Probability estimates for the distribution of Kolmogorov distance in the worst direction

Abstract

ConsiderD n the maximum Kolmogorov distance betweenP n andP among all possible one-dimensional projections, whereP n is an empirical measure based ond-dimensional i.i.d vectors with spherically symmetric probability measureP. We show in this paper that $$c_1 \lambda ^2 \exp ( - 2\lambda ^2 ) \leqslant P(\sqrt n D_n > \lambda )$$ for large λ,d≥2 and an appropriate constantc 1. From this, when dimensiond is fixed, we give a negative answer to Huber's conjecture,P(D n >e)≤N exp(−2nɛ 2), whereN is a constant depending only on dimensiond.