HALTON SEQUENCES AVOID THE ORIGIN

Transcription

1 HALTON SEQUENCES AVOID THE ORIGIN ART B. OWEN STANFORD UNIVERSITY Abstract. The n th point of the Halton sequence in [0, 1] d is shown to have components whose product is larger than Cn 1 where C > 0 depends on d. This property makes the Halton sequence very well suited to quasi-monte Carlo integration of some singular functions that become unbounded as the argument approaches the origin. The Halton sequence avoids a similarly shaped (though differently sized region around every corner of the unit cube, making it suitable for functions with singularities at all corners. Convergence rates are established for quasi-monte Carlo integration based on growth conditions of the integrand, and measures of how the sample points avoid the boundary. In some settings the error is O(n 1+ɛ while in others the error diverges to infinity. Star discrepancy does not suffice to distinguish the cases. Key words: Discrepancy, Integrable Singularities, Quasi-Monte Carlo, Randomized quasi-monte Carlo AMS Subject: 65D32, 65C05 1. Introduction. Quasi-Monte Carlo integration in d dimensions has an asymptotic error rate of O(n 1 [log(n] d 1 when n function evaluations are used, and the integrand has bounded variation in the sense of Hardy and Krause. Here we consider integrands that are unbounded, approaching ± as the argument approaches the boundary of the unit cube. Such integrands cannot be of bounded variation in the sense of Hardy and Krause. Unbounded integrands are extremely common in applications where quasi-monte Carlo might be used to replace Monte Carlo sampling. For these cases the standard QMC theory does not indicate whether QMC is superior to Monte Carlo. This article finds some rates of convergence for QMC integration of singular integrands. The necessary ingredients are conditions on the speed with which the QMC sample points approach the boundary of [0, 1] d, and growth conditions on the integrand and some of its partial derivatives, near the boundary. It is common for the QMC error rate to be written as O(n 1+ɛ, holding for all ɛ > 0, hiding the logarithmic powers. Under some of the conditions presented here, the QMC error is still O(n 1+ɛ even for unbounded integrands. In other settings the QMC error diverges to infinity with n. The difference between these outcomes depends not on the discrepancy of the sample points, but on subtle properties concerning how far the sample points are from the boundary of the unit cube. In particular, the Halton sequence (Halton 1960 avoids the corners of the unit cube, especially the origin, in a way very suitable for unbounded integrands with growth to infinity governed by a power law. Some background and notation are presented in Section 2. Included are three notions of how points can avoid the origin (or avoid all corners of [0, 1] d, some growth conditions for singular functions on [0, 1] d, and Sobol s low variation extension of a function from a subset of [0, 1] d to all of [0, 1] d. We show in Section 3 that the Halton sequences avoid sampling from a hyperbolic region near the origin. In fact all 2 d corners of the unit cube are avoided by the Halton sequence, though stronger results are proved for the origin and the corner opposite it. Section 4 shows that the Halton sequence avoids a larger region about the origin than independent uniform random points do. Random points avoid larger regions about corners other than (0,..., 0 and (1,..., 1 than the Halton points are proven to do. Section 5 presents error rates for QMC for unbounded functions, assuming growth conditions on the integrand, and an avoidance pattern for the sample points. In particular, randomized quasi-monte 1

2 Carlo methods generally avoid the boundary of [0, 1] d in a satisfactory way. The study of QMC for unbounded integrands began with Sobol (1973a. He presents a thorough treatment of the case with d = 1. There he shows how to use the van der Corput sequence to integrate x A with error O(n A 1 log(n for A < 1 and to how to integrate x 1 [log(x] γ for γ < 1 with error O([log(n] γ 1. For multidimensional quadrature, he considers integrands that are products of negative powers of their arguments, and low discrepancy points that avoid a hyperbolic region around the origin. Sobol (1973a also exhibits a QMC rule that avoids the origin this way. The present work extends Sobol s in several ways. Rates of convergence are given for integration of multidimensional functions that may become singular, subject to a power law growth bound, as the argument approaches any of the 2 d corners, not just the origin. To be relevant such a rate requires the existence of points that simultaneously avoid hyperbolic regions around all 2 d corners of [0, 1] d. The Halton sequence is shown to do so, and so are some randomized QMC points. The present paper makes error rates very explicit for the multidimensional case, and some unbounded functions arising in computational finance are found to be estimated with the O(n 1+ɛ error rate. Hartinger, Kainhofer, and Tichy (2004 extend the results of Sobol (1973a to consider integration of products where one factor is a function singular near the lower corner and the other is a bounded probability density function h. They also show how to sample points with a low discrepancy relative to the density function h. Some numerical examples based on the results in Sobol (1973a are given by de Doncker and Guan (2003. They find that a form of error extrapolation improves the accuracy for some power law integrands singular at the origin. 2. Background. We suppose that the integrand is a real valued, Lebesgue measurable function f(x. The argument x is a point in [0, 1] d. Components of x are written with superscripts, so that x = (x 1,..., x d. We assume that I = f(xdx [0,1] d exists. Some transformation of the problem may have been made to render it in this form. Very often, the problem originates as an integral over an unbounded domain such as R d. Then it is common for the transformation to an integrand over [0, 1] d to introduce a singularity at the boundary. Change of variable formulas, sometimes called importance sampling, can often be used to yield a bounded, and even a periodic function on [0, 1] d (Sloan and Joe Sometimes the change of variable techniques remove the singularity at the cost of making the integrand spiky, exchanging one sort of difficulty for another. Thus it remains of interest to study the effects of singular integrands on QMC accuracy. An alternative approach, of working directly with integrals over R d, is taken by several authors, including Genz and Monahan (1998, Mathé and Wei (2003, and Hickernell, Sloan, and Wasilkowski (2004. The quasi-monte Carlo estimate of I is Î = 1 n n f(x i (2.1 i=1 for points x i [0, 1] d. For f of bounded variation in the sense of Hardy and Krause, Hlawka s theorem gives Î I D n(x 1,..., x n V HK (f (2.2 2

3 where Dn is the anchored (star discrepancy of x 1,..., x n and V HK is the total variation of f in the sense of Hardy and Krause. A recent survey of multidimensional total variation appears in Owen (2005. Matoušek (1998a gives a thorough discussion of discrepancy. Niederreiter (1992 presents constructions of points x 1,..., x n for which Dn(x 1,..., x n = O(n 1 [log(n] d 1 as well as infinite sequences along which Dn(x 1,..., x n = O(n 1 [log(n] d. The significance of (2.2 is that functions f with V HK (f < can be integrated with an error asymptotically much better than that of Monte Carlo sampling. Monte Carlo sampling has a root mean square error of O(n 1/2 when f(x 2 dx <. [0,1] d When V HK (f =, as it is for unbounded integrands, then (2.2 is not informative. In d dimensions as in one, a workable approach to integrating singular functions is to avoid the singularity (Davis and Rabinowitz We will suppose that x 1,..., x n K = K n [0, 1] d and that sup x Kn f(x <. We also require an extension f n = f of f from K n to [0, 1] d such that f(x = f(x for x K n. Then a three epsilon argument gives I Î f f dx + Dn(x 1,..., x n V HK ( f + 1 n [0,1] n f(x i f(x i d i=1 = f f dx + Dn(x 1,..., x n V HK ( f. (2.3 [0,1] d K The dependence of K and f upon n is usually suppressed Avoiding the origin and other corners. For d = 1 and f with a singularity at x = 0, one avoids the singularity by avoiding the origin, that is by sampling within K = [ɛ, 1] for 0 < ɛ < 1. There are several useful ways of avoiding the origin for d 1. The following regions K orig min (ɛ = {x [0, 1]d min 1 j d xj ɛ}, K orig prod (ɛ = {x [0, 1]d x j ɛ}, 1 j d K orig max(ɛ = {x [0, 1] d max 1 j d xj ɛ}, and, all of which reduce to [ɛ, 1] when d = 1, have been studied. The set Kmax orig excludes a cube containing the origin, while K orig prod excludes a hyperbolic region, and K orig min excludes a region that is L-shaped for d = 2. By Theorem 3 of Sobol (1973a points x 1 through x 2 m of an LP τ sequence (a (τ, m, d-net in base 2 belong to K orig prod (2 d τ /n where n = 2 m. Although Sobol (1973a wrote about K orig prod, an especially unfortunate misprint, repeated in several places, gives the impression that he wrote about K orig min. Klinger (1997 considered Korig max and showed that Halton points as well as certain digital nets cannot put two points into [0, ɛ] d for small ɛ. The first point of a Halton sequence, as well as that of some nets, is at the origin, and so skipping over that point gives a quadrature rule inside Kmax. orig The regions above distinguish the origin from all of the other 2 d 1 corners of [0, 1] d. Simple rearrangements of x i or of f can be employed if the singularity of f is at a corner other than the origin, or if the points x i are particularly effective in avoiding some other corner of [0, 1] d. In applications, singular behavior might appear 3

4 in many or in all corners of the unit cube. Then we may require a region K that avoids all the corners of the cube. There are multiple ways of avoiding all the corners, such as sampling within K corn min (ɛ = {x [0, 1] d min 1 j d min(xj, 1 x j ɛ}, Kprod(ɛ corn = {x [0, 1] d min(x j, 1 x j ɛ}, 1 j d K corn max (ɛ = {x [0, 1] d max 1 j d min(xj, 1 x j ɛ}, or, for small ɛ > 0, which all reduce to [ɛ, 1 ɛ] for d = Growth conditions on f. Here we introduce growth conditions for functions on (0, 1] d that may become singular as x approaches the origin. For a set u {1,..., d} of indices, the symbol u f(x represents ( j u / xj f(x, with the convention that f(x = f(x. The first growth condition is that u f(x B (x j Aj 1j u (2.4 holds for some A j > 0, some B <, and all u {1,..., d}. The second condition bounds growth for functions on (0, 1 d that may become singular as x approaches any corner. It is u f(x B min(x j, 1 x j Aj 1j u (2.5 where once again A j > 0, and B < and all u {1,..., d}. Larger values of A j correspond to more severe singularities. When max j A j 1 the upper bound for f is not even integrable. When max j A j < 1/2, then f 2 is integrable and Monte Carlo sampling has a root mean square error of O(n 1/2. The conditions above exclude A j = 0, a value that one might have expected to use for functions whose value (and partial derivatives remain bounded as x j varies for fixed values of the other coordinates. We may take arbitrarily small positive A j > 0 in such cases. Excluding A j = 0 simplifies the presentation at the expense of masking logarithmic factors in the error rates. if The conditions (2.4 and (2.5 are easy to verify in some applications. For example ( d f(x = g h j (x j (2.6 for functions g and h j from R to R, then u f(x takes the comparatively simple form g ( u ( d h j(x j j u h j (xj. Some integrands in the valuation of call and put 4

5 options respectively (see Glasserman (2004 take the form ( f C (x = max 0, L α l f l (x K, l=1 ( f P (x = max 0, K ( f l (x = exp β l0 + L l=1 and, α l f l (x, with, d β lj Φ 1 (x j, for scalars α l, β lj, and K, where Φ represents the standard Gaussian cumulative distribution function. We assume, as is common, that all α l > 0 and at least some of the β lj for j 1 are nonzero. Then f C is unbounded. By contrast f P is in the range [0, K ]. The unbounded integrand f C can be rewritten as f C (x = f P (x + L l=1 α lf l (x K. A quadrature rule that is good for integrating f P and f l will then be good for f C, by linearity of integration. Each of the f l (x takes the form (2.6 with g( = exp(, and ( u f l (x = exp β l0 + d β lj Φ 1 (x j d dx j Φ 1 (x j. (2.7 This function f l satisfies (2.5 for arbitrarily small A j > 0. To see why, note that the quantile function Φ 1 ( satisfies Φ 1 (ɛ = 2 log(ɛ + o(1 and Φ 1 (1 ɛ = 2 log(ɛ + o(1 as ɛ 0 (Patel and Read 1996, Chapter 3.9. So exp(βφ 1 (x = O(min(x, 1 x A/2 for any A > 0. Also, letting ϕ(x = (2π 1/2 exp( x 2 /2 denote the Gaussian probability density, d 1 dx Φ 1 (x = ϕ(φ 1 (x = ( 2/2 2π e 2 log(x+o(1 = O(min(x, 1 x 1 A/2 for any A > 0. Papageorgiou (2003 shows that there exist ways to construct points x 1,..., x n with an integration error of O(n 1+ɛ for f l. Because f l satisfies (2.5 for arbitrarily small A j > 0, Theorem 5.2 below shows that f l can be integrated with error O(n 1+ɛ by the Halton sequence and by other QMC schemes that obey easily checked (or imposed corner avoidance conditions. While f P is bounded, it is typical for V HK (f P = (Owen It is noteworthy that the bounded part f P may pose greater difficulty to QMC integration of f C than the unbounded parts f l do Low variation extensions. Here we consider how to extend a function from a set K [0, 1] d to all of [0, 1] d, such that the extended function has low variation. The extension, due to I. M. Sobol, was not published, but was used to establish Theorem 2 of Sobol (1973a. Some additional notation is required. For a set u 1 : d {1,..., d}, the complement 1 : d u is denoted by u and the cardinality is denoted by u. For x [0, 1] d the symbol x u denotes the u -tuple consisting of all x j with j u. The domain of x u is written as [0 u, 1 u ] or as [0, 1] u. For x, z [0, 1] d, let x u : z u denote 5 j u

6 the point y [0, 1] d with y j = x j for j u and y j = z j for j u. When the value x u is held fixed at z u, then f defines a function of x u on [0, 1] u. The expression f(x u ; z u with value f(x u : z u denotes this function. The argument x u precedes the semi-colon and the parameter z u follows it. Two points x, z [0, 1] d define a d- dimensional bounding box, rect[x, z] = {y [0, 1] d min(x j, z j y j max(x j, z j }. We suppose that the set K contains an anchor point c such that x K implies that rect[x, c] K. We also suppose that K has positive d dimensional volume. The case where K has zero volume is included in Owen (2005. We also require that u f(x exists for all u 1:d. Under these conditions we may write f(x = f(c + u and then the low variation extension is f(x = f(c + u [c u,x u ] [c u,x u ] u f(z u :c u dz u, (2.8 1 z u :c u K u f(z u :c u dz u. (2.9 When x j < c j for some j, then the integrals in (2.8 and (2.9 are the corresponding integrals over rect[c u, x u ] multiplied by ±1. The factor is negative if and only if there are an odd number of j with c j > x j. Owen (2005 shows that the variation of f, in the sense of Vitali satisfies V [0,1] d( f 1:d f(x dx. (2.10 K For u 1:d and b [0, 1] d define K u (b u = {x u [0, 1] u x u : b u K}. Then for c = (1,..., 1, it follows from a result in Owen (2005 that V HK ( f u K u(1 u u f(x u : 1 u dx u. ( Halton sequences. Let b 2 and i 0 be integers. The integer i may be written in a base b expansion as k=1 a i,k,bb k 1 where each a i,k,b {0, 1,..., b 1}, and a i,k,b is zero for all but finitely many k. The radical inverse function is defined by φ b (i = a i,k,b b k. k=1 A Halton sequence (Halton 1960 contains points x i = (φ p1 (i,..., φ pd (i for a sequence of non-negative integers i, where p 1,..., p d are relatively prime integers. We will suppose that the p j are all primes. Typically they are the first d primes. If a Halton sequence is started at i = 0 then the first point is the origin, which is often a singular point of the integrand. If the values x 1,..., x n are used instead, then a Halton sequence avoids the origin: Theorem 3.1. For n 1, let x n = (φ p1 (n,..., φ pd (n be the n th point in a 6

7 Halton sequence, where p 1,..., p d are distinct prime numbers. Then x j n 1 n (1 x j n 1 n + 1 If d = 1, then min(x 1 n, 1 x 1 n p 1 1 /(n + 1, and for all d 1, p 1 j, and, (3.1 min(x j n, 1 x j 1 n n(n + 1 p 1 j. (3.2 p 1 j. (3.3 Proof: Let P 1 < P 2 < < P rn be all of the prime numbers smaller than or equal to max(n + 1, p 1,..., p d. Then n = r n r=1 P r ar for integers a r 0. The base P r expansion of n ends in a r zeros. Therefore φ Pr (n Pr ar 1. Define r(j by p j = P r(j. Then x j n p a r(j 1 j = P a r(j r(j p 1 j 1 n p 1 j, establishing (3.1. Now suppose that n + 1 = r n r=1 P r br for integers b r 0. Then there are b j trailing p j 1 s in the base p j representation of n, so that 1 x j n p bj 1 j and (1 x j n p b r(j 1 j = P b r(j r(j establishing (3.2. If d = 1 then min(x 1 n, 1 x 1 n min(p 1 1 min(x j n, 1 x j n establishing (3.3. P max(a r(j,b r(j r(j p 1 j 1 n + 1 p 1 j p 1 j, /n, p 1 1 /(n + 1. Finally 1 n(n + 1 p 1 j, Of the 2 d corners of (0, 1 d, Theorem 3.1 gives a stronger result for two of them, (0,..., 0 and (1,..., 1. For functions that are singular at one corner and perhaps all of the faces joining it, it may be advantageous to arrange for that singularity to be at (0,..., 0 or (1,..., 1. Similarly functions singular, or most singular, at two opposite corners, should perhaps be transformed to have their singularities at these special corners. The lower bound in (3.1 is attained for n = 1 and any d 1. For d = 1, the lower bound in (3.1 is attained repeatedly for n = p r 1 while that in (3.2 is attained repeatedly for n = p r d 1 1. Figure 3.1 plots the value of min 1 i n φ p j (i versus n for 1 n and d = 2, 3, 4, 5, 8, 11, 14, 17, 20. The omitted dimensions (6, 7, 9,..., 19 show a similar pattern. The lower bounds (3.1 are superimposed as reference lines. For small d, the attained minima run nearly parallel to, and just above, the theoretical bounds, for the sample sizes plotted. For larger d there can be 7

8 1e 27 1e 21 1e 15 1e 09 1e 03 1e+00 1e+03 1e+06 1e+09 Fig The figure illustrates the closest approach of Halton points to the origin of [0, 1] d. The horizontal axis represents the sample size n for 1 n The vertical axis is min 1 i n Q d φp j (i where φp is the radical inverse function in base p and p j is the j th prime. Curves are shown, top to bottom, for dimensions d = 2, 3, 4, 5, 8, 11, 14, 17, 20. A dot is plotted at each n for which a new minimum is attained. Dotted reference lines with slope 1 represent the lower bound from Theorem 3.1 for the plotted quantity. a considerable wait before a point is found that comes closer to the origin than x 1 does. For d = 20 the wait is longer than The rate O(n 2 in (3.3 of Theorem 3.1 follows because a point cannot be closer than O(n 1 to either the lower or the upper corner of (0, 1 d. The O(n 2 rate may be quite conservative, as it guards against any single point being close to both of those opposite corners. The value of d min(xj i, 1 xj i was computed for the first ten billion Halton points in dimensions 2, 3,..., 20 and the cumulative minima are plotted in Figure 3.2. A reference line equal to n 1 has been added. The sequences appear to decay faster than n 1, and the decay appears to be steeper for higher dimensions. Table 3.1 contains estimated rates for dimensions 2 through 20. The rate for d = 1 is equal to n 1, from Theorem 3.1. For each dimension d, coefficients β d0 and β d1 were chosen to minimize ( ( log n 2 min(x j n, 1 x j n β d0 β d1 log(n with the sum taken over values n with 1 < n for which a new minimum was attained. The estimated rate is O(n r where r = β d1. For small d the rates appear 8

9 1e 33 1e 26 1e 19 1e 12 1e 05 1e+00 1e+03 1e+06 1e+09 Fig The figure illustrates the closest approach of Halton points to the corners of [0, 1] d. The horizontal axis represents the sample size n for 1 n The vertical axis is Q min d 1 i n min(φp (i, 1 j φp (i where j φp is the radical inverse function in base p and p j is the j th prime. Curves are shown, top to bottom, for dimensions d = 2, 3, 4, 5, 8, 11, 14, 17, 20. A dot is plotted at each n for which a new minimum is attained. The reference line above the points corresponds to 1/n. to be close to 1, but the possibly conservative theoretical rate r = 2 fits more closely for larger d. These calculations do not show that the asymptotic rate is other than n 2, even for d = 2. Nor do they prove it is other than n 1 for large d. Even if the true asymptote is n 1 or n 2, Figure 3.2 and Table 3.1 show that for the range of n and d illustrated there, a more accurate approximation can be computed using an intermediate power of n. Exact calculations of this sort can be made for larger values of n than one ordinarily uses in a quasi-monte Carlo calculation. d For large d and small n, an estimate of the form min 1 i n min(xj i, 1 xj i Cn r does not appear reasonable. For instance with d = 20 one must wait until d 2: d 11: Table 3.1 Q Estimated rates of convergence in n for min d 1 i n min(xj i, 1 xj i using 1 < n The rate is estimated to be n r where r varies with dimension d as shown. When d = 1 it is known that r = 1. 9

10 n = 2,185,874 to find a point x n with d min(xj n, 1 x j n < d min(xj 1, 1 xj 1. A power law Cn r provides a good description for n 2,185,874 but not for 1 n < 2,185, Other sequences. This section compares the origin and corner avoidance of the Halton points to that of other sampling methods. A natural benchmark for the Halton points is the behavior of independent random U[0, 1] d values. The Halton points avoid the origin, and the corner opposite the origin, more strongly than these random points do. But random points avoid the other corners more strongly than the Halton points have been proven to do. We use below the fact that 2 log( d xj has the χ 2 (2d distribution when x U[0, 1]d. We also compare Halton sequences to some other QMC methods Random points. For each n 1 let E n be an event, such as x n being close to the origin. We write E n (i.o if the event E n occurs infinitely often. If n=1 Pr(E n <, then by the Borel-Cantelli theorem (Chung 1974, Chapter 4.2, Pr(E n i.o. = 0. Similarly if E n are independent and n=1 Pr(E n =, then Pr(E n i.o. = 1. Lemma 4.1. For i = 1, 2,..., let x i be U[0, 1] d random vectors. Then for C > 0 and r > 1, ( Pr min 1 i n x j i Cn r i.o. = 0. (4.1 If the x i are also assumed to be independent, then for C > 0 and r 1, ( Pr x j n Cn r i.o. = 1. (4.2 Proof: Suppose that x n U[0, 1] d. Then Pr ( ( ( x j n Cn r = Pr 2 log x j n 2r log(n 2 log(c = Pr (χ 2 (2d 2r log(n 2 log(c = = 2r log(n 2 log(c r log(n log(c z d 1 e z/2 2 d Γ(d y d 1 e y Γ(d dy Cn r Γ(d (r log(n log(cd 1. (4.3 For r > 1 expression (4.3 is summable over 1 n <. With probability 1 the event E i = d xj i Ci r happens only finitely often. The probability that d xj i = 0 ever happens is zero. Therefore, for each i with j xj i Ci r there are only finitely many n i with j xj i Cn r. Thus (4.1 is established. 10 dz

11 For r 1 and large enough n, ( Pr x j i Cn r Cn r Γ(d, which has an infinite sum. Therefore (4.2 holds. For a finite C > 0 the Halton points x 1,..., x n avoid the region {x j xj Cn 1 } while independent uniform points x n enter that region infinitely often, with probability one. The Halton points are known to avoid the region {x j min(xj, 1 x j Cn r } for r = 2, and from empirical evidence, a smaller dimensionally dependent value of r 1 might be more appropriate. Similarly, independent random points x i U[0, 1] d, enter {x j min(xj, 1 x j Cn r } only finitely often for r > 1 but enter it infinitely often with probability one, for r Other QMC points. Sobol (1973a notes that points of a Sobol sequence avoid the origin in the hyperbolic sense. His proof appears in Sobol (1973b. Avoidance of other corners is not considered. A rank one lattice (Sloan and Joe 1994 has x j i = {igj /n}, for i = 1,..., n, where g = (g 1,..., g d is a vector of integers chosen jointly with the sample size n. The expression {z} denotes the fractional part z z of z. Like the Halton sequence these lattice rules include a point at the origin. Unlike the Halton sequence, one cannot simply ignore the origin by starting the sequence at a different place; any n consecutive points from a rank one lattice will contain a point at the origin. It is however possible to shift the lattice points, or indeed any other points, so that they avoid the origin. Linear shifts and random shifts modulo one are considered below. 5. Error rates. Here we present theorems giving the rates of convergence for Î I under various growth conditions on f and boundary avoidance conditions for x i. There are three cases: one for QMC points that belong to K orig min or Kcorn min, one for QMC points that belong to K orig prod or Kcorn prod, and one for randomized QMC points Avoiding L-shaped regions. The decomposition (2.3 provides an upper bound on Î I. To bound the first (truncation term in (2.3 we must bound the integral of f(x f(x where f is a low variation extension of f from K to [0, 1] d. For x K it is easy to see that f(x = f(x. For general x [0, 1] d K the following bound is useful. Lemma 5.1. Let c = (1,..., 1 K [0, 1] d where [x, 1] K whenever x K. Suppose that f satisfies (2.4, and that f is given by (2.9. Then for x [0, 1] d K, f(x f(x B (x j Aj, (5.1 where B = B d (1 + 1/A j. 11

12 Proof: Subtracting (2.9 from (2.8 term by term, yields f(x f(x B 1 z u :1 u K (z j Aj 1 dz u u [x u,1 u ] j u B (x j Aj A u j u j ( = B (1 + (xj Aj 1 A j B (1 + A j (x j Aj A j. The second inequality follows by replacing each upper limit of [x u, 1 u ] by, and the third uses 1 (x j Aj. The value of Lemma 5.1 is that f(x f(x obeys a bound similar to that obeyed by f(x, except that B is replaced by B. The other ingredient in bounding Î I is an estimate of the extent to which x 1,..., x n avoid the origin, or the corners. The closest approach to the boundary is taken below to be ɛ n = Cn r. In most cases of interest r 1, for otherwise the x i cannot have a small discrepancy. For Halton sequences we can restrict our attention to r 2 as well. Where empirical investigations warrant, some values of r between 1 and 2 might be appropriate. Theorem 5.2. Let f(x be a real valued function on (0, 1] d that satisfies condition (2.4. Suppose that x 1,..., x n K orig min (ɛ n, where 0 < ɛ n Cn r < 1. Then Î I C 1D n(x 1,..., x n n r P d Aj + C 2 n r(maxj Aj 1, (5.2 for finite C 1 and C 2. Similarly if f(x is a real valued function on (0, 1 d that satisfies condition (2.5, and x 1,..., x n Kmin corn(ɛ n, where 0 < ɛ n Cn r < 1/2, then (5.2 holds. Proof: To handle the origin avoidance case, let f n be the low variation extension of f from [ɛ n, 1] d to [0, 1] d with anchor (1,..., 1, and suppose that x 1,..., x n K orig min (ɛ n. Then f n (x i = f(x i for 1 i n, and so Î I (0,1 d f(x f n (x dx + D n(x 1,..., x n V HK ( f n. (5.3 From Lemma 5.1, f(x f n (x dx B (0,1 d B B (0,1 d [ɛ n,1] d ɛn d 0 (x j Aj dx j (x j Aj dx j d (1 A j 1 ɛ 1 Aj n (0,1 {j} k {j} (x k A k dx k ( B (1 A j 1 dc 1 minj Aj n r(maxj Aj 1. 12

13 Next, equation (2.11 gives, V HK ( f u f(x u :1 u dx u u [ɛ n,1] u B (x j Aj 1 dx u u [ɛ n,1] u j u ( B ɛ Aj n 1 1 A j ( BC P d Aj A 1 j n r P d Aj, establishing (5.2 under the origin avoidance conditions. For the corner avoidance conditions, let f be the low variation extension of f from [ɛ n, 1 ɛ n ] d to [0, 1] d with anchor c = (1/2,..., 1/2. Equation (5.3 also holds in this case. Now consider the 2 d congruent subcubes [0 u, c u ] [c u, 1 u ] of [0, 1] d where u ranges over all subsets of 1 : d. Then f(x f(x dx is a sum of 2 d terms, each of which is below a finite multiple of n r(maxj Aj 1. Similarly V HK ( f is no larger than the sum of 2 d Hardy-Krause variations from the 2 d subcubes, each of which is below a finite multiple of n r P j Aj. Corollary 5.3. Suppose that x 1,..., x n have Dn(x 1,..., x n Dn 1+ɛ for any ɛ > 0, where D depends on ɛ but not on n. Let f(x be a real valued function on (0, 1] d that satisfies condition (2.4. If x 1,..., x n K orig min (Cn r for 0 < C < and r 1, then as n, Î I = O(n 1+ɛ+r P d Aj. (5.4 If f(x is a real valued function on (0, 1 d satisfying condition (2.5, and x 1,..., x n Kmin corn(cn r for 0 < C < and r 1, then (5.4 holds as n. Proof: It suffices to plug the bound for Dn into (5.2 and notice that when r 1 the first term dominates. The conditions of Corollary 5.3 are satisfied by the Halton sequence. They are also satisfied by low discrepancy points linearly adjusted to avoid the origin (or corners. Hlawka and Mück (1972 show that when x j i xj i ɛ then D n( x 1,..., x n Dn(x 1,..., x n (1 + 2ɛ d 1. Thus, if x 1,..., x n have discrepancy O(n 1+ɛ then so do the origin avoiding points x i = ɛ n +(1 ɛ n x i, interpreted componentwise, when ɛ n = Cn 1 for 0 < C <. For corner avoidance we may take x i = ɛ n + (1 2ɛ n x i instead. For functions like f l in Section 2 where A j can be arbitrarily close to zero, the rate is O(n 1+ɛ for the Halton sequence or indeed for any low discrepancy points after a linear adjustment to avoid the singularity. Monte Carlo sampling attains a root mean squared error of O(n 1/2 when max j A j < 1/2. Low discrepancy sequences that are confined to K orig min (Cn 1 (respectively Kmin corn(cn 1 are asymptotically superior to Monte Carlo for functions satisfying (2.4 (respectively (2.5 when d A j < 1/2. When j A j > 1/2 Monte Carlo can be superior to low discrepancy sampling. Consider the function f(x = d (xj Aj and suppose that x 1 = (ɛ n,..., ɛ n for ɛ n = Cn 1. Then Î C d n P j Aj 1. In an extreme setting with j A j > 1 and 13

14 max j A j < 1/2 quasi-monte Carlo sampling can have Î I while Monte Carlo sampling has root mean square error O(n 1/ Avoiding hyperbolic regions. The previous section shows how low discrepancy points that avoid an L-shaped region around the origin can give worse integration error than Monte Carlo points. A significant improvement can be obtained by avoiding hyperbolic regions. The error bound in Theorem 5.2 has a component with rate depending on j A j and another with rate depending on max j A j. The first component dominates when r 1. To reduce the first component, we consider points that avoid the origin more strongly, by staying out of the region where j xj < ɛ n. The next Lemma, from Sobol (1973a is used below. Lemma 5.4 (Sobol (1973a. Let A 1,..., A d be distinct real numbers, none of which equal 1. Let K = K(ɛ = {x [0, 1] d d xj ɛ} and put φ(a = d (a A j. Then, as ɛ 0. [0,1] d K (x j Aj dx = 1 d φ(1 + ɛ 1 Aj (A j 1φ (A j = O(ɛ1 maxj Aj (5.5 In Lemma 5.4 it is assumed that the A j are distinct. This allows one to avoid considering logarithmic powers of ɛ. If either (2.4 or (2.5 holds for some A j, then it also holds for larger A j. We can then increase some of the A j in order to make them not equal. Theorem 5.5. Let f(x be a real valued function on (0, 1] d that satisfies condition (2.4. Suppose that x 1,..., x n [0, 1] d satisfy d xj i ɛ n where 0 < ɛ n cn r < 1. Then for any η > 0, Î I C 1D n(x 1,..., x n n η+r maxj Aj + C 2 n r(maxj Aj 1, (5.6 holds, for finite C 1 and C 2, that may depend on η. If f(x is a real valued function on (0, 1 d that satisfies condition (2.5, and x 1,..., x n satisfy d min(xj i, 1 xj i ɛ n where 0 < ɛ n cn r < 2 d, then (5.6 holds. In both cases (5.6 holds with η = 0, when there is a unique maximum among A 1,..., A d. Proof: The proof proceeds as in Theorem 5.2. Suppose that the first (origin avoidance conditions hold, and let f be the low variation extension of f from K orig prod (ɛ n to [0, 1] d with anchor (1,..., 1. Then the bound (5.3 holds once again. The truncation error satisfies Q j xj ɛ n f(x f(x dx B Q j xj ɛ n (x j Aj 1 maxj Aj dx = O(ɛn, by Lemma 5.4. Thus the truncation error is below C 2 n r(maxj Aj 1 for some finite C 2. Now for each nonempty u 1:d, let m(u = arg max j u A j, making an arbitrary choice when the maximum is not unique, and put u = u {m(u}. When all of the 14

15 A j are distinct, V HK ( f 1 Q u [0,1] u j u xj ɛ n u f(x u :1 u dx u B 1 Q u [0,1] u j u xj ɛ n (x j Aj 1 dx u j u B 1 Q ɛ A m(u n u [0,1] u j u x j ɛ n (x j Am(j Aj 1 dx u A m(u j u B u ɛ A m(u n A m(u 1 A j u m(j A j C 1 n r maxj Aj, (5.7 for some finite C 1. If A j = A k < max l A l for some j k, then it is possible to increase some of the A j, so that A 1,..., A d are distinct, while leaving max l A l unchanged. Then equation (5.7 still holds for some C 1 <. If two or more A j are equal to max l A l then the A j can be increased to distinct values, while raising the maximum A l by no more than η. This establishes (5.6 under the origin avoidance conditions. For the corner avoidance conditions, let f be the extension of f from {x [0, 1] d j min(xj, 1 x j ɛ n } to [0, 1] d with anchor c = (1/2,..., 1/2. Then split [0, 1] d into 2 d subcubes [0 u, c u ] [c u, 1 u ] for u 1 : d. The truncation error is a sum of 2 d truncation errors like those from the first part of the theorem. The Hardy-Krause variation of f is no larger than the sum of 2 d variations from within the subcubes. Each of these variations is O(n η+r maxj Aj for any η > 0 and is O(n r maxj Aj if the largest A j is unique. Corollary 5.6. Suppose that x 1,..., x n have Dn(x 1,..., x n Dn 1+ɛ for any ɛ > 0, where D depends on ɛ but not on n. Let f(x be a real valued function on (0, 1] d that satisfies condition (2.4. Suppose that x 1,..., x n K orig prod (Cn r, where 0 < C < and r 1. Then as n, Î I = O(n 1+ɛ+r maxj Aj. (5.8 Similarly if f(x is a real valued function on (0, 1 d that satisfies condition (2.5, and x 1,..., x n K corn prod (Cn r where 0 < C < and r 1, then (5.8 holds. Proof: Fix ɛ > 0, and employ (5.6 with η = ɛ/2 and D n(x 1,..., x n Dn 1+ɛ/2. The Halton sequence satisfies Corollary 5.6 with r = 2 for corner avoidance and r = 1 for origin avoidance. For arbitrary low discrepancy points x 1,..., x n the simple linear adjustment from the previous section is inadequate to avoid detailed analysis of the boundary behavior of points, when d > r. A point at a corner must be moved a distance proportional to n r/d to enter K corn prod (Cn r. If all points were moved an amount proportional to n r/d the discrepancy could be adversely affected. If instead one sought to show that only a very small fraction of points need to be moved, then one has to engage in a detailed analysis to count the number of points being moved. Suppose that max j A j < 1/2, so that the root mean square error in Monte Carlo is O(n 1/2. The error bound in the Halton sequence is asymptotically smaller than 15

16 O(n 1/2 when max j A j < 1/(2r. For origin avoidance, or for d = 1, we find r = 1 and then the Halton sequence is superior to Monte Carlo. For corner avoidance using r = 2 we find the Halton sequence is provably superior to Monte Carlo when max j A j < 1/4 but not necessarily when 1/4 < max j A j < 1/2. In many practical problems, max j A j can be taken as an arbitrarily small positive value. Then the Halton points are asymptotically superior to Monte Carlo Randomized quasi-monte Carlo. Randomized versions of quasi-monte Carlo points have been used to derive sample based error estimates. Typically the randomized points satisfy x i U[0, 1] d but the x i are dependent in a way that keeps Dn(x 1,..., x n small. For example, Cranley and Patterson (1976 replaced each x j i by x j i = {xj i + U j } where (U 1,..., U d U[0, 1] d, and Owen (1995 scrambled the digits in a base b expansion of x i. L Ecuyer and Lemieux (2002 provide a recent survey of randomized QMC methods. The first clause in Lemma 4.1 shows that randomized QMC points avoid a hyperbolic region around the origin, so it is not surprising that randomized QMC points are also well suited to unbounded integrands satisfying (2.5. Theorem 5.7. Suppose that x 1,..., x n are random points in [0, 1] d for which each x i U[0, 1] d, and E(Dn(x 1,..., x n = O(n 1+ɛ for all ɛ > 0. Then if f satisfies (2.5, E( Î I = O(n 1+ɛ+maxj Aj (5.9 for all ɛ > 0. Proof: For C > 0 let K n = Kprod corn(cn 1 and let f n be the low variation extension of f from K n to [0, 1] d. Then the three epsilon argument behind (2.3 yields E( Î I 2 [0,1] d K n f(x f n (x dx + E(D n(x 1,..., x n V HK ( f n. The factor of 2 arises because f n (x i does not always equal f(x i. From Lemma 5.4, we find that [0,1] d K n f(x f n (x dx = O(n 1+ɛ+maxj Aj, where some of the A j might have to be increased by an amount between 0 and ɛ to break ties. For ɛ > 0, the proof of Theorem 5.5 for the case r = 1 shows that V HK ( f n = O(n maxj Aj 1+ɛ/2. Similarly by hypothesis E(D n(x 1,..., x n Cn 1+ɛ/2, so E(D n(x 1,..., x n V HK ( f n = O(n 1+maxj Aj+ɛ. To see that the condition on E(Dn(x 1,..., x n is not void one can consider the scrambled nets presented in Owen (1995, or the space efficient alternative scramblings of Matoušek (1998b. They remain nets with probability one under scrambling. Then by theorems in Niederreiter (1992 we have that Pr(Dn(x 1,..., x n Cn 1+ɛ = 1 for any ɛ > 0 and some C < depending on ɛ but not on n. It then follows that E(Dn(x 1,..., x n = O(n 1+ɛ. The root mean square of Dn(x 1,..., x n is also O(n 1+ɛ. Hickernell (1996 has studied moments of discrepancy under randomization. For suitably smooth integrands, some randomizations of QMC can lead to improved accuracy. Owen (1997 gives conditions on f under which certain scrambled nets attain a root mean square error of O(n 3/2+ɛ. We cannot however expect unbounded functions subject only to growth conditions like (2.5 to be suitably smooth. The implied constant in that asymptotic root mean square error includes a factor equal to 1:d f(x 2 dx. We cannot expect that factor to be finite for [0,1] d unbounded functions. If it is finite then [0,1] d 1:d f(x dx must also be finite and so 16

17 then [0,1] d 1:d f(xdx must be finite. But this latter integral is an alternating sum of f at the 2 d corners of [0, 1] d which we expect to include some infinite function values. While the rate in Theorem 5.7 does not greatly improve on the one in Corollary 5.6 for quasi-monte Carlo points, it is not worse, and is sometimes better. Randomized QMC points have an error rate that corresponds to the case r = 1 for QMC points. They always beat the Monte Carlo rate when max j A j < 1/2. 6. Epilogue. Some more results have become available while this paper was circulating as a preprint. Most notably, Hartinger, Kainhofer, and Ziegler (2004 have found that generalized Niedierreiter sequences, including most of the commonly used digital nets avoid hyperbolic regions around the origin at the r = 1 rate. They find that d min(xj n, 1 x j n n 1 [log(n] 1 holds infinitely often but for any ɛ > 0 there is a C ɛ > 0 with d min(xj n, 1 x j n > C ɛ n 1 ɛ for all n 1. For the Faure sequence they find r 2 for some mixed corners and r 3/2 for the corner opposite the origin. Owen (2004 applies the methods of this paper to integrands with isolated point singularties at unknown locations. The mean absolute error in randomized QMC attains the rate o(n 1/2 when the growth conditions on the integrand imply it has finite variance, beating the MC rate that applies to such functions. Acknowledgments. I thank I. M. Sobol for communicating his low variation function extension technique, and for other helpful comments. I thank Reinhold Kainhofer for many useful remarks including one that lead to a smaller constant in (3.3. This work was supported by the U.S. National Science Foundation grand DMS References. Chung, K.-L. (1974. A course in probability theory (2nd ed.. New York: Academic Press. Cranley, R. and T. Patterson (1976. Randomization of number theoretic methods for multiple integration. SIAM Journal of Numerical Analysis 13, Davis, P. J. and P. Rabinowitz (1984. Methods of Numerical Integration (2nd Ed.. San Diego: Academic Press. de Doncker, E. and Y. Guan (2003. Error bounds for the integration of singular functions using equidistributed sequences. Journal of Complexity 19 (3, Genz, A. and J. Monahan (1998. Stochastic integration rules for infinite regions. SIAM journal on scientific computing 19 (2, Glasserman, P. (2004. Monte Carlo methods in financial engineering. New York: Springer. Halton, J. (1960. On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numerische Mathematik 2, Hartinger, J., R. F. Kainhofer, and R. F. Tichy (2004. Quasi-Monte Carlo algorithms for unbounded, weighted integration problems. Journal of Complexity 20 (5, Hartinger, J., R. F. Kainhofer, and V. Ziegler (2004. On the corner avoidance properties of various low-discrepancy sequences. Technical report, Graz University of Technology, Department of Mathematics. Hickernell, F. J. (1996. The mean square discrepancy of randomized nets. ACM Trans. Model. Comput. Simul. 6,

18 Hickernell, F. J., I. H. Sloan, and G. W. Wasilkowski (2004. On tractability of weighted integration over bounded and unbounded regions in R s. Mathematics of Computation 74, 1. Hlawka, E. and R. Mück (1972. Über eine Transformation von gleichverteilten Folgen. II. Computing 9, Klinger, B. (1997. Numerical integration of singular integrands using lowdiscrepancy sequences. Computing 59, L Ecuyer, P. and C. Lemieux (2002. A survey of randomized quasi-monte Carlo methods. In M. Dror, P. L Ecuyer, and F. Szidarovszki (Eds., Modeling Uncertainty: An Examination of Stochastic Theory, Methods, and Applications, pp Kluwer Academic Publishers. Mathé, P. and G. Wei (2003. Quasi-Monte Carlo integration over R d. Mathematics of Computation 73, Matoušek, J. (1998a. Geometric Discrepancy : An Illustrated Guide. Heidelberg: Springer-Verlag. Matoušek, J. (1998b. On the L 2 discrepancy for anchored boxes. Journal of Complexity 14, Niederreiter, H. (1992. Random Number Generation and Quasi-Monte Carlo Methods. Philadelphia, PA: S.I.A.M. Owen, A. B. (1995. Randomly permuted (t, m, s-nets and (t, s-sequences. In H. Niederreiter and P. J.-S. Shiue (Eds., Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, New York, pp Springer-Verlag. Owen, A. B. (1997. Scrambled net variance for integrals of smooth functions. Annals of Statistics 25 (4, Owen, A. B. (2004. Randomized QMC and point singularities. Technical report, Stanford University, Department of Statistics. Owen, A. B. (2005. Multidimensional variation for quasi-monte Carlo. In J. Fan and G. Li (Eds., International Conference on Statistics in honour of Professor Kai-Tai Fang s 65th birthday. Papageorgiou, A. (2003. Sufficient conditions for fast quasi-monte Carlo convergence. Journal of Complexity 19, Patel, J. K. and C. B. Read (1996. Handbook of the Normal Distribution (Second ed.. Marcel Dekker. Sloan, I. H. and S. Joe (1994. Lattice Methods for Multiple Integration. Oxford: Oxford Science Publications. Sobol, I. M. (1973a. Calculation of improper integrals using uniformly distributed sequences. Soviet Math Dokl 14 (3, Sobol, I. M. (1973b. On the use of uniformly distributed sequences for approximate computation of improper integrals. In Theory of cubature formulas and applications to certain problems in mathematical physics, pp Novosibirsk Nauka. (In Russian. 18