ABSTRACT
Recall a sequence of random vectors { ~~~} is discretized if for every € > 0 there exists an integer q and a sequence (C11 ) of events such that Pe(C11 )?:: 1-€ and the image {~11 (w): wE e11 } of e11 under~~~ has at most q elements for all 11. A possible way to discretize a d-dimensional yl/1-consistent estimate is to round the estimate to the closest point on the grid {(a1, ••• ,att)fyl/1:a11 ... ,ad= 0, l, -1,2, -2, ... }. More sophisticated methods are described in Fabian and Hannan (1982). We should stress that the following results do not depend on the way the estimates are discretized. We also believe that discretization can be avoided at the expense of more complicated proofs.
