## The Annals of Probability

- Ann. Probab.
- Volume 14, Number 2 (1986), 582-597.

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

Kenneth S. Alexander and Ronald Pyke

#### 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 $X_j$.

#### Article information

**Source**

Ann. Probab., Volume 14, Number 2 (1986), 582-597.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176992532

**Digital Object Identifier**

doi:10.1214/aop/1176992532

**Mathematical Reviews number (MathSciNet)**

MR832025

**Zentralblatt MATH identifier**

0595.60027

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60B10: Convergence of probability measures

**Keywords**

Partial-sum processes metric entropy weak convergence set-indexed processes Gaussian processes

#### Citation

Alexander, Kenneth S.; Pyke, Ronald. A Uniform Central Limit Theorem for Set-Indexed Partial-Sum Processes with Finite Variance. Ann. Probab. 14 (1986), no. 2, 582--597. doi:10.1214/aop/1176992532. https://projecteuclid.org/euclid.aop/1176992532