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April, 1986 A Uniform Central Limit Theorem for Set-Indexed Partial-Sum Processes with Finite Variance
Kenneth S. Alexander, Ronald Pyke
Ann. Probab. 14(2): 582-597 (April, 1986). DOI: 10.1214/aop/1176992532

Abstract

Given a class $\mathscr{A}$ of subsets of $\lbrack 0, 1\rbrack^d$ and an array $\{X_j: \mathbf{j} \in \mathbb{Z}^d_+\}$ of independent identically distributed random variables with $EX_j = 0, EX^2_j = 1$, the (unsmoothed) partial-sum process $S_n$ is given by $S_n(A) := n^{-d/2}\sum_{j \in n A}X_j, A \in \mathscr{A}$. If for the metric $\rho(A, B) = |A \Delta B|$ the metric entropy with inclusion $N_1(\varepsilon, \mathscr{A}, \rho)$ satisfies $\int^1_0(\varepsilon^{-1} \log N_I(\varepsilon, \mathscr{A}, \rho))^{1/2} d\varepsilon < \infty$, then an appropriately smoothed version of the partial-sum process converges weakly to the Brownian process indexed by $\mathscr{A}$. This improves on previous results of Pyke (1983) and of Bass and Pyke (1984) which require stronger conditions on the moments of $X_j$.

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Kenneth S. Alexander. Ronald Pyke. "A Uniform Central Limit Theorem for Set-Indexed Partial-Sum Processes with Finite Variance." Ann. Probab. 14 (2) 582 - 597, April, 1986. https://doi.org/10.1214/aop/1176992532

Information

Published: April, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0595.60027
MathSciNet: MR832025
Digital Object Identifier: 10.1214/aop/1176992532

Subjects:
Primary: 60F05
Secondary: 60B10

Keywords: Gaussian processes , Metric entropy , partial-sum processes , set-indexed processes , weak convergence

Rights: Copyright © 1986 Institute of Mathematical Statistics

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Vol.14 • No. 2 • April, 1986
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