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.
Citation
John H. J. Einmahl. Martien C. A. van Zuijlen. "Strong Bounds for Weighted Empirical Distribution Functions Based on Uniform Spacings." Ann. Probab. 16 (1) 108 - 125, January, 1988. https://doi.org/10.1214/aop/1176991888
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