A Uniform Law of the Iterated Logarithm for Classes of Functions

Let $\{\xi_k, k \geqslant 1\}$ be a sequence of random variables uniformly distributed over $\lbrack 0, 1\rbrack$ and let $\mathscr{F}$ be a class of functions on $\lbrack 0, 1\rbrack$ with $\int^1_0 f(x) dx = 0$. In this paper we give upper and lower bounds for $\sup_{f \in \mathscr{F}}|\sigma_{k \leqslant N}f(\xi_k)|$ for the class of functions of variation bounded by 1 and for the class of functions satisfying a Lipschitz condition.