Replication and stacking in ergodic theory
American Mathematical Monthly
Van der Corput sequences, Kakutani transforms and one-dimensional numerical integration
Journal of Computational and Applied Mathematics
A quasi-Monte Carlo approach to particle simulation of the heat equation
SIAM Journal on Numerical Analysis
Fast generation of low-discrepancy sequences
Journal of Computational and Applied Mathematics
Computational investigations of low-discrepancy sequences
ACM Transactions on Mathematical Software (TOMS)
On the L2-discrepancy for anchored boxes
Journal of Complexity
Generating and Testing the Modified Halton Sequences
NMA '02 Revised Papers from the 5th International Conference on Numerical Methods and Applications
Good permutations for deterministic scrambled Halton sequences in terms of L2-discrepancy
Journal of Computational and Applied Mathematics
On the optimal Halton sequence
Mathematics and Computers in Simulation
Computation of the endogenous mortgage rates with randomized quasi-Monte Carlo simulations
Mathematical and Computer Modelling: An International Journal
Mathematical and Computer Modelling: An International Journal
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It is a well-known fact that the Halton sequence exhibits poor uniformity in high dimensions. Starting with Braaten and Weller in 1979, several researchers introduced permutations to scramble the digits of the van der Corput sequences that make up the Halton sequence, in order to improve the uniformity of the Halton sequence. These sequences are called scrambled Halton, or generalized Halton sequences. Another significant result on the Halton sequence was the fact that it could be represented as the orbit of the von Neumann-Kakutani transformation, as observed by Lambert in 1982. In this paper, I will show that a scrambled Halton sequence can be represented as the orbit of an appropriately generalized von Neumann-Kakutani transformation. A practical implication of this result is that it gives a new family of randomized quasi-Monte Carlo sequences: random-start scrambled Halton sequences. This work generalizes random-start Halton sequences of Wang and Hickernell. Numerical results show that random-start scrambled Halton sequences can improve on the sample variance of random-start Halton sequences by factors as high as 7000.