## The Annals of Probability

- Ann. Probab.
- Volume 12, Number 2 (1984), 335-360.

### The Bounded Law of the Iterated Logarithm for the Weighted Empirical Distribution Process in the Non-I.I.D. Case

Michael B. Marcus and Joel Zinn

#### Abstract

Using a simple symmetrization procedure, an upper bound is obtained for the probability distribution of various kinds of weighted empirical distribution processes where the underlying real valued random variables are not identically distributed. These probability bounds are used to obtain bounded laws of the iterated logarithm for empirical processes with different kinds of weighting. They are also used to obtain a one sided version of Daniel's theorem in the non-i.i.d. case.

#### Article information

**Source**

Ann. Probab., Volume 12, Number 2 (1984), 335-360.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993294

**Digital Object Identifier**

doi:10.1214/aop/1176993294

**Mathematical Reviews number (MathSciNet)**

MR735842

**Zentralblatt MATH identifier**

0538.60009

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)

Secondary: 60F15: Strong theorems 62F25: Tolerance and confidence regions 62F12: Asymptotic properties of estimators

**Keywords**

Empirical distribution process law of the iterated logarithm Daniel's theorem

#### Citation

Marcus, Michael B.; Zinn, Joel. The Bounded Law of the Iterated Logarithm for the Weighted Empirical Distribution Process in the Non-I.I.D. Case. Ann. Probab. 12 (1984), no. 2, 335--360. doi:10.1214/aop/1176993294. https://projecteuclid.org/euclid.aop/1176993294