These results require that the \(\phi\)-mixing coefficients of \(\{X_ i:i\geq 1\}\) be summable, that \(H(V/n)^{1/2}\) be bounded or convergent to 0, and that the entropies of the classes \(G_ n\) be suitably controlled. The proofs combine the techniques of metric entropy with bracketing, generalizations of inequalities due to Bernstein and Ottaviani to \(\phi\)- mixing sequences, and blocking techniques for \(\phi\)-mixing sequences.
reviewed by: Andre R. Dabrowski
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60F99
Mathematics Subject Classification ID: 62H15
zbMATH DE Number: 3975995
zbMATH Keywords: rates of convergence
zbMATH Keywords: \(\phi \)-mixing random variables
zbMATH Keywords: metric entropy
zbMATH Keywords: blocking techniques
author: J. E. Yukich
MaRDI profile type: Publication
cites work: Probability inequalities for empirical processes and a law of the iterated logarithm
cites work: Q5560061
cites work: Q3346754
cites work: Propri�t�s de convergence presque compl�te du pr�dicteur � noyau
cites work: Central limit theorems for empirical measures
cites work: Q4181698
cites work: Some limit theorems for empirical processes (with discussion)
cites work: A functional central limit theorem for weakly dependent sequences of random variables
cites work: Q5590550
cites work: Q3946851
cites work: Q5774894
cites work: Convergence of sums of mixing triangular arrays of random vectors with stationary rows
cites work: Limit theorems and inequalities for the uniform empirical process indexed by intervals
cites work: The oscillation behavior of empirical processes
cites work: A law of the logarithm for kernel density estimators
cites work: The oscillation behavior of empirical processes: The multivariate case
cites work: Q3723398
cites work: Laws of large numbers for classes of functions
full work available at URL: 0047-259x(86)90076-x
OpenAlex ID: W1966425380
description: scientific article; zbMATH DE number 3975995