A Functional Law of the Iterated Logarithm for Weighted Empirical Distributions

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 X₁, X₂, ⋯ be independent random variables each having the uniform distribution on 0, 1, with Fₙ the empirical df at stage n. The weight function w, defined on 0, 1, is assumed to be bounded on interior intervals and to satisfy some smoothness conditions. Then convergence of the integral ∫¹₀ w²(t)/log log(t⁻¹(1 - t)⁻¹)dt is seen to be a necessary and sufficient condition for the sequence ₙ: n 3\, defined by Uₙ(t) = n1/2w(t)(Fₙ(t) - t)(2 log log n)1/2 to be uniformly compact on a set of probability one, with set of limit points Kw = : f ∈ K\. K is the set set of absolutely continuous functions on 0, 1 with $f(0) = 0 = f(1)$ and ∫¹₀ f'(t) ² dt 1.