Chung-type functional laws of the iterated logarithm for tail empirical processes
The tail empirical process is defined by \(h^{-1/2}\alpha_n(h_n u)\) for \(0\leq u \leq 1\), where \(\alpha_n\) denotes the uniform empirical process based upon \(n\) independent uniform \((0,1)\) random variables. The sequence of positive constants is assumed to fulfill the following assumptions: (H.1) \(h_n\to 0\) monotonically, \(nh_n\) increases monotonically; (H.2) \(nh_n/\log_2 n\to \infty\) or (H.3) \(nh_n/(\log_2 n)^3\to\infty\). \textit{D. M. Mason} [Ann. Inst. Henri Poincaré, Probab. Stat. 24, No.~4, 491-506 (1988; Zbl 0664.60038)] proved under (H.1) and (H.2) that the sequence of functions \(f_n = (2h_n\log_2 n)^{-1/2}\alpha_n(h_n\cdot)\) is almost surely compact with respect to the topology defined by the sup-norm \(\|\cdot\|\), and gave a characterization of the corresponding limit set \({\mathbb{K}}\). Let \((B[0,1],{\mathcal U})\) denote the set \(B[0,1]\) of all bounded functions \(f\) on \([0,1]\), endowed with the uniform topology \({\mathcal U}\), induced by the sup-norm \(\|f\|= \sup_{0\leq u \leq 1}|f(u)|\). Let \(AC_0[0,1]\) denote the set of all absolutely continuous functions \(f\) on \([0,1]\), with Lebesgue derivative \(\dot{f}(u)\), and such that \(f(0)= 0\). For each \(f\in B[0,1]\), set \(|f|_{\mathbb{H}} = (\int_0^1\dot{f}(u)^2)^{1/2}\) if \(f\in AC_0[0,1]\), and \(|f|_{\mathbb{H}} = \infty\) otherwise. Let \({\mathbb{H}} = \{f\in B[0,1]: |f|_{\mathbb{H}}<\infty\}\) denote the Hilbert subspace of \(AC_0[0,1]\) with the norm \(|\cdot|_{\mathbb{H}}\). Then \({\mathbb{K}} = \{f\in AC_0[0,1]{:} |f|_{\mathbb{H}}\leq 1\}\). An estimate of the rate of this limit law is obtained. The result reads as: Assume that (H.1) and (H.3) hold. Then, for each \(f\in{\mathbb{K}}\), we have \[ \liminf_{n\to\infty}(\log_2 n)\|(2\log_2 n)^{-1/2}\*h_n^{-1/2}\*\alpha_n\*(h_n\cdot) - f\|=\Bigl(4\sqrt{1-|f|^2_{\mathbb{H}}}\Bigr)^{-1}\pi \quad \text{a.s}. \]
functional laws of the iterated logarithm
scientific article; zbMATH DE number 1552676