DYADIC SHIFT RANDOMIZATION IN CLASSICAL DISCREPANCY THEORY

Dyadic shifts D circle plus T of point distributions D in the d-dimensional unit cube U-d are considered as a form of randomization. Explicit formulas for the L-q-discrepancies of such randomized distributions are given in the paper in terms of Rademacher functions. Relying on the statistical independence of Rademacher functions, Khinchin's inequalities, and other related results, we obtain very sharp upper and lower bounds for the mean L-q-discrepancies, 0 < q <= infinity. The upper bounds imply directly a generalization of the well-known Chen theorem on mean discrepancies with respect to dyadic shifts (Theorem 2.1). From the lower bounds, it follows that for an arbitrary N-point distribution D-N and any exponent 0 < q <= 1, there exist dyadic shifts D-N circle plus T such that the L-q-discrepancy Lq [D-N circle plus T] > c(d, q.) (log N) ((1/2/)) ((d-1)) Theorem 2.2). The lower bounds for the L-infinity-discrepancy are also considered in the paper. It is shown that for an arbitrary N-point distribution D-N, there exist dyadic shifts D-N circle plus T such that L-infinity[D-N circle plus T] > c(d) (log N) ((1/2)d) (Theorem 2.3).