Translator Disclaimer
June, 1977 A Functional Law of the Iterated Logarithm for Empirical Distribution Functions of Weakly Dependent Random Variables
Walter Philipp
Ann. Probab. 5(3): 319-350 (June, 1977). DOI: 10.1214/aop/1176995795

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.

Citation

Download Citation

Walter Philipp. "A Functional Law of the Iterated Logarithm for Empirical Distribution Functions of Weakly Dependent Random Variables." Ann. Probab. 5 (3) 319 - 350, June, 1977. https://doi.org/10.1214/aop/1176995795

Information

Published: June, 1977
First available in Project Euclid: 19 April 2007

zbMATH: 0362.60047
MathSciNet: MR443024
Digital Object Identifier: 10.1214/aop/1176995795

Subjects:
Primary: 60F15
Secondary: 10K05

Rights: Copyright © 1977 Institute of Mathematical Statistics

JOURNAL ARTICLE
32 PAGES


SHARE
Vol.5 • No. 3 • June, 1977
Back to Top