Functional Laws of the Iterated Logarithm for the Increments of Empirical and Quantile Processes

Abstract

Let $\{\alpha_n(t), 0 \leq t \leq 1\}$ and $\{\beta_n(t), 0 \leq t \leq 1\}$ be the empirical and quantile processes generated by the first $n$ observations from an i.i.d. sequence of uniformly distributed random variables on (0,1). Let $0 < a_n < 1$ be a sequence of constants such that $a_n \rightarrow 0$ as $n \rightarrow \infty$. We investigate the strong limiting behavior as $n \rightarrow \infty$ of the increment functions $\{\alpha_n(t + a_ns) - \alpha_n(t), 0 \leq s \leq 1\}$ and $\{\beta_n(t + a_ns) - \beta_n(t), 0 \leq s \leq 1\},$ where $0 \leq t \leq 1 - a_n$. Under suitable regularity assumptions imposed upon $a_n$, we prove functional laws of the iterated logarithm for these increment functions and discuss statistical applications in the field of nonparametric estimation.