## The bounded law of the iterated logarithm for the weighted empirical distribution process in the non-i.i.d. case.(English)Zbl 0538.60009

The following theorem is a consequence of a more general result obtained in the paper. Theorem. Let $$\{X_ k\}^{\infty}_{k=1}$$ be a sequence of independent, non-negative r.v., $$\psi$$ (t), $$t\geq 0$$, be a non- negative, nondecreasing function and $$\{B_ n\}$$ be an increasing sequence of positive real numbers with $$\lim_{n\to \infty}B_ n=\infty$$. Assume that for some $$\lambda,\beta \in R^ 1$$, $$\lim \sup_{n\to \infty}B_ n^{-1}| \sum^{n}_{k=1}\epsilon_ k\psi(X_ k)|<\lambda$$ a.s. and $$\lim \sup_{n\to \infty}\sup_{\psi(t)>2\lambda B_ n}B_ n^{- 1}\psi(t)\sum^{n}_{k=1}P(X_ k\geq t)<\beta$$. Then $$\lim \sup_{n\to \infty}\sup_{t>0}B_ n^{-1}| \psi(t)\sum^{n}_{k=1}(I_{(X_ k\geq t)}-P(X_ k\geq t))|<1120\lambda +20\beta$$ a.s. The proof is based on the upper bounds on the probability distribution of the weighted empirical process. These bounds are also used to obtain a one sided version of Daniel’s theorem in the non-i.i.d. case.
Reviewer: A.J.Rachkauskas

### MSC:

 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 60F15 Strong limit theorems 60F25 $$L^p$$-limit theorems 62F12 Asymptotic properties of parametric estimators
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