## Access

You are not currently logged in.

Access JSTOR through your library or other institution:

## If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Journal Article

# 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

#### Select the topics that are inaccurate.

Cancel
Preview not available

## 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.

• 582
• 583
• 584
• 585
• 586
• 587
• 588
• 589
• 590
• 591
• 592
• 593
• 594
• 595
• 596
• 597