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.