In this paper we develop a framework to study the dependence structure of scrambled -nets. It relies on values denoted by , which are related to how many distinct pairs of points from lie in the same elementary -interval in base b. These values quantify the equidistribution properties of in a more informative way than the parameter t. They also play a key role in determining if a scrambled set is negative lower orthant dependent (NLOD). Indeed, this property holds if and only if for all , which in turn implies that a scrambled digital -net in base b is NLOD if and only if . Through numerical examples we demonstrate that these values are a powerful tool to compare the quality of different -nets, and to enhance our understanding of how scrambling can improve the quality of deterministic point sets.
Funding source: Natural Sciences and Engineering Research Council of Canada
Award Identifier / Grant number: 238959
Funding source: Austrian Science Fund
Award Identifier / Grant number: F5506-N26
Award Identifier / Grant number: F5509-N26
Funding statement: The authors wish to acknowledge the support of the Natural Science and Engineering Research Council (NSERC) of Canada for its financial support via grant #238959. The first author is also partially supported by the Austrian Science Fund (FWF): Projects F5506-N26 and F5509-N26, which are parts of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”.
Proof of Lemma 2.5.
When either x or y is 1, from Lemma 2.4 we know that
and thus in this case. So for the remainder of the proof, we assume . Let
be the base-b digital expansion of x and y chosen so that only finitely many digits are non-zero. Recall that for . When , then for we have
and . We also define , and for . Without loss of generality we assume . There are four cases.
Case 1: . In this case
Case 2: . In this case, becomes
Case 3: . We use the calculation in Case 2 and the identities and to simplify :
Multiply by to get
which will be shown to be non-negative. Note that by assumption and since their base b expansions differ for the first time at the st digit, we always have .
Case 3a: . The assumption implies and . We estimate
Case 3b: . The assumption implies and . We estimate
Case 3c: . The assumption implies . We estimate
Case 4: . In this case we need to show that
is greater than or equal to zero. Using the identities
Now substituting into (A.1) and simplifying, we get
By multiplying the above by we see that to finish the proof we need to show that
Case 4a: . We have
Case 4b: . Since , we have
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