## The Annals of Probability

- Ann. Probab.
- Volume 3, Number 5 (1975), 762-772.

### A Functional Law of the Iterated Logarithm for Weighted Empirical Distributions

#### Abstract

Finkelstein's (1971) functional law of the iterated logarithm for empirical distributions is extended to cases where the empirical distribution is multiplied by a weight function, $w$. We let $X_1, X_2, \cdots$ be independent random variables each having the uniform distribution on $\lbrack 0, 1 \rbrack$, with $F_n$ the empirical df at stage $n$. The weight function $w$, defined on $\lbrack 0, 1 \rbrack$, is assumed to be bounded on interior intervals and to satisfy some smoothness conditions. Then convergence of the integral $\int^1_0 w^2(t)/\log \log(t^{-1}(1 - t)^{-1})dt$ is seen to be a necessary and sufficient condition for the sequence $\{U_n: n \geqq 3\}$, defined by $$U_n(t) = \frac{n^{\frac{1}{2}}w(t)(F_n(t) - t)}{(2 \log \log n)^{\frac{1}{2}}}$$ to be uniformly compact on a set of probability one, with set of limit points $$K_w = \{wf: f \in K\}$$. $K$ is the set set of absolutely continuous functions on $\lbrack 0, 1 \rbrack$ with $f(0) = 0 = f(1)$ and $$\int^1_0 \lbrack f'(t) \rbrack^2 dt \leqq 1.$$

#### Article information

**Source**

Ann. Probab., Volume 3, Number 5 (1975), 762-772.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996263

**Digital Object Identifier**

doi:10.1214/aop/1176996263

**Mathematical Reviews number (MathSciNet)**

MR402881

**Zentralblatt MATH identifier**

0347.60030

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 60F20: Zero-one laws

**Keywords**

Law of the iterated logarithm weighted empirical distribution functions relatively compact sequence Strassen-type limit set

#### Citation

James, Barry R. A Functional Law of the Iterated Logarithm for Weighted Empirical Distributions. Ann. Probab. 3 (1975), no. 5, 762--772. doi:10.1214/aop/1176996263. https://projecteuclid.org/euclid.aop/1176996263