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A Uniform Central Limit Theorem for Set-Indexed Partial-Sum Processes with Finite Variance

Kenneth S. Alexander and Ronald Pyke
The Annals of Probability
Vol. 14, No. 2 (Apr., 1986), pp. 582-597
Stable URL: http://www.jstor.org/stable/2243928
Page Count: 16
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A Uniform Central Limit Theorem for Set-Indexed Partial-Sum Processes with Finite Variance
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Abstract

Given a class A of subsets of [ 0, 1]d and an array {Xj: j ∈ Zd +} of independent identically distributed random variables with EXj = 0, EX2 j = 1, the (unsmoothed) partial-sum process Sn is given by Sn(A) := n-d/2∑j ∈ n AXj, A ∈ A. If for the metric ρ(A, B) = |A Δ B| the metric entropy with inclusion N1(ε, A, ρ) 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 A. This improves on previous results of Pyke (1983) and of Bass and Pyke (1984) which require stronger conditions on the moments of Xj.

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