Low-discrepancy point sets for non-uniform measures

Acta Arithmetica

Volumn 163, Issue 4, 2014, Pages 345-369

  Aistleitner, Christoph ^a   Dick, Josef ^a  

^a UNIVERSITY OF NEW SOUTH WALES   (Australia)

Author keywords

Beck Fiala theorem; Discrepancy; Inverse of discrepancy; Quasi Monte carlo method

Indexed keywords

EID: 84907386116     PISSN: 00651036     EISSN: None     Source Type: Journal    
DOI: 10.4064/aa163-4-4     Document Type: Article

References (40)

1. C. Aistleitner, Covering numbers, dyadic chaining and discrepancy, J. Complexity 27 (2011), 531-540.
2. C. Aistleitner and M. Hofer, Probabilistic discrepancy bounds for Monte Carlo point sets, Math. Comp. 83 (2014), 1373-1381.
3. K. S. Alexander, Probability inequalities for empirical processes and a law of the iterated logarithm, Ann. Probab. 12 (1984), 1041-1067;
4. Correction, ibid. 15 (1987), 428-430.
5. W. Banaszczyk, Balancing vectors and Gaussian measures of n-dimensional convex bodies, Random Structures Algorithms 12 (1998), 351-360.
6. J. Beck, Some upper bounds in the theory of irregularities of distribution, Acta Arith. 43 (1984), 115-130.
7. J. Beck and W. W. L. Chen, Irregularities of Distribution, Cambridge Univ. Press, Cambridge, 2008.
8. J. Beck and J. Spencer, Integral approximation sequences, Math. Program. 30 (1984), 88-98.
9. D. Bilyk, On Roth's orthogonal function method in discrepancy theory, Unif. Distrib. Theory 6 (2011), 143-184.
10. D. Bilyk, M. T. Lacey, and A. Vagharshakyan, On the small ball inequality in all dimensions, J. Funct. Anal. 254 (2008), 2470-2502.
11. L. Brandolini, L. Colzani, G. Gigante, and G. Travaglini, On the Koksma-Hlawka inequality, J. Complexity 29 (2013), 158-172.
12. B. Chazelle, The discrepancy method, Cambridge Univ. Press, Cambridge, 2000.
13. P. O. Chelson, Quasi-random techniques for Monte Carlo methods, PhD thesis, The Claremont Graduate Univ., 1976.
14. L. Devroye, L. Györfi, and G. Lugosi, A Probabilistic Theory of Pattern Recognition, Appl. Math. (New York) 31, Springer, New York, 1996.
15. J. Dick, F. Y. Kuo, and I. H. Sloan, High-dimensional integration: The quasi-Monte Carlo way, Acta Numer. 22 (2013), 133-288.
16. J. Dick and F. Pillichshammer, Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration, Cambridge Univ. Press, Cambridge, 2010.
17. B. Doerr, A lower bound for the discrepancy of a random point set, J. Complexity 30 (2014), 16-20.
18. M. Drmota and R. F. Tichy, Sequences, Discrepancies and Applications, Lecture Notes in Math. 1651, Springer, Berlin, 1997.
19. E. Giné and R. Nickl, An exponential inequality for the distribution function of the kernel density estimator, with applications to adaptive estimation, Probab. Theory Related Fields 143 (2009), 569-596.
20. P. Glasserman, Monte Carlo Methods in Financial Engineering, Appl. Math. (New York) 53, Springer, New York, 2004.
21. G. Harman, Variations on the Koksma-Hlawka inequality, Unif. Distrib. Theory 5 (2010), 65-78.
22. J. Hartinger and R. Kainhofer, Non-uniform low-discrepancy sequence generation and integration of singular integrands, in: Monte Carlo and Quasi-Monte Carlo Methods 2004, Springer, Berlin, 2006, 163-179.
23. D. Haussler, Sphere packing numbers for subsets of the Boolean n-cube with bounded Vapnik-Chervonenkis dimension, J. Combin. Theory Ser. A 69 (1995), 217-232.
24. S. Heinrich, E. Novak, G. W. Wasilkowski, and H. Woźniakowski, The inverse of the star-discrepancy depends linearly on the dimension, Acta Arith. 96 (2001), 279-302.
25. A. Hinrichs, Covering numbers, Vapnik-Červonenkis classes and bounds for the star-discrepancy, J. Complexity 20 (2004), 477-483.
26. E. Hlawka, Gleichverteilung und Simulation, Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 206 (1997), 183-216.
27. E. Hlawka and R. Mück, A transformation of equidistributed sequences, in: Applications of Number Theory to Numerical Analysis (Montreal, 1971), Academic Press, New York, 1972, 371-388.
28. E. Hlawka und R. Mück, Über eine Transformation von gleichverteilten Folgen. II, Computing (Arch. Elektron. Rechnen) 9 (1972), 127-138.
29. L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley-Interscience, New York, 1974.
30. L. Lovász, J. Spencer, and K. Vesztergombi, Discrepancy of set-systems and matrices, Eur. J. Combin. 7 (1986), 151-160.
31. P. Massart, About the constants in Talagrand's concentration inequalities for empirical processes, Ann. Probab. 28 (2000), 863-884.
32. J. Matousek, Geometric Discrepancy. An Illustrated Guide, Algorithms Combin. 18, Springer, Berlin, 2010.
33. N. Sauer, On the density of families of sets, J. Combin. Theory Ser. A 13 (1972), 145-147.
34. C. Schretter and H. Niederreiter, A direct inversion method for non-uniform quasirandom point sequences, Monte Carlo Methods Appl. 19 (2013), 1-9.
35. M. Talagrand, Sharper bounds for Gaussian and empirical processes, Ann. Probab. 22 (1994), 28-76.
36. M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces, Inst. Hautes Études Sci. Publ. Math. 81 (1995), 73-205.
37. V. N. Vapnik and A. Ja. Červonenkis [A. Ya. Chervonenkis], The uniform convergence of frequencies of the appearance of events to their probabilities, Teor. Veroyatnost. i Primenen. 16 (1971), 264-279 (in Russian).
38. N. Vayatis and R. Azencott, Distribution-dependent Vapnik-Chervonenkis bounds, in: Computational Learning Theory (Nordkirchen, 1999), Lecture Notes in Comput. Sci. 1572, Springer, Berlin, 1999, 230-240.
39. X. Wang and K. S. Tan, How do path generation methods affect the accuracy of quasi-Monte Carlo methods for problems in finance?, J. Complexity 28 (2012), 250-277.
40. X. Wang and K. S. Tan, Pricing and hedging with discontinuous functions: Quasi- Monte Carlo methods and dimension reduction, Management Sci. 59 (2013), 376-389.