Summary: [For the entire collection see Zbl 0752.00061.]
Consider \(D_ n\), the maximum Kolmogorov distance between \(P_ n\) and \(P\) among all possible one-dimensional projections, where \(P_ n\) is an empirical measure based on \(d\)-dimensional i.i.d vectors with elliptically symmetric probability measure \(P\). For the supremum of Gaussian process, \(\sup_{\mathcal F}| W_ P(f)|\), indexed by the class of half-spaces of \({\mathcal F}\), which is the weak limit of \(\sqrt nD_ n\), it is shown that \[ \sum^ \infty_{m=1}\left({1\over 2}m^ 2\lambda^ 2-{1\over 8}\right)\exp(-2m^ 2\lambda^ 2)\leq P\bigl(\sup_{\mathcal F}| W_ P(f)|>\lambda\bigr)\leq c\lambda^{2(d-1)}\exp(-2\lambda^ 2) \] for large \(\lambda\), \(d\geq 2\) and an appropriate constant \(c\). From this, when dimension \(d\) is fixed, I give a negative answer for Huber’s conjecture \(P(D_ n>\varepsilon)\leq N\exp(-2n\varepsilon^ 2)\), where \(N\) is a constant depending only on dimension \(d\).