## The Annals of Probability

- Ann. Probab.
- Volume 9, Number 2 (1981), 323-329.

### The Empirical Discrepancy Over Lower Layers and a Related Law of Large Numbers

#### Abstract

Let $\{X_k\}$ be a sequence of independent random variables which are centered at their means; let $\{T_k\}$ be an i.i.d. sequence of $\beta$-dimensional random vectors with common distribution $\mu$; and let $\{X_k\}$ and $\{T_k\}$ be independent. With $\mathscr{L}$ the collection of lower layers, a necessary and sufficient condition for the almost sure convergence of $\sup_{L \in \mathscr{L}}|\sum^n_{k = 1} \chi L (T_k)/n - \mu(L)|$ to zero is given. In addition, this condition on $\mu$ is shown to imply that $\sup_{L \in \mathscr{L}}|\sum^n_{k = 1} X_{k\chi L}(T_k)|/n \rightarrow 0$ a.s. provided the $X_k$ satisfy a first moment-like condition. Rates of convergence are also investigated.

#### Article information

**Source**

Ann. Probab., Volume 9, Number 2 (1981), 323-329.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994475

**Digital Object Identifier**

doi:10.1214/aop/1176994475

**Mathematical Reviews number (MathSciNet)**

MR606996

**Zentralblatt MATH identifier**

0462.60032

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 62G05: Estimation

**Keywords**

Glivenko-Cantelli theorem lower layers isotone regression maxima of partial sums

#### Citation

Wright, F. T. The Empirical Discrepancy Over Lower Layers and a Related Law of Large Numbers. Ann. Probab. 9 (1981), no. 2, 323--329. doi:10.1214/aop/1176994475. https://projecteuclid.org/euclid.aop/1176994475