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DYADIC SHIFT RANDOMIZATION IN CLASSICAL DISCREPANCY THEORY
Author(s) -
Skriganov M. M.
Publication year - 2015
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s002557931500008x
Subject(s) - mathematics , unit cube , exponent , generalization , upper and lower bounds , distribution (mathematics) , independence (probability theory) , combinatorics , cube (algebra) , discrete mathematics , mathematical analysis , statistics , philosophy , linguistics
Dyadic shifts D ⊕ 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 ⩽ ∞ . 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 shiftsD N ⊕ T such that the L q ‐discrepancyL q [ D N ⊕ T ] > c d , q( log N ) ( 1 / 2 ) ( d ‐ 1 )(Theorem 2.2). The lower bounds for the L ∞ ‐discrepancy are also considered in the paper. It is shown that for an arbitrary N ‐point distribution D N , there exist dyadic shiftsD N ⊕ T such thatL ∞ [ D N ⊕ T ] > c d( log N ) ( 1 / 2 ) d(Theorem 2.3).