## The Annals of Probability

- Ann. Probab.
- Volume 5, Number 3 (1977), 319-350.

### A Functional Law of the Iterated Logarithm for Empirical Distribution Functions of Weakly Dependent Random Variables

#### Abstract

Let $\{n_k, k \geqq 1\}$ be a sequence of random variables uniformly distributed over $\{0, 1\}$ and let $F_N(t)$ be the empirical distribution function at stage $N$. Put $f_n(t) = N(F_N(t) - t)(N\log\log N)^{-\frac{1}{2}}, 0 \leqq t \leqq 1, N \geqq 3$. For strictly stationary sequences $\{n_k\}$ where $n_k$ is a function of random variables satisfying a strong mixing condition or where $n_k = n_k x \mod 1$ with $\{n_k, k \geqq 1\}$ a lacunary sequence of real numbers a functional law of the iterated longarithm is proven: The sequence $\{f_N(t), N \geqq 3\}$ is with probability 1 relatively compact in $D\lbrack 0, 1\rbrack$ and the set of its limits is the unit ball in the reproducing kernel Hilbert space associated with the covariance function of the appropriate Gaussian process.

#### Article information

**Source**

Ann. Probab., Volume 5, Number 3 (1977), 319-350.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176995795

**Digital Object Identifier**

doi:10.1214/aop/1176995795

**Mathematical Reviews number (MathSciNet)**

MR443024

**Zentralblatt MATH identifier**

0362.60047

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 10K05

**Keywords**

Functional law of the iterated logarithm empirical distribution functions mixing random variables lacunary sequences reproducing kernel Hilbert space uniform distribution mod 1

#### Citation

Philipp, Walter. A Functional Law of the Iterated Logarithm for Empirical Distribution Functions of Weakly Dependent Random Variables. Ann. Probab. 5 (1977), no. 3, 319--350. doi:10.1214/aop/1176995795. https://projecteuclid.org/euclid.aop/1176995795