Detailed proofs of Propositions 3.1 and 3.2

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1 Detailed proofs of Propositios 3. ad 3. Proof of Propositio 3. NB: itegratio sets are geerally omitted for itegrals defied over a uit hypercube [0, s with ay s d. We first give four lemmas. The proof of Propositio 3. is preseted i Sectio... Techical lemmas Let X ad X two distict poits of a Lati hypercube of size i [0, d. For ay fuctio f defied o [0, d, cosider Y,LHS fx ad Y,LHS fx. I Theorem i Stei 987, Stei gives the followig result Theorem. If f is a square itegrable fuctio the as teds to +, we have CovY,LHS, Y,LHS d σi + o i with σ i Var [ [Y X i, i,..., d, Y fx, X uiformly distributed o [0, d. I this subsectio, we prove a aalogous result with more geeral settigs ad without the asymptotic assumptio o see Lemma below. Notatio ad defiitios hypercubes of side /, { Q s Q [0, s Q For s ad i N, defie the partitio of [0, s i elemetary s { [α i, β i, α i i 0,,..., } }, β i α i +. For ay square itegrable fuctio g defied o [0, s, s d, defie the sequece with geeral term u g gxdx s, N. 3 Q Q Q s Outlie The first lemma is the aalogous result for Lebesgue itegrability of a result give i quatio A. i Stei 987 for Riema itegrability. The secod lemma gives a importat iequality which allows to work without asymptotic assumptio o. The third oe cosists i simplifyig itegrals uder LHS usig the ANOVA decompositio. Lemma provides the expected iequalities. Lemma. Let s N. If g is a cotiuous fuctio o [0, s, the sequece u g coverges to g xdx as teds to +.

2 Proof. Notig that u g g xdx where We ca rewrite g as g x x [0, s, g x We ow prove that g g + Q Q s gydy s Q x, 5 Q s Q x gydy, where Q x is the set Q i Q s cotaiig x. 0. Ideed, as g is cotiuous o a compact, it is uiformly cotiuous o this compact. Thus g x gx s gy gx Q xdy coverges uiformly to zero. From this latter covergece, we also deduce that for large eough, g g +. We the coclude by applyig the domiated covergece theorem, as [0, s g x + dx <. Lemma. The sequece u g is domiated by g xdx. Proof. Let N, the result is proved by showig that the sequece of geeral term v k g u k g is icreasig. I this case, by Lemma, we have lim v k g g xdx, ad sice v k is icreasig, all the terms of this sequece are domiated by g xdx, hece v 0 g u g g xdx. To prove that the sequece v k g is icreasig, ote that v k+ g k+ s k s Q Q s k+ Q Q s k s gxdx Q P PQ, k+ where PQ, k+ Q k+ Q. The by Jese iequality, we have s P PQ, k+ ad we coclude that v k+ g v k g. For 0 x, x defie P P gxdx 6 gxdx gxdx 7 Q r x, x { if x x 0 otherwise, 8 where is the floor fuctio. We ow ed with the followig result Lemma 3. Let v be a subset of {,..., d}, we have fx fx r x i, x i dx dx f w x w f w x w r x i, x i dx dx. 9 i v w v i v

3 Proof. By the ANOVA decompositio, we have fx fx r x i, x i dx dx f w x w f w x w r x i, x i dx dx. i v w {,..,d} w {,..,d} i v 0 The ote that a certai umber of terms i the member o the right-had side vaishe. If w v c w v c the suppose without ay loss of geerality that there exists k w \v. We have f w x w f w x w r x i, x i dx dx f w x w dx k f w x w r x i, x i dx {k} cdx i v }{{} i v I ad ote that, by a basic property of the ANOVA decompositio, I 0. If w v c w v c ad w w, the suppose without ay loss of geerality that there exists k w \ w. I this case, we have r x i, x i dx dx f w x w f w x w i v f w x w r x k, x k dx k dx k f w x w } {{ } I ad ote that by the defiitio of r, we have f w x w r x k, x k dx k dx k ad thus I 0. The coclusio of the lemma follows. i v\{k} r x i, x i dx {k} cdx {k} c f w x w dx k 3 Let u be a o-empty subset of {,... d} ad cosider X LH, d ad Z LH, d. For ay fuctio f defied o [0, d, cosider We have the followig result Y,LHS fx ad Yu,LHS fx u, Z uc. Lemma. If f is a square itegrable fuctio the we have w u w odd Proof. Recall that for 0 x, x, σ w w CovY,LHS, Y,LHS u r x, x w u w eve { if x x 0 otherwise, σ w w. 5 where is the floor fuctio. For x x,..., x d i [0, d, defie x v x i,..., x i v where v {i,..., i v }. Due to the oit desity of X u, X u uder LHS see McKay et al. 979 or Stei 987 ad by Lemma 3, we have CovY,LHS, Y,LHS u + fxdx u fx fx r x i, x i dx dx 3 i u 6

4 u v fx fx r x i, x i dx dx v u i v u v u v f w x w f w x w r x i, x i dx dx w v i v u v u v w v f w x w f w x w r x i, x i dx w dx w. 7 w v i w The ote that for ay fuctio of w deoted by Aw, we have v u v w Aw v v w Aw w v v u w v u w u w k+ w w Aw k w u k0 u w w Aw 8 w u Hece, we deduce that CovY,LHS, Y,LHS u w u w w w Fially by the defiitio of r, we have 0 f w x w f w x w r x i, x i dx w dx w i w ad by Lemma, this gives 0 f w x w f w x w i w The latter iequalities ad 9 lead to Lemma. f w x w f w x w i w r x i, x i dx w dx w. Q Q w Remark. Note that if u {,..., d}, the resultig iequalities are w w odd σ w w CovY,LHS, Y,LHS w w eve 9 f w x w dx w 0 Q r x i, x i dx w dx w σ w w. σ w w. This cosists of a o-asymptotic equivalet of the theorem due to Stei preseted at the begiig of this sectio.. Proof of Propositio 3. i This is a cosequece of the strog law of large umbers for LHS give i Theorem 3 i Loh 996.

5 ii The proof cosists i traslatig the origial proof give for simple radom samplig LHS see Propositio. i Jao et al. 03 for LHS. Cocerig S u,, it is easy to show that where V V ad V equal to Y,LHS [Y Y,LHS u S LHS u, ΦV 3 [Y, Y,LHS [Y, Y,LHS u [Y, Y,LHS [Y T ad with Φx, y, z, t x yz t y. The we deduce from Theorem i Loh 996 that V µ L N 0, Γ where µ τ u, 0, 0, σ T ad Γ is the covariace matrix of R V A see details i q. 3 i Loh 996 defied by i {,..., }, A i is the additive part see i the mai documet of V i. 5 Thus the Delta method see Theorem 3. i Va der Vaart 998 gives SLHS u, S u L N 0, g T Γg 6 where g Φµ. Developig the term g T Γg does ot seem to provide ay useful iformatio. However, deotig σlhs this term, ad σ IID the aalogous quatity i the CLT for simple radom samplig, we ca show that σlhs σ IID. Ideed we first ote that, for simple radom samplig, the variace give i Jao et al. 03 reads σiid Var[ V S u V ad for LHS, it is easy to show that Hece σ LHS Var[ R S u R σ IID σ LHS + Var[ A S u A ad the coclusio of ii for see Proof of 0 i Jao et al. 03 for details. iii First we have S LHS u, σ 7 σ. 8 σ 9 follows. Cocerig ŜLHS u,, the proof follows the same lies,lhs [ τ u, [Y + τ u [ Y,LHS Yu,LHS l l [ Y,LHS Yu l,lhs CovY,LHS, Yu,LHS + [Y 30 5

6 ad thaks to Lemma, this gives with Cocerig σ,lhs, we have ad otig that we coclude that with As for τ [,LHS u,,lhs [ σ u, ad σ,lhs Y,LHS + Y,LHS u,lhs [ τ u, τ u + B, 3 τ u B, 0. 3 [ Y,LHS [Y [ Y,LHS Y l,lhs l l CovY,LHS, Y,LHS + [Y CovY,LHS, Y,LHS Cov Y,LHS, Y,LHS {,...,d}, we have [ Y,LHS + Y,LHS u +,LHS [ σ 33 3 σ + B, 35 σ B, l l [Y + τ u + [Y + [Y + σ + τ u + [Y + The it is easy to coclude that [ Y,LHS + Yu,LHS Y l,lhs + Yu l,lhs CovY,LHS, Y,LHS + CovY,LHS, Y,LHS CovY,LHS, Y,LHS + CovY,LHS, Y,LHS u,lhs [ τ u,,lhs [ σ u. 37 τ u + B,3 38 σ + B,3 39 with σ + τ u B, Proof of Propositio 3. We first give three lemmas. The proof of Propositio 3. is preseted i Sectio.. 6

7 . Techical lemmas Lemma 5. Let d N, if d the + d d +. Proof. If d, the result is obvious. Otherwise, for ay x > 0, cosider the fuctio g d defied by g d x + d d +. x x We show that a if there exists x 0 > 0 such that g d x 0 0 the for all x x 0, g d x 0 b g d d / 0 ad the coclusio follows. Cocerig a ote that g d x + d x + Ox d x x x + Ox 3 ad the that g d is egative as x teds to +. Moreover for ay d >, g d is first decreasig ad the icreasig. Ideed, we have g dx d + d + d + x x x ad we deduce that g d x 0 0 with x 0 d+ d /d > 0 5 ad is egative o the left-had side ad positive o the right-had side. The coclusio of a follows. Cocerig b, it is easy to check that it is true for d ad, ad for d 3 we have d g d d k d k d d d k0 d k d d + 3 k d k3 d 3 + d k3 k! d k d + 3 d + 3 3d d 3 + d k3 d 3 + d k3 d + k 3d 3 d k d k d k d 3 k d k d 3 + d 3 6 7

8 ad the coclusio follows. Let u be a o-empty subset of {,... d} ad cosider X followig result LH, d. We have the Lemma 6. If f is a square itegrable fuctio, we have [ fx fx u, X u c [Y + τ u + B u, 7 where [Y v u c v odd v B u, [Y v u c v eve v. 8 Proof. First due to the oit desity of X u, X uc uder LHS see McKay et al. 979 or Stei 987 we have [ fx fx u, X u c fx, x fx, x d u d u v fx, x fx, x r x i, x i dx dx v u c i v }{{} Ix i u c r x i, x i dxdx dx We ow deote f x : y fx, y ad the by 7 ad 8 we have Ix v f x x f x x r x i, x i dx dx v u c i v v w v f x,w x w f x,w x w r x i, x i dx w dx w v u c w v i w d u w w f x,w x w f x,w x w r x i, x i dx w dx w. 50 i w dx. 9 Hece by we have for all w, 0 f x,w x w f x,w x w i w r x i, x i dx w dx w f x,w x w dx w w f x x dx w 5 ad ote that f x, x f x, x i r x i, x i dx dx f x,. 5 Fially, ote that ad f x, dx τ u + [Y 53 f xx dx dx [ Y 5 8

9 ad coclude that [ fx fx u, X u c d u Ixdx τ u + [Y + B u, 55 with [Y v u c v odd v B u, [Y v u c v eve v. 56 Lemma 7. The iequalities i quatio 8 imply that Proof. By 8, we have w B u w, d u + d u + + [ Y. 57 w B u w, [ Y w w v u w c [ Y w + d u w [ Y [ + d u w [ Y [ + d u + d u + w d u 58 ad the coclusio follows by applyig twice Lemma 5.. Proof of Propositio 3. i The proof is divided ito two parts. I the first oe, we oly cosider cotiuous fuctios, ad i the secod oe, we exted the result to the larger class of fuctios such that f is itegrable. First part: Cosistecy is obvious as i Propositio 3., except for the term So deote Z the Lati hypercube defied by Y,RLHS Y,RLHS u. 59 Z Z + U 60 9

10 where the U s are idepedet radom vectors uiformly distributed i [0, [ d idepedet from all the permutatios ad shifts i the defiitio of Z, ad is the floor fuctio. We ca write Y,RLHS Yu,RLHS fx u, Z u cfx u, Z uc as fx u, Z u cfx u, Z u c + fx u, Z u c fx u, Z u c fx u, Z u c. 6 The first term o the right-had side is a estimator as described i Sectio 3.. i the mai documet sice by the costructio proposed i q. 60, Z.. ad Z.. are two idepedet LHS; so Propositio 3. states it coverges to [Y + τ u almost surely. The secod term o the right-had side coverges to 0 sice, as f is bouded, by cotiuity o a compact it is bouded by sup f fx u, Z u c fx u, Z u c 6 ad by uiform cotiuity of f due to Heie-Cator theorem this quatity teds to 0 as teds to +. Thus the sum i the right-had side, i.e. Y,RLHS Yu,RLHS, coverges to [Y + τ u almost surely. Secod part: Sice the space of cotiuous fuctios o [0, d deoted C [0, d is dese i L [0, d, let f m m N be a sequece i C [0, d such that [ f m X fx coverges to 0 as m teds to +, where X is uiformly distributed o [0, d. Now let ε > 0 ad M Mε N such that [ fm X fx < ε 65 [f X. 63 We ca write Y,RLHS Y,RLHS u as fx u, Z u cfx u, Z u c + fx u, Z u c f M X u, Z u c f MX u, Z u c + fx u, Z u c fx u, Z u c f MX u, Z u c + fx u, Z u c f M X u, Z u c fx u, Z u c. 6 As oted i the proof of i i Propositio 3., the first term o the right-had side of 6 coverges to τ u + [Y almost surely as teds to + i.e. P ε > 0, N N, > N, Sice f M is uiformly cotiuous o [0, d, we have that fx u, Z u cfx u, Z u c τ u [Y ε <. 65 A sup fm X u, Z u c f MX u, Z u c 66 0

11 coverges almost surely to 0 as teds to +. Moreover, sice f is itegrable, we have that fx u coverges to [ Y as teds to +. Hece P ε > 0, N N, > N, P. fx u, Z u c f M X u, Z u c f MX u, Z u c ε < ε > 0, N N, > N, A fx u, Z u c < ε For the third ad the fourth terms o the right-had side of 6, we apply twice the same proof. First the Cauchy-Schwartz iequality gives P ε > 0, N 3 N, > N 3, P ε > 0, N 3 N, > N 3, fx u, Z u c fx u, Z u c f MX u, Z u c ε < / f X u, Z u c fx u, Z u c f MX u, Z 67 u c / ε < 68 The ote that f X u, Z u c ad fx u, Z u c f MX u, Z u c coverge almost surely to [Y ad [f M X fx where X is uiformly distributed o [0, d respectively. Ad deduce that there exists N N such that for all > N, we have f X u, Z u c < [Y ad fx u, Z u c f MX u, Z u c < [fm X fx almost surely. As a cosequece, deduce from q. 63 that P ε > 0, N 3 N, > N 3, fx u, Z u c fx u, Z u c f MX u, Z u c ε < P ε > 0, N 3 > N, > N 3, ε 65 < ε Fially, qs gives ad we have the coclusio. P ε > 0, N N, > N, ii As i i, the oly term to treat is Y,RLHS Y,RLHS u 69 < ε 70 Y,RLHS Y,RLHS u, 7 so asymptotic ormality is show i the same way by usig the decompositio i 6. We always obtai the sum of a term already cosidered i Sectio 3 i the mai documet which coverges i law to a ormal distributio ad a term which coverges to 0 i probability, ad RLHS the coclusio follows from Slutsky s lemma. We oly detail the proof for S u,, it is exactly.

12 the same for ŜRLHS u,. So ote that followig the proof of ii i Propositio 3. ad the otatio above, it is sufficiet to show that to prove the asymptotic ormality of fx u, Z u c [Y fx u, Z u c fx u, Z u c P 0 7 S RLHS u,. So cosider ε, η > 0 ad prove that there exists N N such that for all > N, the quatity P P fx u [Y fx u, Z u c fx u, Z u c > ε 73 is less tha η. First as f 6 is itegrable, there exists a costat K > 0 such that P fx u, Z u c > K < η/. Hece P P fx u, Z u c [Y fx u, Z u c fx u, Z u c > ε fx u, Z uc K + P K + [Y < P fx u, Z u c [Y fx u, Z u c fx u, Z u > ε fx u, Z uc > K fx u, Z u c fx u, Z u c > ε + η. 7 Now ote that the space of cotiuous fuctios o [0, d, deoted by C[0, d, is dese i L 6 [0, d ad let f m m N be a sequece i C[0, d such that [ f m X fx 6 coverges to 0 as m teds to + where X is uiformly distributed o [0, d. It is easy to ote that there exists M M such that P f M X fx > / < η/. Thus we get from q. 7 that P < i K + [Y P + f M X u, Z u c fx u, Z u c Ai + η fm X u, Z u c f MX u, Z u c + f MX u, Z u c fx u, Z u c 75 where A f M X u, Z u c fx u, Z u c > fm X u, Z u c fx u, Z u c > A f M X u, Z u c fx u, Z u c > fm X u, Z u c fx u, Z u c < ad A 3 ad A are the complemetary evets of A ad A, respectively. So we deduce K + [Y P < P fm X u, Z u c f MX u, Z u c + f MX u, Z u c fx u, Z u c + f M X u, Z u c fx u, Z u c A3 + PA + PA + PA + η K + [Y < P + f M X u, Z u c f MX u, Z u c > ε + η. 78

13 Now by a other desity argumet, ote that there exists a sequece of Lipschitz cotiuous fuctios with costat, deoted f M,q q N, such that sup [0, d f M,q x f M x coverges to 0 as q teds to +. The there exists Q Q N such that sup [0, d f M,Q x f M x < / ad deduce that K + [Y P < P + fm,q X u, Z u c f M,QX u, Z u c + f M,QX u, Z u c f MX u, Z u c + f M,Q X u, Z u c f MX u, Z u c > ε + η < 5K + [Y P > ε + η. 79 ad the coclusio follows. with ad iii First ote that sice X LH, d ad Z, Z RLH, d, we have Y,RLHS fx, Z,RLHS uc ad Yu X π U,π,..., π d U d,π fx u, Z uc 80 d Z π U,π,..., π d U d,πd Z π U,π,..., π d U d,π d where the π i s, the π is, the π i s, the U i, s ad the U i,s are idepedet radom variables uiformly distributed o Π see Defiitio i the mai documet, Π, Π, [0, ad [0,, respectively. Moreover ote that if for a idex i i {,..., d}, we have π i π i the Z i Z i Z i ; ad if π i π i the U i,πi ad U i,π i are idepedet ad therefore Z i ad are two distict poits of a Lati hypercube of size i [0,. For {,..., }, deote by e the set of itegers i u c such that π i π i. Thus we have [ Y,RLHS Yu,RLHS equal to d u π u {,...,} u f π u c π {,...,} d u u c {,...,} d u {ew} f..., π i u i,..., π k u k,..., π l u l,... }{{ }}{{ }}{{ } i u k u c w l u c w c..., π i u i,..., π k u k,... }{{ }}{{ } i u k u c du du u w c 8 where for all i u e, π i π i ad u i u i. Ad otig that d u w d {ew} f..., π i u i,..., π k u k,... π u π u c π u c }{{}}{{ } i u k u c {,...,} u {,...,} d u {,...,} d u 3

14 f..., π i u i,..., π k u k,..., π l u l,... }{{ }}{{ }}{{ } i u k u c w l u c w c is equal to [ fx fx u w, X u w c, Lemma 6 gives [ Y,RLHS Yu,RLHS du du u w c 85 w [Y + τ u w + B u w,. 86 By Lemmas 5 ad 7, ad otig that [Y + τ u w [Y, we obtai [ Y,RLHS Yu,RLHS [Y + τ u + B u, 87 where B u, d u + + d u + [Y. 88 Followig the same proof, it is easy to show that for l, we have [ Y,RLHS Yu l,rlhs [Y + B, 89 where Thus otig that we coclude that [ τ,rlhs u, B, d + + [Y,RLHS Y,RLHS,RLHS [ τ u, d + u τ u τ u + B, + B u, [Y. 90 [Y,RLHS Y,RLHS u 9 where the biases are O as specified i Propositio 3.. Cocerig σ,rlhs, ote that σ,rlhs σ,lhs ad the coclusio follows from iii i Propositio 3.. Cocerig τ,rlhs ad σ,rlhs, we have [ [ Y,RLHS + Y,RLHS u + Y,RLHS +Y,RLHS u l l equal to u, [ Y,RLHS + Yu,RLHS Y l,rlhs + Yu l,rlhs [ Y,RLHS +[Y,RLHS Yu,RLHS + [Y,RLHS Y,RLHS +[Y,RLHS Yu,RLHS. The usig otatio itroduced i Sectio 3 i the mai documet, ote that [ Y,RLHS Y,RLHS [ Y,LHS Y,LHS Cov Y,LHS, Y,LHS +[Y Cov Y,LHS, Y,LHS {,...,d} +[Y ad by 88, 90 ad Lemma, we deduce [ Y,RLHS + Yu,RLHS τ u + [Y + B, + B, 9

15 where B, σ ad B, is specified i 90. The it is easy to coclude that 93,RLHS [ τ u,,rlhs [ σ τ u τ u + B, + B, + B u, 9 σ τ u + B, + B, 95 where the biases are O as specified i Propositio 3.. Refereces Jao, A., Klei, T., Lagoux, A., Nodet, M., ad Prieur, C. 03. Asymptotic ormality ad efficiecy of two sobol idex estimators. SAIM: Probability ad Statistics to appear. Loh, W. L O lati hypercube samplig. The Aals of Statistics, 5: McKay, M. D., Coover, W. J., ad Beckma, R. J A compariso of three methods for selectig values of iput variables i the aalysis of output from a computer code. Techometrics, :39 5. Stei, M Large sample properties of simulatios usig lati hypercube samplig. Techometrics, 9:3 5. Va der Vaart, A. W Asymptotics Statistics. Cambridge Uiversity Press. 5