Discrepancy Bounds for a Class of Negatively Dependent Random Points Including Latin Hypercube Samples
aa r X i v : . [ m a t h . S T ] F e b Discrepancy Bounds for a Class of NegativelyDependent Random Points Including Latin HypercubeSamples
Michael Gnewuch ∗ Nils Hebbinghaus † February 10, 2021
Abstract
We introduce a class of γ -negatively dependent random samples. We prove thatthis class includes, apart from Monte Carlo samples, in particular Latin hypercubesamples and Latin hypercube samples padded by Monte Carlo.For a γ -negatively dependent N -point sample in dimension d we provide proba-bilistic upper bounds for its star discrepancy with explicitly stated dependence on N , d , and γ . These bounds generalize the probabilistic bounds for Monte Carlosamples from [Heinrich et al., Acta Arith. 96 (2001), 279–302] and [C. Aistleitner,J. Complexity 27 (2011), 531–540], and they are optimal for Monte Carlo and Latinhypercube samples. In the special case of Monte Carlo samples the constants thatappear in our bounds improve substantially on the constants presented in the latterpaper and in [C. Aistleitner, M. T. Hofer, Math. Comp. 83 (2014), 1373–1381]. Discrepancy measures are well established and play an important role in fields like com-puter graphics, experimental design, empirical process theory, learning theory and ma-chine learning, random number generation, optimization (in particular, stochastic pro-gramming), and numerical integration or stochastic simulation, see, e.g., [6, 8, 9, 11, 12,20, 21, 22, 25, 41, 43, 45, 47, 50] and the literature mentioned therein.The prevalent and most intriguing discrepancy measure is arguably the star discrep-ancy , which is defined in the following way:Let P ⊂ [0 , d be an N -point set. (We always understand an “ N -point set” as a“multi-set”, i.e., it consists of N points, but those points do not have to be pairwise ∗ Institut f¨ur Mathematik, Universit¨at Osnabr¨uck, Germany ( [email protected] ). † Institut f¨ur Informatik, Christian-Albrechts-Universit¨at zu Kiel, Germany( [email protected] ). local discrepancy of P with respect to a Lebesgue-measurabletest set T ⊆ [0 , d by D N ( P, T ) := (cid:12)(cid:12)(cid:12)(cid:12) N | P ∩ T | − λ d ( T ) (cid:12)(cid:12)(cid:12)(cid:12) , where | P ∩ T | denotes the cardinality of the finite set P ∩ T and λ d denotes the d -dimensional Lebesgue measure on R d . For vectors x = ( x , x , . . . , x d ), y = ( y , y , . . . , y d ) ∈ R d we write [ x, y ) := d Y j =1 [ x j , y j ) = { z ∈ R d | x j ≤ z j < y j for j = 1 , . . . , d } . The star discrepancy of P is then given by D ∗ N ( P ) := sup y ∈ [0 , d D N ( P, [0 , y )) . The star discrepancy is intimately related to quasi-Monte Carlo integration via theKoksma-Hlawka inequality ([36, 39]): For every N -point set P ⊂ [0 , d we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z [0 , d f ( x ) d λ d ( x ) − N X p ∈ P f ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ D ∗ N ( P )Var HK ( f ) , where Var HK ( f ) denotes the variation of the integrand f in the sense of Hardy and Krause,see, e.g., [2, 47]. The Koksma-Hlawka inequality is sharp, see again [47]. (An alternativesharp version of the Koksma-Hlawka inequality can be found in [34]; it says that the worst-case error of equal-weight quadratures based on a set of sample points P over the normunit ball of the Sobolev space of dominated mixed smoothness of order one is exactlythe star discrepancy of P .) The Koksma-Hlawka inequality shows that equal-weightquadratures based on sample points with small star discrepancy yield small integrationerrors. (Deterministic equal-weight quadratures are commonly called quasi-Monte Carloquadratures ; for a survey we refer to [11].) For the very important task of high-dimensionalintegration, which occurs, e.g., in computational finance, physics or quantum chemistry,it is therefore of interest to know tight bounds for the smallest achievable star discrepancy D ∗ ( N, d ) := inf { D ∗ N ( P ) | P ⊂ [0 , d , | P | = N } or, equivalently, for the inverse of the star discrepancy N ∗ ( ε, d ) := inf { N ∈ N | D ∗ ( N, d ) ≤ ε } , which is the minimum number of sample points that guarantee a discrepancy bound ofat most ε , and to be able to construct integration points that satisfy those bounds. Toavoid the “curse of dimensionality” it is crucial that such bounds scale well with respectto the dimension d .For fixed d the best known asymptotic upper bounds for D ∗ ( N, d ) are of the form2 ∗ ( N, d ) ≤ C d ln( N ) d − N − , N ≥ , (1)see [30] or, e.g., the books [12, 47]. For larger d those bounds give us no helpful informationfor moderate values of N , since the function f ( N ) := ln( N ) d − N − is increasing for N ≤ e d − . Additionally, for d ≥ N much larger than e d − are needed before f ( N ) is below the common “Monte Carlo rate” 1 / √ N – in dimension d = 10, e.g., weneed N to be larger than 1 . · . Moreover, the constant C d may grow unfavorablyas d gets larger. Actually, it is known for some N -point constructions P that the constant C ′ d in the representation D ∗ ( P ) ≤ ( C ′ d ln( N ) d − + o (ln( N ) d − )) N − of (1) tends to zero as d approaches infinity, see, e.g., [47, 48, 5] or [23]. But the behaviorof the “whole constant” C d in (1) is unfortunately not known. That is why the asymptoticupper bound (1) is not helpful for high-dimensional integration and we have to look forpre-asymptotic bounds, i.e., bounds that give us useful information already for a moderatenumber of points N .The best known upper and lower bounds for the smallest achievable star discrepancywith explicitly given dependence on the number of sample points N as well as on thedimension d are of the following form: On the one hand, for all d, N ∈ N there exists an N -point set P ⊂ [0 , d satisfying D ∗ N ( P ) ≤ C r dN (2)for some constant C >
0, implying N ∗ ( ε, d ) ≤ (cid:6) C dε − (cid:7) (3)for all d ∈ N , ε ∈ (0 , c, ε > N ∗ ( ε, d ) ≥ cdε − (4)for all 0 < ε ≤ ε , d ∈ N , showing that for all N -point sets P ⊂ [0 , d necessarily D ∗ N ( P ) ≥ min (cid:26) ε , c dN (cid:27) . (5)Notice that (3) and (4) show that the inverse of the star discrepancy depends essentiallylinearly on the dimension d (in the sense that we have an upper bound for it that dependslinearly on d and also a lower bound that depends linearly on d ). The upper bounds(2) and (3) were proved in [33] without providing an estimate for the constant C . Anestimate was given in [1], who showed that C ≤ .
65. In the course of this paper wewill improve his estimate to C ≤ . ε in3rbitrary dimension d by a factor of more than 14, cf. (3). All the results mentioned sofar are based on probabilistic arguments and do not provide an explicit (deterministic)point construction that satisfies (2). The lower bounds (4) and (5) were established in[35]. Notice that there is a gap between the upper and lower bounds (2) and (5), and (3)and (4), respectively.Already 15 years ago Heinrich posed the following problems in [32, Problem 1 & 2]:(P1) For each N, d ∈ N give a (deterministic) construction of an N -point set P ⊂ [0 , d satisfying (2) for some positive constant C not depending on N or d .(P2) Does any of the various known (deterministic) constructions of low discrepancy pointsets satisfy an estimate like (2)?(P3) Determine the order of the smallest possible star discrepancy as a function of thenumber of points N and the dimension d .(P4) Determine α :=sup (cid:8) α | ∃ c, k ≥ ∀ N, d ∈ N ∃ P ⊂ [0 , d : | P | = N ∧ D ∗ N ( P ) ≤ c d k N α (cid:9) . (The so-called exponent of tractability of the star discrepancy τ is related to α inthe following way: α = 1 /τ ; see, e.g., [33].)Similar problems were stated in [49, 50] as Open Problems 6, 7, and 42.Problems (P1) to (P4) turned out to be very difficult to solve. It is, for instance,obvious that problem (P3) is a very hard problem, since it contains the so-called greatopen problem of discrepancy theory to find the precise order of the smallest possible stardiscrepancy in N for fixed dimension d ≥
3. But also the other problems turned out to bevery difficult to answer and have not been solved so far. For problem (P4) it is known that1 / ≤ α < . Conjecture 1.1 (Wo´zniakowski) . α = 1 / . If this conjecture is true, then, due to the essentially linear dependence of the inverseof the star discrepancy on the dimension d , the estimate (2) is actually the best possiblebound (apart from logarithmic factors) that is polynomial in N − as well as in d .One reason for the difficulty of Problems (P1) and (P2) is that already the problemof calculating the star discrepancy of an arbitrary N -point set is N P -hard, see [29], and,in the language of parametrized complexity theory, W [1]-hard, see [24].Heinrich also posed weaker versions of Problem (P1) and (P2), cf. [32]. Those weakerversions were at least formally solved with the help of derandomized algorithms thatgenerate deterministic N -point sets P that satisfy D ∗ N ( P ) ≤ C r dN p ln(1 + N ) , (6)4ee [17, 15, 19], or D ∗ N ( P ) ≤ C r d N p ln(1 + N/d ) , (7)see [16, 18], for some small constants C , C . Those algorithms derandomize probabilisticexperiments, in which random point sets P satisfying (6) or (7) with high probabilityare generated. As numerical experiments in [18, 19] showed, those algorithms work wellin dimensions up to d = 21, but for much larger dimension their running times areprohibitive. (For a more extensive discussion, see also [28].)To get closer to a solution of the problems stated by Heinrich, we propose to studythe following related randomized problems:(R1) What kind of randomized point constructions satisfy (2) in expectation and/or withhigh probability?(R2) Are there randomized point constructions that satisfy (2) in expectation and/or withhigh probability and have more evenly distributed lower dimensional projections orsatisfy better asymptotic discrepancy bounds than Monte Carlo points?(R3) Are there randomized point constructions that lead to a better estimate than (2) orthat can even be used to disprove Wo´zniakowski’s conjecture?We believe that the problems (R1), (R2), and (R3) are important and interesting intheir own rights. Moreover, an answer to question (R1) or (R2) would draw us closerto a solution of problem (P1) (since we may derandomize promising randomized pointconstructions) and of problem (P2) (since we get a hint, which known deterministic con-structions are worth to be examined closer). An affirmative answer to question (R3) maylead, due to the probabilistic method, to progress in problem (P3) and in problem (P4).Let us explain this a little bit more. As mentioned, the upper bound (2) was provedvia probabilistic arguments. Indeed, Monte Carlo points, i.e., independent random pointsuniformly distributed in [0 , d , satisfy this bound with high probability (cf. also Corol-lary 4.5).In [13] it was shown that the star discrepancy of Monte Carlo point sets X behaveslike the right hand side in (2). More precisely, there exists a constant K > X is bounded from below by E [ D ∗ N ( X )] ≥ K r dN (8)and additionally we have the probabilistic discrepancy bound P D ∗ N ( X ) < K r dN ! ≤ exp( − Ω( d )) . (9)The upper bound (2) is thus sharp for Monte Carlo points, showing that they cannotbe employed to improve the upper discrepancy bound (2) or to disprove Wo´zniakowski’sconjecture. 5learly, there are other random point constructions that look more promising thansimple Monte Carlo point sets as, e.g., Latin hypercube samples, see [46], scrambled( t, m, s )-nets, see [54, 56, 55], or randomly shifted lattice rules, see [60]. In general, theserandom points will not be stochastically independent, which raises the next problem:(R4) How to analyze random point constructions whose single points are not stochasti-cally independent?Obviously, this problem is not only of interest for analyzing the star discrepancy, but forstochastic simulation in general.In this paper we want to address the problems (R1), (R2), and (R4). We proceedas follows. First we introduce a negative dependence property of random point setsthat is based on negative orthant dependence and that we call γ -negative dependence, seeDefinitions 2.1 and 3.5. This property allows us to use large deviation bounds of Hoeffding-and Bernstein-type, see Section 2. In Section 3 we prove that the class of γ -negativelydependent random points contains, in particular, Latin hypercube samples and Latinhypercube samples padded by Monte Carlo in arbitrary dimension d , see Theorem 3.6.With Theorem 3.7 we provide a generalization of Theorem 3.6 that may be used to verify γ -negative dependence for random point sets different from (padded or unpadded) Latinhypercube samples. In Section 4 we show that for each γ = e ρd , ρ ∈ N , all γ -negativedependent random point sets P satisfy (2) with high probability and, surprisingly, theconstant C = C ( γ ) depends only mildly on γ , see Theorem 4.4. In particular, ourresult generalizes the results obtained in [1, 4, 33], since the latter results can be seenas probabilistic discrepancy bounds for Monte Carlo point sets. In Corollary 4.5 weprovide discrepancy bounds with explicit constants and explicit success probabilities forthe special instances of Monte Carlo point sets and Latin hypercube samples (paddedby Monte Carlo or not). In the special case of Monte Carlo point sets these constantsand success probabilities improve substantially on the ones derived in [1] and [4]. InRemark 4.6 we explain that the probabilistic discrepancy bound (2) is actually sharpfor Latin hypercube samples; this result follows directly from new lower probabilisticdiscrepancy bounds in [14].Let us close the introduction with some remarks on our notation.For N ∈ N we denote the set { , , . . . , N } by [ N ]. We denote the Lebesgue measureon R by λ . By P , E , and V we always mean probability, expectation, and variance,respectively. If not specified otherwise, all random variables are defined on a probabilityspace (Ω , Σ , P ). The concept of negative dependence was introduced in [40] for pairs of random variables.In the literature one finds several contributions on rather demanding notions of negativedependence as, e.g., negative association introduced in [38]; a survey can be found in [59].Sufficient for our purpose is the following notion for Bernoulli or binary random variables,i.e., random variables that only take values in { , } .6 efinition 2.1. Let γ ≥
1. We call binary random variables T , T , . . . , T N upper γ -negatively dependent if P ^ j ∈ u T j = 1 ! ≤ γ Y j ∈ u P ( T j = 1) for all u ⊆ [ N ], (10)and lower γ -negatively dependent if P ^ j ∈ u T j = 0 ! ≤ γ Y j ∈ u P ( T j = 0) for all u ⊆ [ N ]. (11)We call T , T , . . . , T N γ -negatively dependent if both conditions (10) and (11) are satisfied.If γ = 1, we usually will suppress the explicit reference to γ .A similar notion, called λ -correlation, can be found in [57]. 1-negative dependence isusually called negative orthant dependence, cf. [7].Notice that, in particular, independent binary random variables are negatively depen-dent. Furthermore, it is easily seen that for N = 2 and γ = 1 the notions of upper andlower γ -negative dependence are equivalent, cf. [40].We are interested in binary random variables T i , i = 1 , . . . , N , of the form T i = A ( X i ), where A is a Lebesgue-measurable subset of [0 , d (whose characteristic functionis denoted by A ), and X , . . . , X N are randomly chosen points in [0 , d .Panconesi and Srinivasan derived in [57] Chernoff-Hoeffding-type bounds ([10, 37]) for λ -correlated random variables. We will use the following two similar bounds of Hoeffding-and of Bernstein-type; for a proof see, e.g., [31]. Theorem 2.2.
Let γ ≥ , and let T , . . . , T N be γ –negatively dependent binary randomvariables. Put S := P Ni =1 ( T i − E [ T i ]) . We have P ( | S | ≥ t ) ≤ γ exp (cid:18) − t N (cid:19) for all t > . (12) Theorem 2.3.
Let γ ≥ , and let T , . . . , T N be γ -negatively dependent binary randomvariables. Put S := P Ni =1 ( T i − E [ T i ]) and σ := N P Ni =1 V [ T i ] . Then we have P ( | S | ≥ t ) ≤ γ exp (cid:18) − t N σ + 2 t/ (cid:19) for all t > . (13)Let us close this section by mentioning a line of research that has been initiated in[42] and that is related to the one pursued in this paper. Its main goal is to show that forimportant classes of random variables the variance of (suitable) randomized quasi-MonteCarlo estimators is never worse than the variance of the plain Monte Carlo estimator. Theproof techniques used there are based on a pairwise negative dependence condition andHoeffding’s lemma, see [42]. Further results in this direction are provided in [62, 63, 64].7 Latin Hypercube Sampling and Padding
To provide useful examples of non-trivial γ -negatively dependent random variables, weprove in this section a negative dependence result for sample points stemming from aLatin hypercube sample, which may additionally be padded by Monte Carlo. Furtherexamples are provided in the follow-up paper [64]. Definition 3.1. A Latin hypercube sample (LHS) ( X n ) Nn =1 in [0 , d is of the form X n,j = π j ( n −
1) + U n,j N , where X n,j denotes the j th coordinate of X n , π j is a permutation of 0 , . . . , N −
1, uni-formly chosen at random, and U n,j is uniformly distributed in [0 , d permutations π j and the dN random variables U n,j are mutually independent.The definition of Latin hypercube sampling presented above was introduced in [46] forthe design of computer experiments; there is an older version of Latin hypercube samplingpresented in [58], where all uniform random variables U n,j are simply replaced by the value0 . I of an integral I based on Latin hypercube sampleswith N points never leads to a variance greater than that of the corresponding estimatorbased on N − Remark 3.2.
Notice that the one-dimensional projections of Latin hypercube samplesare much more evenly distributed than the one-dimensional projections of Monte Carlopoints. This observation can be put into a quantitative statement by comparing thestar discrepancies of the former and the latter projections: For a one-dimensional Latinhypercube sample of size N the star discrepancy is at most 1 /N , while for one-dimensionalMonte Carlo samples of the same size the star discrepancy is of size 1 / √ N , cf. (8) and(9). Definition 3.3.
Let d, d ′ , d ′′ ∈ N with d = d ′ + d ′′ . Let Y = ( Y k ) k ∈ N be a (deterministic orrandomized) sequence in [0 , d ′ , and let U = ( U k ) k ∈ N be a sequence of independent uni-formly distributed random vectors in [0 , d ′′ . The d -dimensional concatenated sequence X = ( X k ) k ∈ N = ( Y k , U k ) k ∈ N is called a mixed sequence . One also says that X results from Y by padding by Monte Carlo .Padding by Monte Carlo was introduced in [61] to tackle difficult problems in particletransport theory. He suggested to use a mixed sequence resulting from padding a deter-ministic low-discrepancy sequence. Mixed sequences showed a favorable performance inseveral numerical experiments, see, e.g., [51, 52]. The latter papers also provided theo-retical results on probabilistic discrepancy estimates of mixed sequences which have beenimproved in [3, 27]. Padding by LHS (instead of by Monte Carlo) was considered earlierin [53, Example 5]. 8e define the 1-dimensional grid G N by G N := { , /N, . . . , ( N − /N, } . The following lemma on 1-dimensional LHS is the key ingredient in the proof of our mainresult on d -dimensional LHS (with or without padding by Monte Carlo), Theorem 3.6.Theorem 3.6 combined with Theorem 4.4 immediately imply the discrepancy bounds forLHS (with or without padding by Monte Carlo) in Corollary 4.5. Lemma 3.4.
Let ( X n ) Nn =1 be a LHS in [0 , . Let a, b ∈ [0 , with a ≤ b . (a) We have for all ν ∈ { , , . . . , N } that P ν ^ ℓ =1 X ℓ ∈ [ a, b ) ! ≤ ( b − a ) ν . (14)(b) Let I (0)1 := [ a, b ) , I (0)2 := [0 , b ) and I (1)1 := [ b, , I (1)2 := [0 , a ) ∪ [ b, . Then we havefor all σ ∈ { , } , k ∈ { , , . . . , N } , and ν ∈ { , , . . . , k } that P ν ^ ℓ =1 X ℓ ∈ I ( σ )1 ∧ k ^ ℓ = ν +1 X ℓ ∈ I ( σ )2 ! ≤ δλ ( I ( σ )1 ) ν λ ( I ( σ )2 ) k − ν , (15) where δ := ( if a, b ∈ G N or a = 0 ,e else . The constant δ in (15) is optimal in the following sense: for each δ < e and any σ ∈ { , } there exist N ∈ N , a, b ∈ [0 , , k ∈ { , , . . . , N } , and ν ∈ { , , . . . , k } such that (15) does not hold.Proof. Let α := ⌈ N a ⌉ , β := ⌊ N b ⌋ , and let ε a , ε b ∈ [0 ,
1) such that a = α − ε a N and b = β + ε b N . (a) We may assume ν ≥ N − ≥ β − α ≥ ν −
2, since otherwise (14) holds trivially.We first consider the case β − α ≤ N −
2. If ν points fall into [ a, b ), then one of the threedisjoint events that exactly ν points, ν − ν − α/N, β/N )occurs. Therefore P ν := P ν ^ ℓ =1 X ℓ ∈ [ a, b ) ! = ( β − α )( β − α − · · · ( β − α − ν + 1) N ( N − · · · ( N − ν + 1)+ ν ( β − α )( β − α − · · · ( β − α − ν + 2) N ( N − · · · ( N − ν + 2) ε a + ε b N − ν + 1+ ν ( ν −
1) ( β − α )( β − α − · · · ( β − α − ν + 3) N ( N − · · · ( N − ν + 3) ε a N − ν + 2 ε b N − ν + 1 .
9e have to verify that P ν is at most ( b − a ) ν = (cid:0) ( β − α + ( ε a + ε b )) /N (cid:1) ν . Since for fixedsum ε a + ε b the product ε a ε b is largest if ε a = ε b =: ε , we may confine ourselves to thelatter case. Put C := ( β − α )( β − α − · · · ( β − α − ν + 3) N ( N − · · · ( N − ν + 1)and define the function f ν by f ν ( ε ) := C (cid:2) ( β − α − ν + 2)( β − α − ν + 1) + ( β − α − ν + 2)2 νε + ν ( ν − ε (cid:3) × (cid:18) Nβ − α + 2 ε (cid:19) ν ;it suffices to show that | f ν ( ε ) | ≤ ε ∈ [0 , f ′ ν are ε = 1 and ε ≤ ν > ε = 1 if ν = 2 . Hence f ν takes itsmaximum in [0 ,
1] in 0 or in 1. Now f ν (0) = ( β − α )( β − α − · · · ( β − α − ν + 1) N ( N − · · · ( N − ν + 1) (cid:18) Nβ − α (cid:19) ν ≤ , and from( β − α − ν + 2)( β − α − ν + 1) + ( β − α − ν + 2)2 ν + ν ( ν −
1) = ( β − α + 2)( β − α + 1)and β − α + 2 ≤ N we obtain f ν (1) = ( β − α + 2)( β − α + 1)( β − α ) · · · ( β − α − ν + 3) N ( N − · · · ( N − ν + 1) (cid:18) Nβ − α + 2 (cid:19) ν ≤ . The remaining (simpler) case β − α = N − a = 0 and ν = 0, since theyare already covered by (a). Furthermore, due to (a) it suffices to show that P ( σ ) C ( k, ν ) := P k ^ ℓ = ν +1 X ℓ ∈ I ( σ )2 (cid:12)(cid:12)(cid:12) ν ^ ℓ =1 X ℓ ∈ I ( σ )1 ! ≤ δλ ( I ( σ )2 ) k − ν . Let us first consider σ = 0: If b ∈ G N , then b = β/N and P (0) C ( k, ν ) = β − νN − ν β − ( ν + 1) N − ( ν + 1) · · · β − ( k − N − ( k − ≤ (cid:18) βN (cid:19) k − ν = (cid:0) λ ( I (0)2 ) (cid:1) k − ν . If b / ∈ G N , then β ≤ N − P (0) C ( k, ν ) ≤ P k ^ ℓ = ν +1 X ℓ ∈ (cid:20) , β + 1 N (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ν ^ ℓ =1 X ℓ ∈ I (0)1 ! = β + 1 − νN − ν β + 1 − ( ν + 1) N − ( ν + 1) · · · β + 1 − ( k − N − ( k − ≤ (cid:18) βN − (cid:19) k − ν ≤ (cid:18) NN − (cid:19) k − ν b k − ν ≤ (cid:18) N − (cid:19) N − b k − ν ≤ e (cid:0) λ ( I (0)2 ) (cid:1) k − ν . σ = 1: If a, b ∈ G N , then we have a = α/N , b = β/N , and it iseasily verified that (15) holds with δ = 1.If a / ∈ G N , then α ≥ P (1) C ( k, ν ) ≤ P k ^ ℓ = ν +1 X ℓ ∈ [0 , a ) ∪ (cid:20) βN , (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ν ^ ℓ =1 X ℓ ∈ I (1)1 ! = N − β + ( α − − νN − ν N − β + ( α − − ( ν + 1) N − ( ν + 1) · · ·× N − β + ( α − − ( k − N − ( k − k − ν ) N − β + ( α − − νN − ν · · · N − β + ( α − − ( k − N − ( k −
2) 1 − ε a N − ( k − . We want to prove that the last term is less or equal than e (cid:0) ( N − β − ε b + ( α −
1) + (1 − ε a )) /N (cid:1) k − ν . Obviously, it is enough to show this for the case ε b = 1. Put C := N − β + ( α − − νN − ν · · · N − β + ( α − − ( k − N − ( k −
2) 1 N − ( k − f k,ν by f k,ν ( ε ) := C [ N − β + ( α − − ( k −
1) + ( k − ν ) ε ] (cid:18) NN − β + α − ε (cid:19) k − ν ;it suffices to show | f k,ν ( ε ) | ≤ e for all ε ∈ [0 , f ′ k,ν is at least 1, hence f k,ν takes its maximum in [0 ,
1] in 0 or 1. Now f k,ν (0) = N − β + ( α − − νN − ν · · · N − β + ( α − − ( k − N − ( k − × (cid:18) NN − β + ( α − − (cid:19) k − ν ≤ (cid:18) NN − (cid:19) k − ν ≤ e, and f k,ν (1) = N − β + ( α − − ( ν − N − ν · · · N − β + ( α − − ( k − N − ( k − × (cid:18) NN − β + ( α − (cid:19) k − ν ≤ (cid:18) NN − (cid:19) k − ν ≤ e.
11e now show that the constant δ in (15) is optimal: Let σ = 0 and β = N − α , ε a = 0, ε b ∈ (0 , k = N , and ν = 1. For this choice of parameters we get P (cid:16) X ℓ ∈ I (0)1 (cid:17) = b − a and P (0) C ( N,
1) = 1 . Therefore every δ that satisfies (15) for every choice of N and ε b has to fulfill δ ≥ (cid:18) NN − ε b (cid:19) N − ≥ (cid:18) − ε b N − (cid:19) N − ;since the last expression converges to e − ε b for N → ∞ and since we can choose ε b arbitrarily small, this implies δ ≥ e .In the case σ = 1 we can consider a corresponding example: Let α = N − β , ε a = 0, ε b ∈ (0 , k = N , and ν = 1. Again, it is easily verified that (15) cannot holdfor δ < e by choosing N sufficiently large and 1 − ε b sufficiently small.This concludes the proof of Lemma 3.4. Definition 3.5.
For d ∈ N we put C d := { [0 , a ) | a ∈ [0 , d } and D d := { B \ A | A, B ∈ C d } . Let
S ∈ {C d , D d } . We say that the random points X , . . . , X N in [0 , d are S - γ -negativelydependent if for all S ∈ S the random variables S ( X ) , . . . , S ( X N ) are γ -negatively dependent. Theorem 3.6.
Let d, d ′ , d ′′ ∈ N such that d = d ′ + d ′′ . Let ( X n ) Nn =1 be a LHS in [0 , d ′ and ( Y n ) Nn =1 be independently randomized Monte Carlo points in [0 , d ′′ . For n = 1 , . . . , N put Z n := ( X n , Y n ) . For a, b ∈ [0 , d let A := [0 , a ) , B := [0 , b ) , and D := B \ A . Thenthe random variables D ( Z ) , . . . , D ( Z N ) are γ d ′ -negatively dependent, (16) where γ d ′ := d ′ Y i =1 δ i and δ i := ( if a i , b i ∈ G N or a i = 0 ,e else . In particular, the random points ( Z n ) Nn =1 are C d -negatively as well as D d - γ d ′ -negativelydependent.Proof. Let U = ( u n ) Nn =1 be a family of uniformly distributed i.i.d. random points in [0 , d .For c ∈ { , , . . . , d } we define the random point sets b P ( c ) = ( b p n ( c )) Nn =1 by( b p n ( c )) i = ( π i ( n − u n,i N for i ≤ cu n,i for i > c, π i is a randomly chosen permutation. Here the permutations π i , i ∈ [ d ], and the u n , n ∈ [ N ], are mutually independent. Notice that b P ( c ) is an MC point set for c = 0, a c -dimensional LHS padded by MC for 1 ≤ c < d , and a d -dimensional LHS for c = d . Put γ ( c ) := c Y i =1 δ i . We first show via induction that for c = 0 , , . . . , d the random variables D ( b p ( c )) , . . . , D ( b p N ( c )) are upper γ ( c )-negatively dependent . (17)This is clearly satisfied for c = 0, since the random variables D ( b p (0)), . . . , D ( b p N (0))are even independent. Now let c ≥ c −
1. We use the shorthand b P := b P ( c ) and e P := b P ( c −
1) and correspondingnotation for the random points in both sets. We denote by P ∗ c the orthogonal projectiononto all coordinates except of the c th coordinate (i.e., for x ∈ R d we have P ∗ c ( x ) =( x , . . . , x c − , x c +1 , . . . , x d )). Note that P ∗ c ( b p j ) = P ∗ c ( e p j ) for all j ∈ [ N ]. Furthermore,we put A c := [0 , a c ), A ∗ c := P ∗ c ( A ), B c := [0 , b c ), B ∗ c := P ∗ c ( B ), and D c := B c \ A c , D ∗ c := B ∗ c \ A ∗ c . We have b p j ∈ D if and only if one of the following two disjoint eventsoccurs:1. P ∗ c ( b p j ) = P ∗ c ( e p j ) ∈ A ∗ c and b p j,c ∈ D c ,2. P ∗ c ( b p j ) = P ∗ c ( e p j ) ∈ D ∗ c and b p j,c ∈ B c .Since our random point distribution is symmetric, i.e., our random points are exchange-able, we get for k ∈ [ N ] P k ^ j =1 b p j ∈ D ! = k X ν =0 (cid:18) kν (cid:19) P k ^ j =1 b p j ∈ D ∧ ν ^ j =1 P ∗ c ( e p j ) ∈ A ∗ c ∧ k ^ j = ν +1 P ∗ c ( e p j ) ∈ D ∗ c ! = k X ν =0 (cid:18) kν (cid:19) P ν ^ j =1 P ∗ c ( e p j ) ∈ A ∗ c ∧ k ^ j = ν +1 P ∗ c ( e p j ) ∈ D ∗ c ! × P ν ^ j =1 b p j,c ∈ D c ∧ k ^ j = ν +1 b p j,c ∈ B c (cid:12)(cid:12)(cid:12) ν ^ j =1 P ∗ c ( e p j ) ∈ A ∗ c ∧ k ^ j = ν +1 P ∗ c ( e p j ) ∈ D ∗ c ! . (18)Since different components of our random point set b P are mutually independent, for fixed ν ∈ { , , . . . , k } the conditional probability in (18) is equal to P ν ^ j =1 b p j,c ∈ D c ∧ k ^ j = ν +1 b p j,c ∈ B c ! e P if we substitute all occuring points b p j by e p j , we obtain P k ^ j =1 b p j ∈ D ! ≤ δ c P k ^ j =1 e p j ∈ D ! ≤ γ ( c ) λ d ( D ) k , (19)where the first inequality follows from P ν ^ j =1 b p j,c ∈ D c ∧ k ^ j = ν +1 b p j,c ∈ B c ! ≤ δ c ( b c − a c ) ν b k − νc = δ c P ν ^ j =1 e p j,c ∈ D c ∧ k ^ j = ν +1 e p j,c ∈ B c ! which is valid due to (15) for σ = 0, and the second inequality follows from our inductionhypothesis.Now we show that 1 D ( b p ( c )) , . . . , D ( b p N ( c )) are lower γ ( c )-negatively dependent. Thisholds if and only if1 F ( b p ( c )) , . . . , F ( b p N ( c )) are upper γ ( c )-negatively dependent, (20)where F := A ∪ (cid:0) [0 , d \ B (cid:1) . We now verify (20) by induction. Again, the statement isobvious for c = 0. So let c ≥ c −
1. As before we use thenotation b P , e P , etc.. We have b p j ∈ F if and only if one of the following three disjointevents occurs:1. P ∗ c ( b p j ) = P ∗ c ( e p j ) ∈ A ∗ c and b p j,c ∈ A c ∪ [ b c , P ∗ c ( b p j ) = P ∗ c ( e p j ) ∈ D ∗ c and b p j,c ∈ [ b c , P ∗ c ( b p j ) = P ∗ c ( e p j ) ∈ [0 , d − \ B ∗ c and b p j,c ∈ [0 , k ∈ [ N ] P k ^ j =1 b p j ∈ F ! = X ≤ ν ≤ ν ≤ k (cid:18) kν , ν − ν , k − ν (cid:19) × P ν ^ j =1 P ∗ c ( e p j ) ∈ A ∗ c ∧ ν ^ j = ν +1 P ∗ c ( e p j ) ∈ D ∗ c ∧ k ^ j = ν +1 P ∗ c ( e p j ) ∈ [0 , d \ B ∗ c ! × P (cid:18) ν ^ j =1 b p j,c ∈ A c ∪ [ b c , ∧ ν ^ j = ν +1 b p j,c ∈ [ b c , ∧ k ^ j = ν +1 b p j,c ∈ [0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν ^ j =1 P ∗ c ( e p j ) ∈ A ∗ c ∧ ν ^ j = ν +1 P ∗ c ( e p j ) ∈ D ∗ c ∧ k ^ j = ν +1 P ∗ c ( e p j ) ∈ [0 , d \ B ∗ c (cid:19) . (21)14or fixed ν , ν the conditional probability appearing in the sum in (21) is equal to P ν ^ j =1 b p j,c ∈ A c ∪ [ b c , ∧ ν ^ j = ν +1 b p j,c ∈ [ b c , ! (provided that the event on which we condition occurs with positive probability). Sincethis observation and (21) hold also for e P if we substitute all occuring points b p j by e p j , theinequality P k ^ j =1 b p j ∈ F ! ≤ δ c P k ^ j =1 e p j ∈ F ! ≤ γ ( c ) λ d ( F ) k (22)follows from P (cid:18) ν ^ j =1 b p j,c ∈ A c ∪ [ b c , ∧ ν ^ j = ν +1 b p j,c ∈ [ b c , (cid:19) ≤ δ c ( a c + (1 − b c )) ν (1 − b c ) ν = δ c P ν ^ j =1 e p j,c ∈ A c ∪ [ b c , ∧ ν ^ j = ν +1 e p j,c ∈ [ b c , ! , which holds true due to (15) for σ = 1, and our induction hypothesis.This concludes the proof of Theorem 3.6.Studying the proof above it is easy to see that the following generalization of Theo-rem 3.6 is valid. Theorem 3.7.
Let a, b ∈ [0 , d and put A := [0 , a ) , B := [0 , b ) , and D := B \ A . Let ( Z n ) Nn =1 be a set of (not necessarily independent) random points in [0 , d that satisfies thefollowing two conditions: (i) Different components of the random point set are mutually independent. (ii)
For all i ∈ [ d ] and for I (0)1 ,i := [ a i , b i ) , I (0)2 ,i := [0 , b i ) and I (1)1 ,i := [ b i , , I (1)2 ,i :=[0 , a i ) ∪ [ b i , there exists a δ i > such that for all σ ∈ { , } , k ∈ { , , . . . , N } , ν ∈ { , , . . . , k } , and all J ⊆ [ N ] , J ν , J k − ν ⊆ J with | J | = k , | J ν | = ν , | J k − ν | = k − ν and J ν ∩ J k − ν = ∅ , one has P ^ ℓ ∈ J ν Z ℓ,i ∈ I ( σ )1 ,i ∧ ^ ℓ ∈ J k − ν Z ℓ,i ∈ I ( σ )2 ,i ≤ δ i λ ( I ( σ )1 ,i ) ν λ ( I ( σ )2 ,i ) k − ν . (23) Then the random variables D ( Z ) , . . . , D ( Z N ) are γ d -negatively dependent, where γ d := d Y i =1 δ i . (24)15 Probabilistic Discrepancy Bounds
Now we consider the star discrepancy D ∗ N ( X ) (as defined in the introduction) of D d - γ -negatively dependent random points X = ( X n ) Nn =1 (cf. Definition 3.5).To “discretize” the star discrepancy, we define δ –covers as in [17]: Definition 4.1.
For any δ ∈ (0 ,
1] a finite set Γ of points in [0 , d is called a δ –cover of [0 , d , if for every y ∈ [0 , d there exist x, z ∈ Γ ∪ { } such that x ≤ y ≤ z and λ d ([0 , z ]) − λ d ([0 , x ]) ≤ δ . The number N ( d, δ ) denotes the smallest cardinality of a δ –cover of [0 , d .The following theorem was stated and proved in [26]. Theorem 4.2. [26, Thm.1.15] For any d ≥ and δ ∈ (0 , we have N ( d, δ ) ≤ d d d d ! ( δ − + 1) d . Notice that due to Stirling’s formula we have d d /d ! ≤ e d / √ πd . Furthermore, it iseasy to verify that in the case d = 1 the identity N (1 , δ ) = ⌈ δ − ⌉ (25)is established with the help of the δ -Cover Γ := { / ⌈ δ − ⌉ , / ⌈ δ − ⌉ , . . . , } .With the help of δ -covers the star discrepancy can be approximated in the followingsense. Lemma 4.3.
Let P ⊂ [0 , d be an N -point set, δ > , and Γ be a δ -cover of [0 , d . Then D ∗ N ( P ) ≤ max x ∈ Γ D N ( P, [0 , x )) + δ. The proof of Lemma 4.3 is straightforward, cf., e.g., [17, Lemma 3.1].We are ready to state and prove our main result, a general probabilistic discrepancybound for D d - γ -negatively dependent random points. Theorem 4.4.
Let d, N ∈ N and ρ ∈ [0 , ∞ ) . Let X = ( X n ) n ∈ [ N ] be a set of D d - e ρd -negatively dependent random points in [0 , d such that each X n is uniformly distributed.Then for every c > D ∗ N ( X ) ≤ c r dN (26) holds with probability at least − e − (1 . · c − . − ρ ) · d , implying that for every q ∈ (0 , D ∗ N ( X ) ≤ . s . ρ + ln (cid:0) (1 − q ) − (cid:1) d r dN (27) holds with probability at least q . X , . . . , X n in Theorem 4.4 wehave ρ = 0 and our bound (27) improves on the main result of [4], that is, Theorem 1in that paper. This is further illustrated by the next corollary. It follows immediatelyfrom Theorem 4.4, since we may take ρ = 0 for Monte Carlo points and ρ = 1 for Latinhypercube samples (padded by Monte Carlo or not), see Theorem 3.6. Notice in particularthe quantitative improvements in the constant in (28) compared to the constant 9 . Corollary 4.5.
Let d, N ∈ N and let X = ( X n ) n ∈ [ N ] be a random point set in [0 , d .1. If X is a Monte Carlo point set, then there exists a realization P ⊂ [0 , d of X such that D ∗ N ( P ) ≤ . · r dN . (28) The probability that X = ( X n ) n ∈ [ N ] satisfies D ∗ N ( X ) ≤ · r dN and D ∗ N ( X ) ≤ · r dN (29) is at least . and . , respectively.2. If X is a Latin hypercube sample or a Latin hypercube sample padded by MonteCarlo, then there exists a realization P ⊂ [0 , d of X such that D ∗ N ( P ) ≤ . · r dN . (30) The probability that X = ( X n ) n ∈ [ N ] satisfies D ∗ N ( X ) ≤ · r dN and D ∗ N ( X ) ≤ · r dN (31) is at least . and . , respectively.Estimate (28) implies N ∗ ( ε, d ) ≤ (cid:6) . · dε − (cid:7) for all ε > . (32)We may use Theorem 4.4 to prove similar corollaries for other random samples thanMonte Carlo point sets or (padded) Latin hypercube samples; cf. also [64]. Remark 4.6.
We already mentioned in the introduction that the probabilistic discrep-ancy bound (26) is sharp for Monte Carlo point sets. As shown in [14], the same is thecase for Latin hypercube samples: There exists a constant
K > d ≥ N ≥ d the expected star discrepancy of a Latin hypercube sample X is boundedfrom below by E [ D ∗ N ( X )] ≥ K r dN (33)and additionally we have the probabilistic discrepancy bound P D ∗ N ( X ) < K r dN ! ≤ exp( − Ω( d )) , (34)see [14, Theorem 2]. Nevertheless, recall that Latin hypercube samples have a big advan-tage over Monte Carlo samples, namely that their one-dimensional projections are moreevenly distributed, cf. Remark 3.2. Proof of Theorem 4.4.
We adapt the line of proof of [1, Theorem 1] and employ thebounds on the size of minimal δ -covers from Theorem 4.2, dyadic chaining and largedeviation bounds of Hoeffding- and Bernstein-type (but this time the ones for sums of γ -negatively dependent random variables in Theorem 2.2 and 2.3). For a, b ∈ [0 , d with a ≤ b we write ∆( a, b ) := [0 , b ) \ [0 , a ) . We start by putting µ := 13 and c µ := ∞ X ℓ =0 (cid:18)r µ + 12 µ (cid:19) ℓ = 11 − q µ +12 µ . Let c > c > c will be determined laterin the proof. Let K be the smallest natural number that satisfies K ≥ µ and1 √ K K ≤ c c c µ r dN . (35)We choose for each µ ≤ k ≤ K a 2 − k -cover Γ k of minimum size. Furthermore, we putΓ µ − := { } .Let P be an arbitrary realization of X . Due to Lemma 4.3 we can choose for each testbox [0 , y ) ⊆ [0 , d an a K ∈ Γ K ∪ { } such that D N ( P, [0 , y )) ≤ D N ( P, [0 , a K )) + 2 − K . If K > µ , we additionally choose for k = K − , . . . , µ points a k = a k ( a k +1 ) ∈ Γ k ∪ { } recursively, depending only on the previously chosen point a k +1 , such that a k ≤ a k +1 and λ d (∆( a k , a k +1 )) ≤ − k . (36)Finally, we put a µ − = a µ − ( a µ ) := 0 and get ∆( a µ − , a µ ) = [0 , a µ ). Notice that [0 , a K ) = ∪ Kk = µ ∆( a k − , a k ). Hence we have D N ( P, [0 , y )) ≤ K X k = µ D N ( P, ∆( a k − , a k )) + 2 − K . (37)18et us now for µ ≤ k ≤ K define the sets A k by A k := { ∆( a k − ( b ) , b ) | b ∈ Γ k } . Clearly, |A k | ≤ | Γ k | . Moreover, we define events E k by E µ := ( max ∆ µ ∈A µ D N ( X, ∆ µ ) ≤ c r dN ) for k = µ and E k := ( max ∆ k ∈A k D N ( X, ∆ k ) ≤ c c r k − k − r dN ) for µ + 1 ≤ k ≤ K . We put E := K \ k = µ E k . Let us assume that the event E has occured. Then, due to (37) and (35), the realization P of X satisfies for an arbitrary test box [0 , y ) ⊆ [0 , d D N ( P, [0 , y )) ≤ c c K X k = µ +1 r k − k − ! r dN + 2 − K ≤ c c r µ µ K − µ − X j =0 s µ + j j µ + c µ r K K !! r dN ≤ c c r µ µ K − µ − X j =0 (cid:18)r µ + 12 µ (cid:19) j + s µ + ( K − µ ) µ K − µ c µ !! r dN ≤ c c r µ µ K − µ − X j =0 (cid:18)r µ + 12 µ (cid:19) j + (cid:18)r µ + 12 µ (cid:19) K − µ c µ !! r dN = c (cid:18) c c µ r µ µ (cid:19) r dN . Let us now derive a lower bound for the probability P ( E ). For k = µ we may useTheorem 2.2 and a simple union bound to obtain P ( E cµ ) ≤ | Γ µ | e ρd e − c d . (38)For µ + 1 ≤ k ≤ K we first use the definition of K , cf. (35), to get the estimate p ( k − − k ≤ p ( K − K − − k < c c c µ r dN ! − − k . P ( E ck ) ≤ | Γ k | e ρd exp (cid:18) − c c ( k − d / c µ ) (cid:19) . (39)We have P ( E ) = 1 − P ( E c ) ≥ − K X k = µ P ( E ck ) . We put τ µ := c / c µ )and σ := µ − ln(2(2 µ + 1)) −
1. From now on we assume that c ≥ p ( µ + ρ − σ ) / . Let us first consider the case d = 1. Due to (25) we have | Γ k | = 2 k for k = µ, . . . , K .Hence we get from (38) and (39) for k = µ P ( E cµ ) ≤ exp (cid:0) − c + ρ + ( µ + 1) ln(2) (cid:1) and for µ + 1 ≤ k ≤ K P ( E ck ) ≤ exp (cid:0) − (2 c τ µ − ln(2))( k −
1) + ρ + 2 ln(2) (cid:1) . Hence we obtain P ( E ) ≥ − e − (2 c − ρ − ( µ +1) ln(2)) + e ρ +2 ln(2) K X k = µ +1 e − ( c τ µ − ln(2) ) ( k − ! =1 − e − (2 c − ρ − ( µ +1) ln(2)) + e ρ +2 ln(2) e − ( c τ µ − ln(2) ) µ K − µ − X j =0 e − ( c τ µ − ln(2) ) j ! =1 − e − (2 c − ρ − ( µ +1) ln(2)) e − c ( µτ µ − − e − ( c τ µ − ln(2) ) ! ≥ − e − (2 c − ρ − ( µ +1) ln(2)) (cid:18) e − ( µ + ρ − σ )( µτ µ − − e − ( µ + ρ − σ ) τ µ +ln(2) (cid:19) . Choosing τ µ = 0 , ρ ≥
01 + e − ( µ + ρ − σ )( µτ µ − − e − ( µ + ρ − σ ) τ µ +ln(2) < e. Since e − (2 c − ρ − ( µ +1) ln(2)) < e e − (2 c − ρ − µ + σ ) , c = p ( µ + ρ − σ ) / P ( E ) > d ≥
2. Due to Theorem 4.2 we have | Γ k | ≤ √ πd (2 e ) d (2 k + 1) d for k = µ, . . . , K .Hence we get from (38) and (39) for k = µ P ( E cµ ) ≤ r πd e − (2 c − µ − ρ + σ ) d and for µ + 1 ≤ k ≤ K P ( E ck ) ≤ r πd (2 e ) d kd (1 + 2 − k ) d e ρd exp (cid:18) − c c ( k − d / c µ ) (cid:19) ≤ r πd e (1+2 ln(2)+ ϑ + ρ ) d exp (cid:0) − (2 c τ µ − ln(2))( k − d (cid:1) , where ϑ := ln(1 + 2 − µ − ). Put ζ := 2 ln(2) + ϑ . Then we obtain P ( E ) ≥ − r πd (cid:18) e − (2 c − µ − ρ + σ ) d + e (1+ ζ + ρ ) d e − ( c τ µ − ln(2) ) µd K − µ − X j =0 e − ( c τ µ − ln(2) ) jd (cid:19) ≥ − r πd e − (2 c − µ − ρ + σ ) d e − (2 c ( µτ µ − − ln(2)) µ − − ζ − σ ) d − e − (2 c τ µ − ln(2)) d ! ≥ − r πd e − (2 c − µ − ρ + σ ) d (cid:18) e − (( µ + ρ − σ )( µτ µ − − ln(2)) µ − − ζ − σ ) d − e − (( µ + ρ − σ ) τ µ − ln(2)) d (cid:19) . Choosing τ µ = 0 , d = 1, we get1 + e − (( µ + ρ − σ )( µτ µ − − ln(2)) µ − − ζ − σ ) d − e − (( µ + ρ − σ ) τ µ − ln(2)) d < r πd d = 2 and – since the left hand side of inequality (40)is monotonic decreasing in d , while the right hand side is monotonic increasing – holdstherefore for all d ≥
2. Hence the choice c = p ( µ + ρ − σ ) / P ( E ) > Acknowledgment
The authors thank Marcin Wnuk and two anonymous referees for valuable comments.Part of the work of Michael Gnewuch was done while he was a research fellow and avisitor at the School of Mathematics and Statistics of the University of New South Wales21n Sydney and a “Chercheur Invit´e” at the Laboratoire d’Informatique (LIX) of ´EcolePolytechnique in Paris, France. He acknowledges support from the Australian ResearchCouncil ARC and thanks his hosts Josef Dick, Frances Y. Kuo, Ian H. Sloan, and BenjaminDoerr for their hospitality.Another part of his work was done while he visited special semesters and programs atthe Radon Institute for Computational and Applied Mathematics (RICAM) in Linz, Aus-tria, the Institute of Computational and Experimental Mathematics (ICERM) of BrownUniversity in Providence, USA, and the Erwin Schr¨odinger Institute (ESI) in Vienna,Austria.
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