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Báo cáo toán học: "The asymptotic behavior of the average Lp−discrepancies and a randomized discrepancy"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: The asymptotic behavior of the average Lp−discrepancies and a randomized discrepancy. | The asymptotic behavior of the average Lp discrepancies and a randomized discrepancy Stefan Steinerberger Department of Financial Mathematics University of Linz Altenbergstrafie 69 A-4040 Linz Austria stefan.steinerberger@gmail.com Submitted June 7 2010 Accepted Jul 30 2010 Published Aug 9 2010 Mathematics Subject Classification 11K06 11K38 60D05 Keywords discrepancy average Lp discrepancy Abstract This paper gives the limit of the average Lp star and the average Lp extreme discrepancy for 0 1 d and 0 p TO. This complements earlier results by Heinrich Novak Wasilkowski WoZniakowski Hinrichs Novak and Gnewuch and proves that the hitherto best known upper bounds are optimal up to constants. We furthermore introduce a new discrepancy DN by taking a probabilistic approach towards the extreme discrepancy DN. We show that it can be interpreted as a centralized L1 discrepancy D f provide upper and lower bounds and prove a limit theorem. 1 Introduction. This paper discusses two relatively separate problems in discrepancy theory one being well-known and one being introduced. The reason for doing so is that our solution for the former was actually inspired by our investigating the average case for the latter. The paper is structured as follows We introduce the Lp discrepancies known results and a motivation behind a probabilistic approach towards discrepancy in this section give our results in the second section and provide proofs in the last part of the paper. The author is supported by the Austrian Science Foundation FWF Project S9609 part of the Austrian National Research Network Analytic Combinatorics and Probabilistic Number Theory . THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R106 1 Lp discrepancies. In a seminal paper Heinrich Novak Wasilkowski and Wozniakowski 5 used probabilistic methods to estimate the inverse of the star-discrepancy which is of great interest for Quasi Monte Carlo methods. Their approach relies on the notion of the average Lp star discrepancy.