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This article provides a strong law of large numbers for integration on digital nets randomized by a nested uniform scramble.
Über Fouriersche Reihen mit zwei veränderlichen Größen
M. Krause · 1903
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On double Fourier series, and especially those which represent the double zeta-function with real and incommensurable parameters
G. H. Hardy · 1905
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On definitions of bounded variation for functions of two variables
J. A. Clarkson and C. R. Adams · 1933
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The distribution of points in a cube and the accurate evaluation of integrals (in Russian)
I. M. Sobol’ · 1967
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Calculation of improper integrals using uniformly distributed sequences
I. M. Sobol’ · 1973
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Uniform Distribution of Sequences
L. Kuipers and H. Niederreiter · 1974
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Randomization of number theoretic methods for multiple integration
R. Cranley and T. N. L. Patterson · 1976
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Pseudo-random numbers and optimal coefficients
H. Niederreiter · 1977
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An elementary proof of the strong law of large numbers
N. Etemadi · 1981
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Discrépance de suites associées à un système de numération (en dimension s s )
H. Faure · 1982
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Methods of Numerical Integration (2nd Ed.)
P. J. Davis and P. Rabinowitz · 1984
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Non-uniform Random Variate Generation
Luc Devroye · 1986
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Point sets and sequences with small discrepancy
H. Niederreiter · 1987
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Interpolation of Operators
C. Bennett and R. Sharpley · 1988
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Random Number Generation and Quasi-Monte Carlo Methods
H. Niederreiter · 1992
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Continuity of monotone functions
B. Lavrič · 1993
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Random approximation in numerical analysis
S. Heinrich · 1994
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Lattice Methods for Multiple Integration
I. H. Sloan and S. Joe · 1994
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A quasi-Monte Carlo algorithm for the global illumination problem in a radiosity setting
A. Keller · 1995
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Randomly permuted ( t , m , s ) (t,m,s) -nets and ( t , s ) (t,s) -sequences
A. B. Owen · 1995
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Low-discrepancy sequences and global function fields with many rational places
H. Niederreiter and C. Xing · 1996
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Quasirandom points and global function fields
H. Niederreiter and C. Xing · 1996
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Monte Carlo variance of scrambled equidistribution quadrature
A. B. Owen · 1997
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Scrambled net variance for integrals of smooth functions
A. B. Owen · 1997
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Simulation optimization
S. Andradóttir · 1998
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On the L 2 –discrepancy for anchored boxes
J. Matoušek · 1998
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Scrambling Sobol’ and Niederreiter-Xing points
Local antithetic sampling with scrambled nets
A. B. Owen · 2008
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MinT: new features and new results
R. Schürer and W. Ch. Schmid · 2009
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Digital sequences, discrepancy and quasi-Monte Carlo integration
J. Dick and F. Pillichshammer · 2010
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High-dimensional integration: the quasi-Monte Carlo way
J. Dick, F. Y. Kuo, and I. H. Sloan · 2013
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The smoothing effect of integration in ℝ d \mathbb{R}^{d} and the ANOVA decomposition
M. Griebel, F. Y. Kuo, and I. H. Sloan · 2013
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A Panorama of Discrepancy Theory
W. Chen, A. Srivastav, and G. Travaglini, editors · 2014
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A. B. Owen · 1998
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On quasi-Monte Carlo integrations
I. M. Sobol’ · 1998
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Extensible lattice sequences for quasi-Monte Carlo quadrature
F. J. Hickernell, H. S. Hong, P. L’Ecuyer, and C. Lemieux · 2000
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The mean square discrepancy of scrambled ( t , s ) (t,s) -sequences
F. J. Hickernell and R. X. Yue · 2000
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A survey of randomized quasi-Monte Carlo methods
P. L’Ecuyer and C. Lemieux · 2002
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Construction and comparison of high-dimensional Sobol’ generators
I. M. Sobol’, D. Asotsky, A. Kreinin, and S. Kucherenko · 2002
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Calculation of discrepancy measures and applications
C. Doerr, M. Gnewuch, and M. Wahlström · 2014
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Koksma-Hlawka inequality
F. J. Hickernell · 2014
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Functions of bounded variation, signed measures, and a general Koksma-Hlawka inequality
Ch. Aistleitner and J. Dick · 2015
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On the convergence rate of randomized quasi–Monte Carlo for discontinuous functions
Z. He and X. Wang · 2015
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Error bounds of MCMC for functions with unbounded stationary variance
D. Rudolf and N. Schweizer · 2015
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Application of quasi-Monte Carlo methods to elliptic PDEs with random diffusion coefficients: a survey of analysis and implementation
F. Y. Kuo and D. Nuyens · 2016
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A constraint on extensible quadrature rules
A. B. Owen · 2016
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Asymptotic normality of scrambled geometric net quadrature
K. Basu and R. Mukherjee · 2017
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Note on “The smoothing effect of integration in ℝ d \mathbb{R}^{d} and the ANOVA decomposition”
M. Griebel, F. Y. Kuo, and I. H. Sloan · 2017
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Quasi-Monte Carlo for an integrand with a singularity along a diagonal in the square
K. Basu and A. B. Owen · 2018
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High dimensional integration of kinks and jumps–Smoothing by preintegration
A. Griewank, F. Y. Kuo, H. Leövey, and I. H. Sloan · 2018
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BoTorch: Bayesian optimization in PyTorch
M. Balandat, B. Karrer, D. R. Jiang, S. Daulton, B. Letham, A. G. Wilson, and E. Bakshy · 2019
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Probability Theory and Examples
R. Durrett · 2019
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Solvable integration problems and optimal sample size selection
R. J. Kunsch, E. Novak, and D. Rudolf · 2019
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