## The Annals of Probability

### Functional Laws of the Iterated Logarithm for the Increments of Empirical and Quantile Processes

#### Abstract

Let $\{\alpha_n(t), 0 \leq t \leq 1\}$ and $\{\beta_n(t), 0 \leq t \leq 1\}$ be the empirical and quantile processes generated by the first $n$ observations from an i.i.d. sequence of uniformly distributed random variables on (0,1). Let $0 < a_n < 1$ be a sequence of constants such that $a_n \rightarrow 0$ as $n \rightarrow \infty$. We investigate the strong limiting behavior as $n \rightarrow \infty$ of the increment functions $\{\alpha_n(t + a_ns) - \alpha_n(t), 0 \leq s \leq 1\}$ and $\{\beta_n(t + a_ns) - \beta_n(t), 0 \leq s \leq 1\},$ where $0 \leq t \leq 1 - a_n$. Under suitable regularity assumptions imposed upon $a_n$, we prove functional laws of the iterated logarithm for these increment functions and discuss statistical applications in the field of nonparametric estimation.

#### Article information

Source
Ann. Probab., Volume 20, Number 3 (1992), 1248-1287.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176989691

Digital Object Identifier
doi:10.1214/aop/1176989691

Mathematical Reviews number (MathSciNet)
MR1175262

Zentralblatt MATH identifier
0767.60028

JSTOR
links.jstor.org

Subjects
Primary: 60F15: Strong theorems
Secondary: 60F17: Functional limit theorems; invariance principles 62G05: Estimation

#### Citation

Deheuvels, Paul; Mason, David M. Functional Laws of the Iterated Logarithm for the Increments of Empirical and Quantile Processes. Ann. Probab. 20 (1992), no. 3, 1248--1287. doi:10.1214/aop/1176989691. https://projecteuclid.org/euclid.aop/1176989691