Prime Numbers

414 Chapter 8 THE UBIQUITY OF PRIME NUMBERS
integrals that present the more difficult challenge for the qMC methods. That
is, financial integrands are often “smoother” in practice.
Just as interesting as the qMC technique itself is the controversy that
has simmered in the qMC literature. Some authors believe that the Halton
sequence—the one on which we have focused as an example of primebased
qMC—is inferior to, say, the Sobol [Bratley and Fox 1988] or Faure
[Niederreiter 1992] sequences. And as we have indicated above, this assessment
tends to depend strongly on the domain of application. Yet there is some
theoretical motivation for the inferiority claims; namely, it is a theorem [Faure
1982] that the star discrepancy of a Faure sequence satisfies
D ∗ N ≤ 1
D!
p − 1
2lnp
D ln D N
N ,
where p is the least prime greater than or equal to D. Whereas a Ddimensional
Halton sequence can be built from the first D primes, and this
Faure bound involves the next prime, still the bound of Theorem 8.3.5 is
considerably worse. What is likely is that both bounding theorems are not
best-possible results. In any case, the prime numbers once again enter into
discrepancy theory and its qMC applications.
As has been pointed out in the literature, there is the fact that qMC’s
error growth of O (ln D
N)/N is, for sufficiently large D, and sufficiently
small N, or practical combinations of D, N magnitudes, worse than direct
Monte Carlo’s O 1/ √
N . Thus, some researchers do not recommend qMC
methods unconditionally. One controversial problem is that in spite of various
theorems such as Theorem 8.3.5 and the Faure bound above, we still do not
know how the “real-world” constants in front of the big-O terms really behave.
Some recent developments address this controversy. One such development is
the discovery of “leaped” Halton sequences. In this technique, one can “break”
the unfortunate correlation between coordinates for the D-dimensional Halton
sequence. This is done in two possible ways. First, one adopts a permutation on
the inverse-radix digits of integers, and second, if the base primes are denoted
by p0,...,pD−1, then one chooses yet another distinct prime pD and uses only
every pD-th vector of the usual Halton sequence. This is claimed to improve
the Halton sequence dramatically for high dimension, say D =40to400
[Kocis and Whiten 1997]. It is of interest that these authors found a markedly
good distinct prime pD to be 409, a phenomenon having no explanation.
Another development, from [Crandall 1999a], involves the use of a reduced set
of primes—even when D is large—and using the resulting lower-dimensional
Halton sequence as a vector parameter for a D-dimensional space-filling curve.
In view of the sharply base-dependent bound of Theorem 8.3.5, there is reason
to believe that this technique of involving only small primes carries a distinct
statistical advantage in higher dimensions.
While the notion of discrepancy is fairly old, there always seem to appear
new ideas pertaining to the generation of qMC sets. One promising new
approach involves the so-called (t, m, s)-nets [Owen 1995, 1997a, 1997b],