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

Marcus, Michael B.; Zinn, Joel

Ann. Probab. 12, 335-360 (1984).

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

Keywords:

law of the iterated logarithm; weighted empirical process; Daniel’s theorem