Generalized von Neumann–Kakutani transformation and random-start scrambled Halton sequences

Abstract

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.