How many distribution functions are there? Bracketing entropy bounds for high dimensional distribution functions

We establish a new upper bound for the bracketing entropy of the class ${\cal F}_d$ of $d$-dimensional distribution functions with respect to $L_r(Q)$-metrics: $\log N_{[ ]} (\epsilon, {\cal F}_d, L_r (Q)) \le K {\epsilon}^{-1} (\log(1/\epsilon))^d$ where $K$ depends only on $r$ and $d$.