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DISCREPANCY FOR RANDOMIZED RIEMANN SUMS

LUCA BRANDOLINI, WILLIAM CHEN, GIACOMO GIGANTE and GIANCARLO TRAVAGLINI
Proceedings of the American Mathematical Society
Vol. 137, No. 10 (OCTOBER 2009), pp. 3187-3196
Stable URL: http://www.jstor.org/stable/40590549
Page Count: 10
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
DISCREPANCY FOR RANDOMIZED RIEMANN SUMS
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Abstract

Given a finite sequence U N = {u₁,...,u N } of points contained in the d-dimensional unit torus, we consider the L² discrepancy between the integral of a given function and the Riemann sums with respect to translations of U N . We show that with positive probability, the L² discrepancy of other sequences close to U N in a certain sense preserves the order of decay of the discrepancy of U N . We also study the role of the regularity of the given function.

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