Summary of I-binomial scrambling of digital nets and sequences
Yu Zhang
yz08@fsu.edu
In this paper, the author presented a method to reduce the randomness requirement
for Owen’s random scrambling. The author first introduced the basic notation:
i-binomial property. Then the left and right i-binomial scrambling were defined. At
last, the author showed the advantage of the left i-binomial scrambling.
The most popular scheme to form a sample space is Owen’s random scrambling
scheme. Assume b is an integer larger than 2, σ is a mapping from [0, 1) onto itself.
We can map A to σ(A), where A=a1b-1+a2b-2+…, a1, a2 are in {0, 1, …, b-1}. The first
digit of σ(A) is π(a1 ), π is a fixed permutation of (0, 1, …, b-1). So, for each value of
a1, we can define the second digit πa1 (a2 ) by fixing a permutation π(a1 ) of {0,
1, …, b-1}. Then we get the sequence πa1, a2 (a3), πa1, a2 ,a3 (a4 ), …. There are b!
possible permutations and they are mutually independent with each other. Therefore,
in S dimensions, (σ1,…,σs ) is called fully random scrambling if all the permutations are
chosen at random and mutually independent. Then, it can keep the good properties
of (t, m, s)-nets and (t, s)-sequences. The reason why Owen’s scrambling is popular is
it has a fast convergence rate for many smooth functions.
Assume μ is an integer. If the elements of an infinite matrix M=(mr,c) satisfy: for
integer r, c >1, mr,c=μmr,c-1+mr-1,c-1(mod b), then we say M is an i-binomial matrix.
Here, 0-bimonial matrix is important. It has a property that the elements at one
diagonal are the same. If Faure sequences c(d)=L(d)Pd wants to have i-binomial
property, each L(d) should be 0-binomial. Sequences C(d)=PdU has the property, if each
U has the i-binomial property.
If a class of random linear scrambling is essentially equivalent to generalized Faure
sequences with randomly chosen lower triangular matrices L(d), we call it left
i-binomial scrambling. The key point of this paper is it proved that all the results on
the expected errors of the integration problem obtained with Owen’s scrambling also
hold with the left i-binomial scrambling. However, the left i-binomial scrambling is
based on much smaller sample space than the fully random scrambling.