Strong Bounds for Weighted Empirical Distribution Functions Based on Uniform Spacings

Abstract

Let $U_1, U_2,\ldots$ be a sequence of independent rv's having the uniform distribution on (0, 1). Let $\hat{F}_n$ be the empirical distribution function based on the transformed uniform spacings $\mathbb{D}_{i, n} := G(nD_{i, n}), i = 1,2,\ldots,n$, where $G$ is the $\exp(1)$ df and $D_{i, n}$ is the $i$th spacing based on $U_1, U_2,\ldots, U_{n - 1}$. In this paper a complete characterization is obtained for the a.s. behaviour of $\lim \sup_{n \rightarrow \infty}b_nV_{n, \nu}$ and $\lim \sup_{n \rightarrow \infty} b_nW_{n, \nu}$ where $\nu \in \lbrack 0, \frac{1}{2}\rbrack, \{b_n\}^\infty_{n = 1}$ is a sequence of norming constants, $V_{n, \nu} = \sup_{0 < t < 1} \frac{n|\hat{F}_n(t) - t|}{t^{1 - \nu}} \quad\text{and}\quad W_{n, \nu} = \sup_{0 < t < 1} \frac{n|\hat{F}_n(t) - t|}{(1 - t)^{1 - \nu}}.$ It turns out that compared with the i.i.d. case only $W_{n, \nu}$ behaves differently. The results imply, e.g., laws of the iterated logarithm for $\log(n^{\nu - 1}V_{n, \nu})$ and $\log(n^{\nu - 1}W_{n, \nu})$. Of independent interest is the theorem on the lower-upper class behaviour of the maximal spacing, which gives the final solution for this problem and generalizes some recent results in the literature.