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 $$\int^1_0 \lbrack f'(t) \rbrack^2 dt \leqq 1.$$