## 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.

### MSC:

 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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### References:

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