Journal Article

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

Philipp, Walter

1977

Overview

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.

• Publisher: Institute of Mathematical Statistics,The Institute of Mathematical Statistics
• Subject: 10K05 / 60F15 / Distribution functions / empirical distribution functions / Functional law of the iterated logarithm / Hilbert spaces / Integers / lacunary sequences / Logarithms / Martingales / Mathematical vectors / mixing random variables / Random variables / Real numbers / reproducing kernel Hilbert space / uniform distribution mod 1 / Unit ball