Necessity of low effective dimension

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1 Necessity of low effective diension Art B. Owen Stanford University October 2002, Orig: July 2002 Abstract Practitioners have long noticed that quasi-monte Carlo ethods work very well on functions that are nearly superpositions of low diensional functions. The reason is that the low diensional coordinate projections of QMC rules can have very good equidistribution properties at saple sizes for which the original rule itself cannot have good equidistribution. This paper explores a converse proposal: that low effective diension is necessary for QMC to be uch better than MC in high diensions with practical saple sizes. 1 Introduction In soe high diensional applications Quasi-Monte Carlo (QMC ethods of integration give very good results that are not easily explained by the Koksa-Hlawka inequality or other discrepancy bounds. By inspecting the proofs of upper bounds on error one ight suspect that the asyptotes would set in at a nuber n of evaluations that grows at least exponentially with the diension d. Indeed, Sloan and Wozniakowski (1998 show that in a worst case setting, that QMC (using lattices is no better than estiating the integrand by zero until n 2 d. Yet in nuerical exaples, particularly soe arising in coputational finance (Paskov and Traub 1995; Ninoiya and Tezuka 1996, it is soeties observed that QMC ethods in diensions as high as 360 can provide very good accuracy at practically relevant saple sizes n. One explanation that has been offered is that the integrands ay have effective diension saller than d. Specifically, the integrand ay be nearly a su of lower diensional parts as described in Caflisch, Morokoff, and Owen (1997. Then if the QMC rule used has good equidistribution in its low diensional coordinate projections, an accurate result is not surprising. 1

2 As Tezuka (2002 notes, a second class of high diensional integrands for which good results have been seen in QMC are certain isotropic integration probles of Capstick and Keister (1996. Such probles reduce to coputing the expectation of a function of the nor of a d diensional spherical Gaussian rando vector. Papageorgiou and Traub (1997 report good results for QMC on such probles. Recently Owen (2001 has shown that polynoials in the squared nor have effective diension no larger than their degree. Furtherore, nuerical investigation of a 25 diensional isotropic function published in Papageorgiou and Traub (1997 show that it is very nearly a superposition of functions of 3 or fewer of the input variables. Because the isotropic integrands appeared to be also of low effective diension, it is interesting to conjecture that low effective diension is soehow a necessary condition for QMC to work well at values of n below those where the discrepancy bounds apply. It is clear that În I can be sall without f necessarily having low superposition diension. For instance any f continuous on [0, 1] d, of whatever effective diension, always has at least one point x with f(x = I and so n = 1 is copatible with În = I. We will frae the proble in a way that rules out such cases, because no general purpose integration algoriths are based on finding such an x. In this paper we find that low effective diension is necessary for scrabled (0,, d-nets to have a uch saller variance than ordinary Monte Carlo, in high diensions and for practical saple sizes. We do however uncover a surprising free lunch phenoenon: it is possible to have a scrabled net variance of zero on certain nonzero functions of effective diension + 1 or + 2, but this effect requires a saple size n of at least (d 1 d 2. Section 2 introduces soe notation, including the ANOVA decoposition of L 2 [0, 1] d. Section 3 discusses effective diension and forula (5 there shows how low effective diension can lead to upper bounds on quadrature error. We are interested in a converse which we base on scrabled nets as outlined in Section 4. Theore 1 there shows that if a scrabled (t,, d- net (for 1 < d has variance below 0.01/Γ ties that of ordinary Monte Carlo sapling, that the function f is necessarily of superposition diension at ost. The quantity Γ 1 is a iniu gain coefficient for the net, and Section 5 describes how far below unity Γ can be, for (0,, d- nets. Section 6 discusses the results and considers how generally necessity of low superposition diension ight hold. Section 7 proves soe results on iniized gain coefficients. 2

3 2 Notation We consider approxiating I = [0,1] d f(xdx by Î n = 1 n n f(x i (1 i=1 for points x 1,..., x n [0, 1] d. Equation (1 includes Monte Carlo (MC and quasi-monte Carlo (QMC ethods that doinate practice when d is large. If f L 2 [0, 1] d, then siple Monte Carlo sapling with independent x i uniforly distributed on [0, 1] d yields a rando În with ean I and variance σ 2 /n. The error În I is of order n 1/2 in probability, written O p (n 1/2. The law of the iterated logarith establishes that the slightly larger error bound În I = O([n 1 log log(n] 1/2 holds with probability 1. If f BVHK[0, 1] d, the space of functions of bounded variation in the sense of Hardy and Krause, then În I D n(x 1,..., x n f HK (2 The first indications that QMC could be significantly better than MC arose fro constructions of points x 1,..., x n for which Dn = O(n 1+ɛ. The saple size n at which the asyptotic error rates for QMC should set in is thought to grow exponentially with diension d. Sloan and Wozniakowski (1998 show that in a worst case sense QMC is no better than estiating the integrand by zero until n 2 d. Owen (1997b finds that scrabled Faure sequences becoe uch better than Monte Carlo at roughly n d d while Owen (1998b estiates a threshold at roughly n 4 d for Niederreiter-Xing sequences. For f L 2 [0, 1] d there is an analysis of variance (ANOVA decoposition with f(x = f u (x. (3 u {1,...,d} The function f u (x depends on x = (x 1,..., x d only through coponents x j with j u. It also satisfies 1 0 f u(xdx j = 0 whenever j u for any values of x k, k j. The nae ANOVA arises because the variance of f is σ 2 = (f(x I 2 dx, attributable to subsets u of inputs via σ 2 = u σ 2 u (4 where σ 2 = 0 and σ2 u = f u (x 2 dx for u with cardinality u > 0. To rule out trivial cases we assue 0 < σ 2 <. 3

4 3 Effective diension Caflisch, Morokoff, and Owen (1997 define the effective diension of a function in two senses. The function f has effective diension s in the superposition sense if σu σ 2 u s and it has effective diension s in the truncation sense if σu σ 2. u {1,...,s} The idea of effective diension appears in Paskov and Traub (1995 where they reark that certain integrands fro finance are not essentially deterined by just a sall nuber of leading input variables. Sloan and Wozniakowski (1998 introduce classes of functions in which the iportance of each successive variable x j decays as j increases. Such functions can have sall truncation diension relative to their noinal diension. The definitions above capture two notions in which f is alost s diensional. As Caflisch, Morokoff, and Owen (1997 reark, the choice of 99 th percentile is arbitrary. This paper uses the 99 th percentile for definiteness sake. Hickernell (1998 akes the threshold percentile a paraeter in the definition. The quadrature error in a QMC rule x 1,..., x n satisfies the bound Î n I D n, u (x u 1,..., x u n f u (5 u >0 where x u i is the coordinate projection of x i onto the subset u of input variables, D n, u is a discrepancy for n points in [0, 1] u and f u is a copatible nor. A version of forula (5 appears in Caflisch, Morokoff, and Owen (1997 and several versions are given by Hickernell (1998. The upper bound (5 shows how low superposition diension can guarantee good perforance in QMC. Suppose that the function f belongs to a class in which f u is sall whenever u > s for 1 s < d. Then (5 provides for a sall error, when we use a ethod with D n, u sall for u s. Many widely studied discrepancies allow for tight versions of (5. For those discrepancies, given x 1,..., x n we ay find f such that În I = D n, u (x u 1,..., xu n f u and f = f u. Then, given lower bounds on D n, u for large u, low effective diension is necessary for În I to be uniforly sall over functions with f 1. 4

5 4 Scrabled nets Digital nets are defined in Niederreiter (1992. A ethod of scrabling of the was proposed in Owen (1995. The discrepancy of scrabled nets was studied in Hickernell and Yue (2000. The variance forulas for scrabled nets below are fro Owen (1997a. The variance of În when x i are scrabled versions of a i [0, 1] d is 1 n Γ u,κ σu,κ. 2 u >0 κ Here Γ u,κ is the gain coefficient corresponding to the subset u {1,..., d} and the vector κ containing u nonnegative integers, defined as 1 n n Γ u,κ = (bn n(b 1 u i,j,r W i,j,r, (6 where and i=1 j=1 r u N i,j,r = N i,j,r (κ = 1 b k j +1 a r i = bk j +1 a r j W i,j,r = W i,j,r (κ = 1 b k j a r i = bkj a r j are indicator variables designating narrow and wide atches respectively between the coponents a r i and ar j. This forula holds for any a 1,..., a n [0, 1] d but it can siplify for nets, especially nets for which the quality paraeter t is 0. The notation Γ u,κ suppresses the dependence of the gain coefficient on b and. When this dependence ust be ade explicit, the notation Γu,κ b, will be used. The variance of Î n under scrabling is a su of contributions fro every nonepty u and every vector κ {0, 1,... } u. Because σu 2 = κ σ2 u,κ a ethod with all gains Γ u,κ = 1 has the sae variance as Monte Carlo sapling. Theore 1 Let f L 2 [0, 1] d, let În be the quadrature rule (1. Denote the variance of În by Var snet (În or Var c (În depending on whether the x i are a scrabled (t,, d-net with 1 < d in base b or siple Monte Carlo respectively. Suppose that Var(Îrqc ɛvar(îc for 0 < ɛ < 1. Then the function f satisfies u > σ2 u σ 2 ( 1 ɛ in Γ u,κ. u >,κ {0,1,... } u 5

6 Proof: Under the hypothesis of the theore, ɛ Var snet(în Var c (În u > κ Γ u,κ σ 2 u,κ σ 2 ( in Γ u,κ u >,κ u > σ 2 u σ 2. The following corollary is iediate: Corollary 1 If scrabled (t,, d-net integration of f has variance saller than 0.01/ in u >,κ Γ u,κ ties that of ordinary Monte Carlo, then f has effective diension at ost. 5 Lower bounds on gain coefficients To draw any practical conclusions fro Theore 1 and the Corollary requires lower bounds on gain coefficients Γ b, u,κ for u >. Fro the defining property of a (t,, d-net it follows (Owen 1997a that Γ u,κ = 0 when u + κ t. When t > 0 and u + κ > t the net property does not uniquely define Γ u,κ. Fro here on, we restrict attention to the gain coefficients for (t,, d- nets with t = 0. For (0,, d-nets, it is known that [ ( Γ u,κ = 1 + (1 b u ( b k k k j=0 ( ] u ( b j. (7 j Here, and in the following, the integers u and κ are replaced by u and k in expressions for Γ. This reduces clutter and leads to no loss of generality because for a (0,, d-net the gain depends on the subset u and the vector κ only through their cardinality and coponent su respectively. We suppose also that b is a prie power. Most digital net constructions use prie power bases. Niederreiter (1987 shows how to construct digital nets in ore general bases b 2 fro those with prie power bases, but the resulting nets have coparatively sall diension d. It follows easily fro (7 that Γ u,k = 1 for k. Also (0,, d-nets in base b can only exist when d b + 1. Thus for any b and, the interesting gain coefficients can be arranged into a b + 1 by table. As a consequence of Theore 1 we are particularly interested in Γ = Γ b, in Γ b, u>,k 0 u,k, 6

7 Table 1: Shown are the gain coefficients for randoized (0, 4, d-nets in base 16. The value Γ u, κ appears in the row with u on the left and in the colun headed by κ. The largest relevant value of u is 17 because u d b + 1 = 17. The upper left corner of exact zeros is left blank. For κ = 4 the gain is exactly one. The other values have been rounded. Rows 11 through 16 look the sae as rows 10 and 17. for < d. If Γ is near one then a variance reduction of slightly over 100- fold for scrabled (0,, d-nets iplies that f has effective diension at ost. Table 1 shows the gain coefficients for scrabled (0, 4, d nets in base 16. These nets have n = points in [0, 1] d. The sallest gain coefficient for u > 4 is Γ = Γ 6,0 = / = Theore 1 shows that if scrabled net sapling has variance ɛ ties as large as ordinary Monte Carlo sapling then the fraction of the variance of f due to ANOVA coponents of diension 5 or ore is at ost ɛ/ = ɛ To conclude that f has effective diension 5 or less we need to find that the scrabled net variance is no larger than ties the Monte Carlo variance. The values Γ 16,4 u,k in Table 1 below the row for u = 4 are alternately above and below unity as u or k increases. When u > the gain Γ u,k is at least 1 for odd u + k and at ost 1 for even u + k. Moving fro left to right in the table the gains approach one. When u is plus an odd positive nuber, the gains decrease to unity as k increases through even values beginning with 0. Siilarly when u > 0 is odd and k increases through odd values beginning with 1, the gains increase to unity. When u 7

8 is plus an even positive nuber, the trends described above are reversed. Lea 1 below shows that these onotone trends hold in generality. It follows that the search for Γ can be restricted to cases of the for Γ +2r,0 for r 1 or Γ +2r+1,1 for r 0. It also holds in Table 1 that Γ +2r+1,1 Γ +2r+2,0 when r 0 and + 2r + 2 b + 1. Lea 2 below shows that this pattern holds in generality, and so the search for the iniu Γ can ordinarily be restricted to Γ +2r+2,0 for r 0. There is an exception when = b, for then u = +2 = b+2 is inapplicable and so the iniizer ust be Γ b+1,1. Finally the ters Γ +2r+2,0 in Table 1 are nondecreasing as r 0 increases. This too is a general phenoenon (Lea 3 below and so the iniizer is Γ = Γ +2,0 when b 1. Theore 2 Let b 2 be an integer. For a non-negative integer b 1 Γ = in in Γ u,k = Γ +2,0, <u b+1 k 0 and for = b Γ = in in Γ u,k = Γ +1,1. <u b+1 k 0 Proof: See Section 7. The iniizing gain coefficients siplify. They can be surprisingly sall as the next proposition shows. Proposition 1 Let b 2 and 0 be integers. Then ( Γ b, b +2,0 = b 1, for b 1 and, b 1 b 1 ( Γ b, b +1,1 = b, for b. b 1 b In particular Γ b,b 1 b+1,0 = Γ b,b b+1,1 = 0. Proof: The result follows after a short anipulation of binoial coefficients. The alternating su within the square brackets of (7 atches all but u +k ters of u ( u j=0 j ( b j, which equals (1 b u by the binoial theore. For the cases here u + k = 2. This proposition provides an astonishing exaple of an apparently free lunch. For a (0,, d-net certain hypercubical subsets of [0, 1] d are guaranteed to contain a nuber of points equal to b ties their volue. We 8

9 say these sub-cubes are balanced by the net. For < d the balanced sub-cubes have at least d sides of length 1, and in that sense are less than fully d diensional. It is a surprise to be able to integrate exactly soe fully d diensional integrands using nets that do not balance any fully d diensional sub-cubes. The phenoenon in Proposition 1 does not help with high diensional probles. First, the sallest gain coefficients are for diensions only one or two higher than. Second, the nuber of points in such nets is n = b which is either b b 1 or b b. Because b d 1 this phenoenon does not apply for n < (d 1 d 2. To further study Γ we suppose that < b + 1 so that Γ b, = Γ b, +2,0. The sallest values of this gain arise with sall b and large : Proposition 2 If 1 and b + 1 then Γ b, +2,0 increases with b. If b 2 and 0 b 1 then Γ b, +2,0 decreases with. Proof: Although Γ b, +2,0 is defined for integers, we ay interpolate by real values. Then ( b log Γ b, +2,0 = b b b 1 1 b 1 ( + 1 = b(b 1(b 1 0, and, ( ( log Γ b, b 1 +2,0 = log b 1 b 1 1 b 1 1 b 1 0. Very sall values of Γ are possible for n (d 1 d 2. Here we investigate nuerically how sall Γ can be for large diensions and practical saple sizes. We restrict attention to Γ b, b+2,0 as this is the iniu gain when b 1. Consider d 20 and n For d 20 the sallest prie power b d 1 is b = 19. For b = 19 and n 10 7 the largest we can have is log 19 (10 7 = 5. Therefore for d 20 and n 10 7, the iniu gain ust be at least Γ 19,5 7,0 = , which is not uch below one. Table 2 records the results of siilar calculations varying both the lower bound on d and the upper bound on n. 9

10 Table 2: This table shows iniu possible gain coefficients for scrabled (0,, d-nets. The rows are labelled with diensions d and the coluns are labelled with saple sizes n. Let b = b(d be the sallest prie power no saller than d 1. Shown is the sallest value of Γ b, u,k subject to b b(d, b n, < u d, and k 0. Zeros for d = 10 correspond to (0, 8, 10-nets in base 9. These have Γ 9,8 10,0 = 0 and n = 98 = 43, 046, 721. For large diensions and practical saple sizes Γ b, u,k cannot be appreciably saller than 1. In those settings, a variance reduction of just over 100 iplies that f has effective diension at ost. When the saple size can reach b b 2 then Γ b, u,k can becoe surprisingly sall. 6 Discussion The conclusion of this paper is that low effective diension is necessary for scrabled (0,, d-nets to be uch better than Monte Carlo for large d and practical n. A surprising free lunch phenoenon was found in which the scrabled net variance could be zero for soe nonzero functions of effective diension b + 1 when = b or b 1, but the free lunch was only seen for n (d 1 d 2. There are three iportant features to Theore 1. The first is that perforance of scrabled nets is studied relative to Monte Carlo, not absolutely. The second is that it is not asyptotic. The saple sizes used are of the for b for < d, including values below those at which QMC asyptotics are thought to take effect. The third is that it provides a conclusion about the function f itself, without reference to a containing function class. These features are iportant because they capture what surprised any observers: QMC can be uch better than MC for specific functions with large d at surprisingly sall n. A non-asyptotic analysis is essential because Var snet (În/Var c (În 0, for any f L 2 [0, 1] d, regardless of effective 10

11 diension. Siple counterexaples with spiky integrands show that low effective diension cannot be sufficient for good perforance of QMC, either absolutely or relatively. For exaple, let f(x = ɛ 1 1 x 1 ɛ. Then f has truncation and superposition diension both equal to 1 but cannot be integrated well by QMC for n 1/ɛ. It is also easy to see that sall truncation diension is not necessary for QMC to be uch better than MC. The linear function f(x = d j=1 xj is easy for QMC ethods, but has truncation diension at least 0.99d, for any re-ordering of the variables. Truncation diension is an iportant aspect of infinite diensional probles. Owen (1998a shows by a artingale arguent that any square integrable function on [0, 1] necessarily has finite effective diension in the truncation sense, for any threshold less than 100 percent. Recent work of Sloan (2002 on function classes with successively less iportant diensions shows that a sall quadrature error can be obtained for soe functions having high superposition diension. The iportance of input j decays rapidly, as quantified by a series of constants γ j for j 1. Let f be a purely 1000 diensional function in that class involving only diensions 1, 000, 001 through 1, 001, 000. Then f can be integrated with a sall error despite having superposition diension These tractability results are not at odds with the thesis of this paper. The forer study perforance relative to the zero rule Î = 0 while this paper considers perforance relative to Monte Carlo ethods. The function f would have a sall nor in order to fit in the space defined by the γ j sequence. A sall nor for f would also lead to a sall Monte Carlo variance and it is not clear that QMC would beat MC for this f. Low superposition diension is necessary for scrabled (0,, d-nets to beat MC by a wide argin at odest saple sizes in high diensions. But this does not show that low superposition diension is universally necessary for this phenoenon. Perhaps other general purpose QMC ethods can beat MC by a wide argin for odest n and integrands of high superposition diension. To show siilar results for scrabled (t,, d-nets with t > 0 requires lower bounds on gain coefficients Γ u,κ for t > 0. The definition of a (t,, d- net is not sharp enough to deterine gain coefficients when t > 0. Upper bounds on gain coefficients appear in Owen (1998b, Niederreiter and Pirsic (2001, and Yue and Hickernell (2002. A quadrature rule ay be described by a great any nonnegative t-values, one for each of a class of subintervals of [0, 1] d. The net noenclature specifies only the largest of these. When 11

12 the largest is zero then they are all zero. But in general a net with t > 0 can have saller t even t = 0 when projected into a set u of coponents. Schid (2001 describes this phenoenon. Yue (1999 provides gain coefficients for soe leading subsequences of (0, d-sequences in base b that like (λ, 0,, d- nets are not necessarily nets. It is interesting to speculate on whether necessity of low superposition diension ight hold outside of scrabled nets. Heinrich, Hickernell, and Yue (2001 show that scrabled nets are asyptotically optial quadrature rules in various settings. The function classes are defined by the decay of certain Haar wavelet coefficients and three approxiation senses are considered: worst case, rando case, and average case, in their terinology. The rando case setting is closest to the one considered here, but their results are not relative to Monte Carlo and are asyptotic. 7 Proofs We begin by recalling that ( n r = 0 if r < 0 or r > n and that the binoial identity ( ( n r = n 1 ( r + n 1 r 1 holds even when r > n 1 or r < 0. We will also use ( n ( r n = n r + 1 (8 r r 1 but only for 0 < r n + 1. This first lea establishes that certain onotone alternations seen in gain tables hold generally. Lea 1 For non-negative integers b,, r, and s, with b 2, b and + 2r + 1 b + 1, Γ +2r+1,2s Γ +2r+1,2s+2 1, (9 Γ +2r+1,2s+1 Γ +2r+1,2s+3 1. (10 For non-negative integers b,, r, and s, with b 2, b and + 2r b + 1, Γ +2r,2s Γ +2r,2s+2 1, (11 Γ +2r,2s+1 Γ +2r,2s+3 1. (12 Proof: We prove (11. The proofs of the other three propositions use the sae sequence of techniques. If 2s then Γ +2r,2s = Γ +2r,2s+2 = 1, so 12

13 without loss of generality we suppose 2s <. Use u = + 2r to shorten soe interediate expressions. Γ +2r,2s+2 Γ +2r,2s [ ( = (1 b u ( b 2s 2 2s 2 ( ( b 2s + 2s 2s j=0 ( u j ( b j ] 2s 2 j=0 ( u ( b j j [ ( ( = (1 b u ( b 2s 2 ( b 2s 2s 2 2s ( ( ] u u + ( b 2s 1 + ( b 2s 2s 1 2s [( ( = (1 b u ( b 2s 2 b 2 2s 2 2s ( ( ] u u b + b 2 2s 1 2s [( ( = (b 1 u b 2s 2 ( 1 2r+ 2s 2 b 2 2s 2 2s ( ( ( ( ] b b + b 2 + b 2 2s 1 2s 2 2s 2s 1 [( ( ] = (b 1 u b 2s 2 (1 b + (b 2 b. 2s 2 2s 1 If 2s = 1 then the first ter in square brackets vanishes and so the entire factor in square brackets is nonnegative. If 2s < 1 then we apply (8 13

14 within the square brackets, obtaining ( ( (1 b + (b 2 b 2s 2 2s 1 ( ( ( ( 2s = (b 1 b 2s 2 2s 1 ( ( 2r + 2s + 1 = (b 1 b 2s 2 2s 1 1 ( ( b (b 1 2s 2 2s 1 1 ( ( b (b 1 2s , 1 where we have used 2s 1 > 0 and b. It follows that Γ +2r,2s+2 Γ +2r,2s 0 establishing (11. To siplify soe anipulations, we introduce the ter Differences Su Su r yields Su+1 = S u S u = j=0 ( u ( b j. j reduce to sus of r ters. Also the binoial identity bsu 1 and Su+2 = S u 2bSu 1 + b 2 Su 2. The next lea shows that for any sall gain coefficient with k = 1 there is ordinarily one with k = 0 that is no larger. Lea 2 For non-negative integers b,, and r, with b 2, b and + 2r + 1 b + 1, Γ +2r+1,1 Γ +2r+2,0. (13 14

15 Proof: Γ +2r+1,1 Γ +2r+2,0 [ ( + 2r = (1 b 2r 1 ( b 1 1 ] S+2r+1 1 [ ( ] + 2r + 1 (1 b 2r 2 ( b S+2r+2 [ ( ( + 2r + 2r + 1 = (1 b 2r 2 ( b 1 (1 b ( b 1 ] (1 bs+2r S +2r+1 bs+2r+1 1 [ ( ( + 2r + 2r + 1 = (1 b 2r 2 ( b 1 (1 b ( b 1 ( ] + 2r ( b ( + 2r = (1 b 2r 2 ( b 1 (1 b 1 ( + 2r = (b 1 2r 1 b 1 ( 1 2r The final lea shows that the eleents Γ +2r,0 in the k = 0 colun of the gain table increase with r. Lea 3 For nonnegative integers b,, and r, with b 2, 1 b and + 2r + 2 b + 1, Γ +2r+2,0 Γ +2r,0. (14 15

16 Proof: Γ +2r+2,0 Γ +2r,0 [ ( + 2r + 1 = (1 b 2r 2 ( b (1 b 2r [ ( b ( + 2r 1 [ ( + 2r + 1 = (1 b 2r 2 ( b ] ] S+2r+2 ] S+2r ( + 2r 1 (1 b 2 ( b S+2r+2 + (1 b 2 S+2r [ ( ( + 2r r 1 = (1 b 2r 2 ( b (1 b 2 ( b ] S+2r + 2bS+2r 1 b2 S+2r 2 + (1 2b + b2 S+2r = (1 b 2r 2 [ ( b ( + 2r + 1 ] b 2 (S+2r S+2r 2 2b(S +2r S+2r 1 (1 b 2 ( b ( + 2r 1 [( ( + 2r r 1 = (1 b 2r 2 ( b (1 b 2 ( ( ( ] + 2r + 2r + 2r b + b 2 2b. 1 The product outside the square brackets is positive. The factor within square brackets siplifies to ( ( + 2r 1 + 2r (b (1 b 1 1 ( [ + 2r 1 = (b r ] 1 1 b 1 2r + 1 0, because + 2r b 1. Proof of Theore 2: Suppose b 1. For u = + 2r b + 1 with r 0 the iniu value of Γ u,k for k 0 is Γ u,0 by the alternating gain Lea 1. Siilarly for u = + 2r + 1 b + 1 with r 0 the iniu value of Γ u,k for k 0 is Γ u,1. Therefore the iniizing Γ u,k has the for 16

17 Γ +2r,0 or Γ +2r+1,1. But Lea 2 shows that Γ +2r+1,1 Γ +2r+2,0 in this case. Therefore the iniizing Γ u,k has the for Γ +2r,0. Finally Lea 1 show that the iniizing Γ u,k is Γ +2,0. When = b then + 2 > b + 1 and the only eligible Γ u,k entries are Γ b+1,k. Lea 1 then shows that the iniizing value is Γ b+1,1. Acknowledgents I thank Grzegorz Wasilkowski for helpful coents, and I also thank an anonyous reviewer for coents that otivated iportant changes in this paper. This work was supported by the NSF under grant DMS References Caflisch, R. E., W. Morokoff, and A. B. Owen (1997. Valuation of ortgage backed securities using Brownian bridges to reduce effective diension. Journal of Coputational Finance 1, Capstick, S. and B. D. Keister (1996. Journal of Coputational Physics 123 (2, Heinrich, S., F. J. Hickernell, and R. X. Yue (2001. Optial quadrature for Haar wavelet spaces. subitted to Math. Cop. Hickernell, F. J. (1998. Lattice rules: how well do they easure up? In P. Hellekalek and G. Larcher (Eds., Rando and Quasi-Rando Point Sets, pp New York: Springer. Hickernell, F. J. and R. X. Yue (2000. The ean square discrepancy of scrabled (t, s-sequences. SIAM Journal of Nuerical Analysis 38, Niederreiter, H. (1987. Point sets and sequences with sall discrepancy. Monatshefte fur atheatik 104, Niederreiter, H. (1992. Rando Nuber Generation and Quasi-Monte Carlo Methods. Philadelphia, PA: S.I.A.M. Niederreiter, H. and G. Pirsic (2001, Dec.. The icrostructure of (t,,s- nets. Journal of Coplexity 17 (4, Ninoiya, S. and S. Tezuka (1996. Toward real-tie pricing of coplex financial derivatives. Appl. Math. Finance 3,

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