The Annals of Probability

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

Walter Philipp

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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


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