Statistics and Computing

, Volume 29, Issue 1, pp 53–65 | Cite as

Asymptotic normality of extensible grid sampling

  • Zhijian HeEmail author
  • Lingjiong Zhu


Recently, He and Owen (J R Stat Soc Ser B 78(4):917–931, 2016) proposed the use of Hilbert’s space filling curve (HSFC) in numerical integration as a way of reducing the dimension from \(d>1\) to \(d=1\). This paper studies the asymptotic normality of the HSFC-based estimate when using one-dimensional stratification inputs. In particular, we are interested in using scrambled van der Corput sequence in any base \(b\ge 2\) with sample sizes of the form \(n=b^m\), for which the sampling scheme is extensible in the sense of multiplying the sample size by a factor of b. We show that the estimate has an asymptotic normal distribution for functions in \(C^1([0,1]^d)\), excluding the trivial case of constant functions. The asymptotic normality also holds for discontinuous functions under mild conditions. Previously, it was only known that scrambled (0, md)-net quadratures enjoy the asymptotic normality for smooth enough functions, whose mixed partial gradients satisfy a Hölder condition. As a by-product, we find lower bounds for the variance of the HSFC-based estimate. Particularly, for non-trivial functions in \(C^1([0,1]^d)\), the lower bound is of order \(n^{-1-2/d}\), which matches the rate of the upper bound established in He and Owen (2016).


Asymptotic normality Hilbert’s space filling curve Van der Corput sequence Randomized quasi-Monte Carlo Extensible grid sampling 



The authors are enormously grateful to the Editor, the Associate Editor, and three anonymous referees whose suggestions and comments have greatly improved the quality of the paper. The authors also thank Professor Art B. Owen for the helpful comments.


  1. Ambrosio, L., Colesanti, A., Villa, E.: Outer Minkowski content for some classes of closed sets. Math. Ann. 342(4), 727–748 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Basu, K., Mukherjee, R.: Asymptotic normality of scrambled geometric net quadrature. Ann. Stat. 45(4), 1759–1788 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Billingsley, P.: Probability and Measure, 3rd edn. Wiley, New York (1995)zbMATHGoogle Scholar
  4. Butz, A.R.: Alternative algorithm for Hilbert’s space-filling curve. IEEE Trans. Comput. 20(4), 424–426 (1971)CrossRefzbMATHGoogle Scholar
  5. Dick, J., Niederreiter, H.: On the exact \(t\)-value of Niederreiter and Sobol’ sequences. J. Complex. 24(5), 572–581 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Dick, J., Pillichshammer, F.: Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  7. Dick, J., Kuo, F.Y., Sloan, I.H.: High-dimensional integration: the quasi-Monte Carlo way. Acta Numer. 22, 133–288 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Gerber, M., Chopin, N.: Sequential quasi Monte Carlo. J. R. Stat. Soc. Ser. B 77(3), 509–579 (2015)MathSciNetCrossRefGoogle Scholar
  9. Gerber, M., Chopin, N.: Convergence of sequential quasi-Monte Carlo smoothing algorithms. Bernoulli 23(4B), 2951–2987 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. He, Z., Owen, A.B.: Extensible grids: uniform sampling on a space filling curve. J. R. Stat. Soc. Ser. B 78(4), 917–931 (2016)MathSciNetCrossRefGoogle Scholar
  11. He, Z., Wang, X.: On the convergence rate of randomized quasi-Monte Carlo for discontinuous functions. SIAM J. Numer. Anal. 53(5), 2488–2503 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Lawder, J.K.: Calculation of mappings between one and \(n\)-dimensional values using the Hilbert space-filling curve. Research Report JL1/00, Birkbeck College, University of London, London (2000)Google Scholar
  13. L’Ecuyer, P.: Quasi-Monte Carlo methods with applications in finance. Finance Stoch. 13(3), 307–349 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. L’Ecuyer, P., Lemieux, C.: Recent advances in randomized quasi-Monte Carlo methods. In: Dror, M., L’Ecuyer, P., Szidarovszky, F. (eds.) Modeling Uncertainty: An Examination of Stochastic Theory, Methods, and Applications, pp. 419–474. Kluwer Academic, Boston (2002)Google Scholar
  15. L’Ecuyer, P., Munger, D., Tuffin, B.: On the distribution of integration error by randomly-shifted lattice rules. Electron. J. Stat. 4, 950–993 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Loh, W.-L.: On the asymptotic distribution of scrambled net quadrature. Ann. Stat. 31, 1282–1324 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Matoušek, J.: On the \({L}_2\)-discrepancy for anchored boxes. J. Complex. 14(4), 527–556 (1998)CrossRefzbMATHGoogle Scholar
  18. Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992)CrossRefzbMATHGoogle Scholar
  19. Novak, E.: Deterministic and Stochastic Error Bounds in Numerical Analysis. Springer, Berlin (1988)CrossRefzbMATHGoogle Scholar
  20. Owen, A.B.: Randomly permuted \((t, m, s)\)-nets and \((t, s)\)-sequences. In: Niederreiter, H., Shiue, P.J.-S. (eds.) Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, pp. 299–317. Springer, Berlin (1995)CrossRefGoogle Scholar
  21. Owen, A.B.: Scrambled net variance for integrals of smooth functions. Ann. Stat. 25(4), 1541–1562 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Owen, A.B.: Monte Carlo Theory, Methods and Examples (2013). owen/mc/. Accessed Aug 2017Google Scholar
  23. Schretter, C., Niederreiter, H.: A direct inversion method for non-uniform quasi-random point sequences. Monte Carlo Methods Appl. 19(1), 1–9 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Schretter, C., He, Z., Gerber, M., Chopin, N., Niederreiter, H.: Van der Corput and golden ratio sequences along the Hilbert space-filling curve. In: Cools, R., Nuyens, D. (eds.) Monte Carlo and Quasi-Monte Carlo Methods, Springer Proceedings in Mathematics and Statistics, vol. 163, pp. 531–544. Springer, Berlin (2016)Google Scholar
  25. van der Corput, J.G.: Verteilugsfunktionen I. Nederl. Akad. Wetensch. Proc. 38, 813–821 (1935)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of MathematicsSouth China University of TechnologyGuangzhouChina
  2. 2.Department of MathematicsFlorida State UniversityTallahasseeUSA

Personalised recommendations