Chung's Law and The Csáki Function

Abstract

We have found the limit $$L_h = \mathop {\lim \inf }\limits_{T \to \infty } (\log _2 T)^{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} \left\| {\frac{{W(T \cdot )}}{{(2T\log _2 T)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }} - h} \right\|$$ for a Wiener process W and a class of twice weakly differentiable functions h∈C[0, 1], thus solving the problem of the convergence rate in Chung's functional law for the so-called “slowest points”. Our description is closely related to an interesting functional emerging from a large deviation problem for the Wiener process in a strip.