Functional laws of the iterated logarithm for large increments of empirical and quantile processes. (English) Zbl 0767.60028

Let \(U_ 1,U_ 2,\dots\) be i.i.d. r.v.’s with a uniform distribution on (0,1) and let \(\alpha_ n(s)=n^{1/2}(F_ n(s)-s)\) and \(\xi_ n(h,t;s)=\alpha_ n(t+hs)-\alpha_ n(t)\), where \(F_ n(s)=n^{- 1}\#\{U_ i\leq s: 1\leq i\leq s\}\), \(n\geq 1\), are empirical distribution functions. The random sets \[ {\mathcal E}_ n(h_ n)=\{(2h_ n\log\log n)^{-1/2}\xi_ n(h_ n,t;\cdot): 0\leq t\leq 1-h_ n\}, \] are investigated in the space \(B(0,1)\) of all bounded functions on \(\langle 0,1\rangle\) with the usual sup-norm. A sequence \(\{{\mathcal A}_ n\}\) of subsets of \(B(0,1)\) which is relatively compact (i.e. contained in a compact subset \({\mathcal K}\subset B(0,1))\) is said to have limit set \({\mathcal B}\subset B(0,1)\), if \({\mathcal B}\) consists of all limits of convergent subsequences \(f_{n_ j}\in{\mathcal A}_{n_ j}\), \(1\leq n_ 1<n_ 2<\dots\) as \(j\to\infty\), and minimally covers \({\mathcal B}'\subset B(0,1)\), if \({\mathcal B}'\) is the set of all limits of convergent sequences \(f_ n\in{\mathcal A}_ n\) as \(n\to\infty\). The main Theorem 1.3 states that under the assumptions \(0<h_ n<1\), \(h_ n\downarrow 0\), \(n\cdot h_ n\uparrow\infty\) and \(\log(1/h_ n)/\log\log n\to\infty\) as \(n\to\infty\), the sequence \(\{{\mathcal E}_ n(h_ n)\}\) is a.s. relatively compact in \(B(0,1)\) with limit set equal to \({\mathcal S}_{c+1}\), and minimally covers \({\mathcal S}_ c\), where \({\mathcal S}_ \lambda=\{f\in B(0,1): f(0)=0,f\text{ is absolutely continuous and has the Lebesgue derivative } \dot f\) satisfying \(\int^ 1_ 0(\dot f(s))^ 2ds\leq\lambda\}.\) The case when \(0<h_ n<1\), \(h_ n\to h\in(0,1)\) as \(n\to\infty\) is also considered, and all the results are formulated simultaneously for increments of empirical quantile processes \(\beta_ n(s)=n^{1/2}(Q_ n(s)-s)\), formed by means of the empirical quantile functions \(Q_ n(s)=\inf\{t\geq 0: F_ n(t)\geq s\}\), \(0\leq s\leq 1\), \(n\geq 1\). Moreover, various applications of the obtained limit theorems concerning continuous functionals and oscillation moduli of empirical and quantile processes, as well as nonparametric density estimation are described.


60F15 Strong limit theorems
60F17 Functional limit theorems; invariance principles
60G05 Foundations of stochastic processes
62G30 Order statistics; empirical distribution functions
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