Normal Approximation under Local Dependence. By Louis H.Y. Chen and Qi-Man Shao. National University of Singapore and University of Oregon

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1 The Annals of Probablty 2004, Vol. 00, No. 00, Normal Approxmaton under Local Dependence By Lous H.Y. Chen and Q-Man Shao Natonal Unversty of Sngapore and Unversty of Oregon We establsh both unform and non-unform error bounds of the Berry-Esseen type n normal approxmaton under local dependence. These results are of an order close to the best possble f not best possble. They are more general or sharper than many exstng ones n the lterature. The proofs couple Sten s method wth the concentraton nequalty approach. Accepted for publcaton n Annals of Probablty Revsed: August 16, 2003 (Frst verson: May 3, 2002) Receved AMS 1991 subject classfcatons. Prmary 60F05; secondary 60G60. Key words and phrases. Sten s method, normal approxmaton, local dependence, concentraton nequalty, unform Berry-Esseen bound, non-unform Berry-Esseen bound, random feld. Research s partally supported by grant R at the Natonal Unversty of Sngapore Research s partally supported by NSF grant DMS , and grants R and R at the Natonal Unversty of Sngapore. 1

2 2 L.H.Y Chen and Q.-M. Shao 1. Introducton Snce the work of Berry and Esseen n the 1940 s, much has been done n the normal approxmaton for ndependent random varables. The standard tool has been the Fourer analytc method as developed by Esseen (1945). However, wthout ndependence, the Fourer analytc method becomes dffcult to apply and bounds on the accuracy of approxmaton correspondently dffcult to fnd. In such stuatons, a method of Sten (1972) provdes a much more vable alternatve to the Fourer analytc method. Correspondng to calculatng a Fourer transform and applyng the nverson formula, t nvolves dervng a drect dentty and solvng a dfferental equaton. As dependence s the rule rather than the excepton n applcatons, Sten s method has become ncreasngly useful and mportant. A crucal step n the Fourer analytc method for normal approxmaton s the use of a smoothng nequalty orgnally due to Esseen (1945). The smoothng nequalty s used to overcome the dffculty resultng from the non-smoothness of the ndcator functon whose expectaton s the dstrbuton functon. There s a correspondence of ths n Sten s method, whch s called the concentraton nequalty. It s orgnally due to Sten. Its smplest form s used n Ho and Chen (1978). More elaborate versons are proved n Chen (1986, 1998) and Chen and Shao (2001). In Chen and Shao (2001), t s developed also for obtanng non-unform error bounds. Ths paper s concerned wth normal approxmaton under local dependence usng Sten s method. Local dependence roughly means that certan subsets of the random varables are ndependent of those outsde ther respectve neghborhoods. No structure on the ndex set s assumed. Both unform and non-unform error bounds of the Berry-Esseen type are obtaned and shown to be more general or sharper than many exstng results n the lterature. These nclude those of Sher-

3 Normal Approxmaton under Local Dependence 3 gn (1979), Prakasa Rao (1981), Bald and Rnott (1989), Bald, Rnott and Sten (1989), Rnott (1994), and Dembo and Rnott (1996). The approach used n the paper s that of the concentraton nequalty. It s based on the deas of Chen (1986) where the concentraton nequalty s derved dfferently from those n Chen and Shao (2001), due to the non-postvty of the covarance functon. The unform bounds obtaned are mprovements of those n Chen (1986) and the non-unform bounds, whch are proved by followng the technques n Chen and Shao (2001), are new n the lterature. In provng the bounds, an attempt s made to acheve the best possble order for them. For example, the non-unform bounds obtaned are best possble as functons of the varables. The forms of the bounds obtaned are nspred by the results n Chen (1978), where necessary and suffcent condtons are proved for asymptotc normalty of locally dependent random varables (termed fntely dependent n that paper). Such bounds deal successfully wth those cases where the varance of a sum of n random varables grows at a dfferent rate from n. An example due to Erckson (1974) s used to llustrate ths pont. Ths paper s organzed as follows. The man results and ther applcatons are gven n Secton 2. Two unform and one non-unform condtonal concentraton nequaltes are proved n Secton 3. The proofs of the unform bounds are gven n Secton 4 and that of the non-unform bounds n Secton Man results Throughout ths paper let J be an ndex set and {X, J be a random feld wth zero means and fnte varances. Defne W = J X and assume that Var(W ) = 1. Let n be the cardnalty of J, F be the dstrbuton functon of W and Φ be the standard normal dstrbuton functon.

4 4 L.H.Y Chen and Q.-M. Shao For A J, let X A denote {X, A, A c = {j J : j A, and A the cardnalty of A. Adopt the notaton: a b = mn(a, b) and a b = max(a, b). We frst ntroduce dependence assumptons and defne notatons that wll be used throughout the paper. (LD1) For each J there exsts A J such that X and X A c are ndependent. (LD2) For each J there exst A B J such that X s ndependent of X A c and X A s ndependent of X B c. (LD3) For each J there exst A B C J such that X s ndependent of X A c, X A s ndependent of X B c, and X B s ndependent of X C c. (LD4 ) For each J there exst A B B C D J such that X s ndependent of X A c, X A s ndependent of X B c, X A s ndependent of {X Aj, j B c, {X Al, l B s ndependent of {X A j, j C c, and {X Al, l C s ndependent of {X A j, j D c. It s clear that (LD4 ) mples (LD3), (LD3) yelds (LD2) and (LD1) s the weakest assumpton. Roughly speakng, (LD4 ) s a verson of (LD3) for {X A, J. On the other hand, n many cases (LD1) mples(ld2), (LD3) and (LD4*) wth B, C, B, C and D defned as: B = j A A j, C = j B A j, B = j A B j, C = j B B j and D = j C B j. Other forms of local dependence have also been used n the lterature, such as dependency neghborhoods n Rnott and Rotar (1996) where convergence rates of multvarate central lmt theorem were obtaned. Some of ther results may not be covered by our theorems. For each J let Y = j A X j. We defne (2.1) ˆK (t) = X {I( Y t < 0) I(0 t Y ), K (t) = E ˆK (t), ˆK(t) = J ˆK (t), K(t) = E ˆK(t) = J K (t).

5 Snce Var(W ) = 1, we have Normal Approxmaton under Local Dependence 5 (2.2) K(t)dt = EW 2 = Unform Berry-Esseen bounds The Berry-Esseen theorem (Berry (1941) and Esseen (1945), see for example, Petrov (1995)) states that f {X, J are ndependent wth fnte thrd moments, then there exsts an absolute constant C such that sup F (z) Φ(z) C E X 3. z R J Here and throughout the paper C denotes an absolute constant whch may have dfferent values at dfferent places. If n addton X, J are dentcally dstrbuted, then the bound s of the order n 1/2, whch s known to be the best possble. The man objectve of ths subsecton s to obtan general unform Berry-Esseen bounds under varous dependence assumptons wth an am to acheve the best possble orders. We frst present a result under assumpton (LD1). Theorem 2.1. Under (LD1) we have (2.3) sup F (z) Φ(z) r 1 + 4r 2 + 8r 3 + r r r 6, z where r 1 = E J (X Y EX Y ), r 2 = J E X Y I( Y > 1), (2.4) r 3 = J E X (Y 2 1), r 4 = J E{ W X (Y 2 1), r 5 = Var( ˆK(t))dt, r 6 = ( t Var( ˆK(t))dt) 1/2. Snce r 1, r 2, r 3 and r 4 depend on the moments of {X, Y, W, they can be easly estmated (see Remark 2.1). The followng alternatve formulas of r 5 and r 6 may

6 6 L.H.Y Chen and Q.-M. Shao be useful. Let {X, J be an ndependent copy of {X, J and defne Y = j A X j. Then Var( ˆK(t)) =,j J { E X X j {I( Y t < 0) I(0 t Y ){I( Y j t < 0) I(0 t Y j ) X X j {I( Y t < 0) I(0 t Y ){I( Y j t < 0) I(0 t Yj ) and hence r 5 =,j J Smlarly, we have r 2 6 = (1/2) { E X X j I(Y Y j 0)( Y Y j 1) X Xj I(Y Yj 0)( Y Yj 1).,j J { E X X j I(Y Y j 0)( Y 2 Y j 2 1) X Xj I(Y Yj 0)( Y 2 Yj 2 1). In partcular, under assumpton (LD2) and r 5 r 2 6 (1/2),j J,B B j,j J,B B j { E X X j ( Y Y j 1) + X Xj ( Y Yj 1) { E X X j ( Y 2 Y j 2 1) + X Xj ( Y 2 Yj 2 1). Thus, we have a much neater result under (LD2). Theorem 2.2. Let N(B ) = {j J : B j B and 2 < p 4. Assume that (LD2) s satsfed wth N(B ) κ. Then (2.5) sup F (z) Φ(z) z ( κ) ( (E X 3 p + E Y 3 p ) κ ) 1/2 (E X p + E Y p ) J J In partcular, f E X p + E Y p θ p for some θ > 0 and for each J, then (2.6) sup F (z) Φ(z) ( κ) n θ 3 p + 2.5θ p/2 κn, z where n = J.

7 Normal Approxmaton under Local Dependence 7 Note that n many cases κ s bounded and θ s of order of n 1/2. In those cases κ n θ 3 p + θ p/2 κn = O(n (p 2)/4 ), whch s of the best possble order of n 1/2 when p = 4. However, the cost s the exstence of fourth moments. To reduce the assumpton on moments, we need a stronger condton. Theorem 2.3. Suppose that (LD3) s satsfed. Let N(C ) = {j J : C B j, W = j N(C ) c X j, σ 2 = Var( W ) and λ = 1 max J (1/σ ). Then (2.7) sup F (z) Φ(z) 4λ 3/2 (r 2 + r 3 + r 7 + r 8 + r 9 + r 10 + r 11 + r 12 ), z where Z = j B X j, r 7 = J E{ X Y I( X > 1), r 8 = J E{ X I( X 1)( Y 1) Z, (2.8) r 9 = J E{ W X I( X 1)( Y 1)( Z 1), r 10 =,j J,B B j r 11 = J P ( X > 1)E X ( Y 1), { E X X j ( Y Y j 1) + X Xj ( Y Yj 1), r 12 = J E{( W + 1)( Z 1)E X ( Y 1) and (X, Y ) s an ndependent copy of (X, Y ). In partcular, we have

8 8 L.H.Y Chen and Q.-M. Shao Then Theorem 2.4. Let 2 < p 3. Assume that (LD3) s satsfed wth N(C ) κ. (2.9) sup F (z) Φ(z) 75κ p 1 E X p. z J Rnott (1994) and Dembo and Rnott (1996) obtaned unform bounds of order n 1/2 when X s bounded wth order of n 1/2 under a dfferent local dependence assumpton whch appears to be weaker than (LD2). However, ther approach does not seem to be extendable to random varables whch are not necessarly bounded. Remark 2.1. Although r 4 nvolves W, there are several ways to bound t. When there s no addtonal assumpton besdes (LD1), we can use the followng estmate: E W X ( A E W X (Y 2 1) (Xj 2 1)) j A A E W X (Xj 2 1)I( X X j ) + A E W X (Xj 2 1)I( X > X j ) j A j A A E W X j (Xj 2 1) + A E W X (X 2 1) j A j A A E W Y j E X j (Xj 2 1) + A E Y j X j ( X j 1) j A j A + A 2 E W Y E X (X 2 1) + A 2 E Y X ( X 1) A (1 + E Y j )E X j (Xj 2 1) + A E Y j X j ( X j 1) j A j A + A 2 (1 + E Y )E X (X 2 1) + A 2 E Y X ( X 1) Non-unform Berry-Esseen bound Non-unform bounds were frst obtaned by Esseen (1945) for ndependent and dentcally dstrbuted random varables

9 Normal Approxmaton under Local Dependence 9 {X, J. These were mproved to CnE X 1 3 /(1+ x 3 ) by Nagaev (1965). Bkels (1966) generalzed Nagaev s result to F (z) Φ(z) C J E X z 3 for ndependent and not necessarly dentcally dstrbuted random varables. In ths subsecton we present a general non-unform bound for locally dependent random felds {X, J under (LD4 ). Theorem 2.5. Assume that E X p < for 2 < p 3 and that (LD4 ) s satsfed. Let κ = max J max( D, {j : D j ). Then (2.10) F (z) Φ(z) Cκ p (1 + z ) p J E X p m-dependent random felds Let d 1 and Z d denote the d-dmensonal space of postve ntegers. The dstance between two ponts = ( 1,, d ) and j = (j 1,, j d ) n Z d s defned by j = max 1 l d l j l and the dstance between two subsets A and B of Z d s defned by ρ(a, B) = nf{ j : A, j B. For a gven subset J of Z d, a set of random varables {X, J s sad to be an m- dependent random feld f {X, A and {X j, j B are ndependent whenever ρ(a, B) > m, for any subsets A and B of J. Thus choosng A = {j : j m J, B = {j : j 2m J, C = {j : j 3m J, B = {j : j 3m J, C = {j : j 4m J, and D = {j : j 5m J n Theorems 2.4 and 2.5 yelds a unform and a non-unform bound.

10 10 L.H.Y Chen and Q.-M. Shao Theorem 2.6. Let {X, J be an m-dependent random felds wth zero means and fnte E X p < for 2 < p 3. Then (2.11) sup F (z) Φ(z) 75(10m + 1) (p 1)d E X p z J and (2.12) F (z) Φ(z) C(1 + z ) p (11) pd (m + 1) (p 1)d J E X p. Here we have reduced the m-dependent random feld to a 1-dependent random feld by takng blocks and then appled (2.10) to get (2.12). The result (2.11) was prevously obtaned by Shergn (1979) wthout specfyng the absolute constant. For non-unform bounds, results weaker than (2.12) have been obtaned n the lterature. See, for example, Prakasa Rao (1981) and Henrch (1984). However, the result n Prakasa Rao (1981) s far from best possble even for ndependent random felds whle Henrch (1984) s the best possble only for the..d. case. For other unform and non-unform Berry-Esseen bounds for m-dependent and weakly dependent random varables, see Thomrov (1980), Dasgupta (1992), and Sunklodas (1999). In Sunkloads (1999) a lower bound s also gven Examples In ths subsecton we gve three examples dscussed n lterature to llustrate the usefulness of our general results Graph dependency Ths example was dscussed n Bald and Rnott (1989) and Rnott (1994) where some results on unform bound were obtaned. Consder a set of random varables {X, V ndexed by the vertces of a graph G = (V, E). G s sad to be a dependency graph f for any par of dsjont sets Γ 1 and Γ 2 n V such that no edge n E has one endpont n Γ 1 and the other n Γ 2, the sets of random varables {X, Γ 1 and {X, Γ 2 are ndependent. Let D denote

11 Normal Approxmaton under Local Dependence 11 the maxmal degree of G,.e., the maxmal number of edges ncdent to a sngle vertex. Let A = {j V : there s an edge connectng j and, B = j A A j, C = j B A j, B = j A B j, C = j B B j and D = j C B j. An applcaton of Theorems 2.4 and 2.5 yelds the followng theorem. Theorem 2.7. Let {X, V be random varables ndexed by the vertces of a dependency graph. Put W = V X. Assume that EW 2 = 1, EX = 0 and E X p θ p for V and for some θ > 0. Then (2.13) sup P (W z) Φ(z) 75D 5(p 1) V θ p z and for z R, (2.14) P (W z) Φ(z) C(1 + z ) p D 5p V θ p. Whle (2.13) compares favorably wth those of Bald and Rnott (1989), (2.14) s new The number of local maxma on a graph Consder a graph G = (V, E) (whch s not necessary a dependency graph) and ndependently dentcally dstrbuted contnuous random varables {Y, V. For V defne the 0-1 ndcator varable where N = {j V : 1 f Y > Y j for all j N X = 0 otherwse d(, j) = 1 and d(, j) denotes the shortest path dstance between the vertces and j. Note that d(, j) = 1 ff and j are neghbors, so X = 1 ndcates that Y s a local maxmum. Let W = V X be the number of local maxma. If (V, E) s a regular graph,.e., all vertces have the same degree d,

12 12 L.H.Y Chen and Q.-M. Shao then by Bald, Rnott and Sten (1989), EW = V /(d + 1), and σ 2 = Var(W ) = s(, j)(2d + 2 s(, j)) 1 (d + 1) 2,j V,d(,j)=2 where s(, j) = N N j. sup P (W z) Φ((z EW )/σ) Cσ 1/2, z Theorem 2.8 below s obtaned by applyng Theorem 2.2. The unform bound, whch mproves σ 1/2 of Bald, Rnott and Sten (1989) to σ 1, s smlar to that of Dembo and Rnott (1996). However the non-unform bound s new. Theorem 2.8. We have (2.15) sup P (W z) Φ((z EW )/σ) Cd 2 V /σ 3 z and (2.16) P (W z) Φ((z EW )/σ) C(1 + z ) 3 d 5 V /σ 3. Theorem 2.9. We have (2.17) P (W z) Φ((z np)/σ) Cr 2 (1 + z 3 ) np(1 p) dependence wth o(n) varance Ths example was dscussed n Erckson (1974). Defne a sequence of bounded, symmetrc and dentcally dstrbuted random varables X 1,, X n wth EX 2 = 1 as follows. Let X 1 and X 2 be ndependent bounded and symmetrc random varables wth varance 1 and put B 2 k = Var( k =1 X ). For k 2, defne X k+1 = X k f B 2 k > k1/2 and defne X k+1 to be ndependent of X 1,, X k f B 2 k k1/2. It s clear that X 1,, X n s a 1-dependent sequence, B 2 n n 1/2 2 and n =1 X s a sum of B 2 n n 1/2 terms

13 Normal Approxmaton under Local Dependence 13 of ndependent and dentcally dstrbuted random varables. By the Berry-Esseen theorem, sup P (W z) Φ(z/B n ) Cn 1/4 E X 1 3 z and the order n 1/4 s correct. Whle the bound n (2.11) generalzes and mproves many others, t s asymptotcally Cn 1/4 E X 1 3, whch goes to. On the other hand, Theorem 2.3 gves the correct order n 1/4. To see ths, we observe that f X s ndependent of all other random varables, then we can choose A = B = C = { and X = Y = Z. On the other hand f X = X 1 or X +1, then A = B = C = { 1, or {, + 1 respectvely. In ths case Y = Z = 0 dentcally. Consequently the rght hand sde of (2.7) s bounded by CE X 1 /B n 3 number of X whch s ndependent of all the other random varables CE X 1 /B n 3 B 2 n = CE X 1 3 B 1 n Cn 1/4 E X 1 3, as desred. 3. Concentraton nequaltes The concentraton nequalty n normal approxmaton usng Sten s method plays a role correspondng to that of the smoothng nequalty of Esseen (1945) n the Fourer analytc method. It s used to overcome the dffculty caused by the non-smoothness of the ndcator functon whose expectaton s the dstrbuton functon. In ths secton we establsh two unform and one non-unform condtonal concentraton nequaltes. We frst prove Propostons 3.1, 3.2 and 3.3, whch wll be used n the proofs of Theorems 2.1, 2.2, 2.5 respectvely. Esseen (1968), Petrov (1995) and others have obtaned many unform concentraton nequaltes for sums of ndependent random varables. Our unform concentraton nequaltes are dfferent from thers except n the..d. case.

14 14 L.H.Y Chen and Q.-M. Shao Let {X, J be a random feld wth EX = 0 and EX 2 <. Put W = J X. Assume that EW 2 = 1. Proposton 3.1. Assume (LD1). Then for any real numbers a < b, (3.1) P (a W b) 0.625(b a) + 4r r 3 + 4r 5, where r 2, r 3 and r 5 are as defned n (2.4). Proof. Let α = r 3 and defne (b a + α)/2 for w a α (3.2) 1 2α (w a + α)2 (b a + α)/2 f(w) = w (a + b)/2 for a α < w a for a < w b 1 2α (w b α)2 + (b a + α)/2 for b < w b + α (b a + α)/2 for w > b + α Then f s a contnuous functon gven by 1, for a w b f (w) = 0, for w a α or w b + α, lnear, for a α w a or b w b + α Clearly f(w) (b a + α)/2. Wth ths f, Y, and ˆK(t) and K(t) as defned n (2.1), we have (3.3) (b a + α)/2 EW f(w ) = J E{X (f(w ) f(w Y )) = 0 E {X f (W + t)dt J Y = { E f (W + t) ˆK (t)dt J = E f (W + t) ˆK(t)dt := H 1 + H 2 + H 3 + H 4,

15 where Normal Approxmaton under Local Dependence 15 { H 1 = Ef (W ) K(t)dt, H 2 = E (f (W + t) f (W ))K(t)dt, { { H 3 = E f (W + t) ˆK(t)dt t >1, H 4 = E f (W + t)( ˆK(t) K(t))dt. Clearly, by (2.2) (3.4) { H 1 = Ef (W ) 1 K(t)dt t >1 Ef (W )(1 r 2 ) P (a W b) r 2 and (3.5) H 3 J E X Y I( Y > 1) = r 2. By the Cauchy nequalty, (3.6) H 4 (1/8)E [f (W + t)] 2 dt + 2E ( ˆK(t) K(t)) 2 dt To bound H 2, let (b a + 2α)/8 + 2r 5. Then by wrtng H 2 = E we have 1 t 0 = α t α L(α) = sup P (x W x + α). x R 0 f (W + s)dsk(t)dt E t f (W + s)dsk(t)dt {P (a α W + s a) P (b W + s b + α)dsk(t)dt t {P (a α W + s a) P (b W + s b + α)dsk(t)dt, (3.7) H 2 α 1 1 t L(α)ds K(t) dt + α t L(α)ds K(t) dt

16 16 L.H.Y Chen and Q.-M. Shao = α 1 L(α) tk(t) dt 0.5α 1 L(α)r 3 = 0.5L(α). It follows from (3.3) - (3.7) that (3.8) P (a W b) 0.625(b a) α + 2r 2 + 2r L(α). Substtutng a = x and b = x + α n (3.8), we obtan L(α) 1.375α + 2r 2 + 2r L(α) and hence (3.9) L(α) 2.75α + 4r 2 + 4r 5. Fnally combnng (3.8) and (3.9), we obtan (3.1). Proposton 3.2. Assume (LD3). Let ξ = ξ = (X, Y, Z ), where Y = j A X j and Z = j B X j. For Borel measurable functons a ξ and b ξ of ξ such that a ξ b ξ, we have (3.10) P ξ (a ξ W b ξ ) 0.625σ 1 (b ξ a ξ ) + 4σ 2 r σ 3 r 3 + 4σ 3 r 10 a.s., where σ and r 10 are as defned n Theorem 2.3, and P ξ ( ) denotes the condtonal probablty gven ξ. Proof. We use the same notaton as n Theorem 2.3 and follow the same lne of the proof as that of Proposton 3.1. Let f ξ be defned smlarly as n (3.2) such that f ξ ((a ξ + b ξ )/2) = 0 and f ξ s a contnuous functon gven by 1, for a ξ w b ξ f ξ(w) = 0, for w a ξ α or w b ξ + α, lnear, for a ξ α w a ξ or b ξ w b ξ + α

17 Normal Approxmaton under Local Dependence 17 where α = σ 2 r 3. Then, f ξ (w) (b ξ a ξ + α)/2. Put ˆM(t) = j N(C ) c ˆKj (t) and M(t) = j N(C ) c K j (t). Observe that X j and {ξ, W Y j are ndependent for j N(C ) c. Smlar to (3.3), (3.11) 0.5(b ξ a ξ + α)σ E{ W f ξ (W ) = j N(C ) c E ξ {X j (f ξ (W ) f ξ (W Y j )) = E ξ{ f ξ(w + t) ˆM(t)dt := H 1,ξ + H 2,ξ + H 3,ξ + H 4,ξ, where, E ξ ( ) denotes the condtonal expectaton gven ξ, H 1,ξ = E ξ f ξ(w ) M(t)dt, H 2,ξ = E ξ{ (f ξ(w + t) f ξ(w ))M(t)dt, H 3,ξ = E ξ{ f ξ(w + t) ˆM(t)dt, H 4,ξ = E ξ{ f ξ(w + t)( ˆM(t) M(t))dt. t >1 Note that ξ and ˆM(t) are ndependent. Analogous to (3.4), (3.5) and (3.6), (3.12) H 1,ξ P ξ (a ξ W b ξ )σ 2 r 2, (3.13) (3.14) H 3,ξ E ξ { X j Y j I( Y j > 1) j N(C ) c = E{ X j Y j I( Y j > 1) r 2, j N(C ) c H 4,ξ 0.125σ (b ξ a ξ + 2α) + 2σ 1 E ( ˆM(t) M(t)) 2 dt = 0.125σ (b ξ a ξ + 2α) + 2σ 1 ρ, where ρ = Var( ˆM(t))dt.

18 18 L.H.Y Chen and Q.-M. Shao To bound H 2,ξ, defne L ξ (α) = lm sup k x Q P ξ (x 1/k W x + 1/k + α), where Q s the set of ratonal numbers and wth a lttle abuse of notaton, we regard P ξ as a regular condtonal probablty gven ξ. Followng the proof of (3.7) yelds (3.15) H 2,ξ α 1 L ξ (α) tm(t) dt 0.5α 1 L ξ (α)r 3 = 0.5σ 2 L ξ (α). Thus by (3.12) - (3.15), (3.16) P ξ (a ξ W b ξ ) 0.625σ 1 (b ξ a ξ ) ασ 1 + 2σ 2 r 2 + 2σ 3 ρ + 0.5L ξ (α). Now substtute a ξ = x 1/k and b ξ = x + 1/k + α n (3.16). By takng supremum over x Q and then lettng k, we obtan and hence L ξ (α) 1.375ασ 1 + 2σ 2 r 2 + 2σ 3 ρ + 0.5L ξ (α) (3.17) L ξ (α) 2.75ασ 1 + 4σ 2 r 2 + 4σ 3 ρ. Combnng (3.16) and (3.17), we obtan (3.18) P ξ (a ξ W b ξ ) 0.625σ 1 (b ξ a ξ ) + 4σ 2 r σ 3 r 3 + 4σ 3 ρ. It remans to prove that ρ r 10. Let (X j, Y j ) be an ndependent copy of (X j, Y j ). Note that ˆK j (t) and ˆK l (t) are ndependent for l N(B j ) c. Drect computatons yeld ρ = j N(C ) c l N(C ) c = j N(C ) c l N(B j)n(c ) c Cov( ˆK j (t), ˆK l (t))dt Cov( ˆK j (t), ˆK l (t))dt

19 = E j N(C ) c l N(B j)n(c ) c j J = r 10. l N(B j) E Normal Approxmaton under Local Dependence 19 { X l X j I(Y l Y j 0)( Y l Y j 1) X l Xj I(Y l Yj 0)( Y l Yj 1) { X l X j ( Y l Y j 1) + X l X j ( Y l Y j 1) Ths completes the proof of the proposton. For obtanng a non-unform condtonal concentraton nequalty, we need two lemmas on moment nequaltes for locally dependent random felds. Lemma 3.1. Let {X, J be a random feld satsfyng (LD3) and let ξ be a measurable functon of X wth Eξ = 0 and Eξ 4 < for each J. Let T = J ξ and σ 2 = E(T 2 ). Then (3.19) and for a > 0, σ 2 = Eξ ξ j J j A (3.20) ET 4 = 3σ 4 6 J E(ξ ξ A )E(ξ B ξ C ) + 3 J E(ξ ξ A )Eξ 2 B 3 J E(ξ ξ 2 A ξ B ) + J E(ξ ξ 3 A ) +6 J E(ξ ξ A ξ B ξ C ) 3 J E(ξ ξ A ξ 2 B ) (3.21) 3σ J {a 3 Eξ 4 + a 1 Eξ 4 A + a 1 Eξ 4 B + a 1 Eξ 4 C, where ξ A In partcular, we have = ξ j, ξ B = ξ j, ξ C = ξ j. j A j B j C (3.22) σ 2 κ 1 J Eξ 2

20 20 L.H.Y Chen and Q.-M. Shao and (3.23) ET 4 3σ κ 3 1 Eξ 4, J where κ 1 = max J max( C, {j J : C j ). Proof. By (LD3), ξ and {ξ j, j A c are ndependent and ths mples (3.19). Note that for each J, ξ and T ξ A are ndependent, {ξ, ξ A and T ξ B are ndependent, and {ξ, ξ A, ξ B and T ξ C ET 4 = { (3.24) E ξ (T 3 (T ξ A ) 3 ) J are ndependent. Therefore = 3E { ξ ξ A T 2 3E { ξ ξa 2 T + E { ξ ξ 3 A J = 3 E { ξ ξ A E(T ξ B ) E { ξ ξ A (T 2 (T ξ B ) 2 ) J J 3 E { ξ ξ 2 A ξ B + E { ξ ξa 3 J J = 3 E { ξ ξ A (σ 2 2Eξ B ξ C + EξB 2 ) J +3 E { ξ ξ A (2ξ B ξ C ξb 2 ) 3 E { ξ ξ 2 A ξ B + E { ξ ξa 3 J J J = 3σ 4 6 E(ξ ξ A )E(ξ B ξ C ) + 3 E(ξ ξ A )EξB 2 J J 3 J E(ξ ξ 2 A ξ B ) + J E(ξ ξ 3 A ) +6 J E(ξ ξ A ξ B ξ C ) 3 J E(ξ ξ A ξ 2 B ). Now the Cauchy nequalty mples that for any a > 0 and any random varables u 1, u 2, u 3, u 4, (3.25)E u 1 u 2 u 3 u 4 (1/4) { a 3 E u a 1 E u a 1 E u a 1 E u 4 4. It follows that the rght hand sde of (3.24) s bounded by 5.5 J {a3 Eξ 4 + a 1 Eξ 4 A + a 1 Eξ 4 B + a 1 Eξ 4 C. Ths proves (3.21).

21 Normal Approxmaton under Local Dependence 21 To prove (3.22) and (3.23), put A 1 = {j J : A j and C 1 C j. Then κ 1 = max J ( C C 1 ). By the C r nequalty and (3.19) σ 2 0.5{Eξ 2 + Eξj 2 J j A 0.5κ 1 Eξ J j J κ 1 Eξ 2. J Smlarly by (3.21) wth a = κ 1, ET 4 3σ {κ 3 1Eξ 4 + κ 1 J 3σ κ 3 1 E ξ 4. J 1 κ3 1 Ths completes the proof of Lemma 3.1. C 1 j j A E ξ j 4 + κ 1 1 κ3 1 Eξ 2 j j B E ξ j 4 + κ 1 = {j J : 1 κ3 1 j C E ξ j 4 Lemma 3.2. Let {X, J be a random feld satsfyng (LD4 ). Let ξ be a measurable functon of X wth Eξ = 0 and Eξ 4 < and η a measurable functon of X A wth Eη = 0 and Eη 4 <. Let T = J ξ and S = J η. Then for any a > 0 E(T 2 S 2 ) 3ET 2 ES 2 +4 J {a 3 Eξ 4 +a 1 Eξ 4 C +a 1 Eξ 4 D +a 1 Eη 4 B +a 1 Eη 4 C +a 1 Eη 4 D, (3.26) where ξ C = ξ j, ξ D = ξ j, η B = η j, η C = η j, η D = η j. j C j D j B j C j D In partcular, we have (3.27) E(T 2 S 2 ) 3ET 2 ES κ 3 2 {E ξ 4 + E η 4, J where κ 2 = max J max( D, {j J : D j ).

22 22 L.H.Y Chen and Q.-M. Shao Proof. By (LD4 ), the followng pars of random varables are ndependent () ξ and (T ξ A )(S η B ) 2 ; () ξ ξ A and (S η B ) 2 ; () ξ η B and (T ξ C )(S η C ). Thus we have E(T 2 S 2 ) = J E(ξ T S 2 ) = ( ) E ξ {T S 2 T (S η B ) 2 + ξ A (S η B ) 2 J = 2E(ξ η B T S) E(ξ ηb 2 T ) + E(ξ ξ A )E(S η B ) 2 J J J = 2 J E{ξ η B (T S (T ξ C )(S η C )) +2 J E(ξ η B )E(T ξ C )(S η C ) J E(ξ η 2 B ξ C ) + J E(ξ ξ A ){E(S 2 ) 2Eη B S + Eη 2 B = 2 J E{ξ η B (ξ C S + η C T ξ C η C ) +2 J E(ξ η B ){E(T S) E(ξ C S) E(η C T ) + E(ξ C η C ) J E(ξ η 2 B ξ C ) + ET 2 E(S 2 ) 2 J E(ξ ξ A )Eη B η C + J E(ξ ξ A )Eη 2 B = 2(E(T S)) 2 + ET 2 ES J E{ξ η B ξ C η D +2 J E{ξ η B η C ξ D 2 J E{ξ η B ξ C η C ) +2 J E(ξ η B ){ E(ξ C η D ) E(η C ξ D ) + E(ξ C η C ) J E(ξ η 2 B ξ C ) 2 J E(ξ ξ A )Eη B η C + J E(ξ ξ A )Eη 2 B 3ET 2 ES J {a 3 Eξ 4 + a 1 Eξ 4 C + a 1 Eξ 4 D + a 1 Eη 4 B + a 1 Eη 4 C + a 1 Eη 4 D, where (3.25) was used to obtan the last nequalty. By the C r nequalty, (3.27) follows drectly from (3.26). We are now ready to state and prove a non-unform concentraton nequalty.

23 Normal Approxmaton under Local Dependence 23 Proposton 3.3. Let {X, J be a random feld satsfyng (LD4 ) and put κ = max J max( D, {j J : D j ). Let ξ be a measurable functon of X satsfyng Eξ = 0 and ξ 1/(4κ). Defne ξ Aj = ξ l, ξ Bj = ξ l, T = ξ j. l A j l B j j J Assume that 1/2 ET 2 2 and let ζ = ζ = (ξ, ξ A, ξ B ). Then for Borel measurable functons a ζ and b ζ of ζ such that b ζ a ζ 0 (3.28) E ζ{ (1 + T ) 3 I(a ζ T b ζ ) C(b ζ a ζ + α) a.s., where α = 16κ 2 j J E ξ j 3. In partcular f a ζ x > 1, then (3.29) P ζ (a ζ T b ζ ) C(1 + x) 3 (b ζ a ζ + α) a.s. Proof. Let C c = J C and let ˆK j,ξ (t) = ξ j {I( ξ Aj t < 0) I(0 t ξ Aj ), ˆM ξ (t) = j C c ˆK j,ξ (t), M ξ (t) = E ˆM ξ (t), T = Snce ξ j 1/(4κ) and 1/2 ET 2 2, we have j C c ξ j = T ξ C. (3.30) ξ C j 1/4, E T ξ C j 2 6 and by (3.23), (3.31) E T ξ C j κ 2 j J E ξ j α. Consder two cases. Case I. α > 1. By (3.23), E ζ{ (1 + T ) 3 I(a ζ T b ζ ) (E ζ 1 + T 4 ) 3/4 8 ( 16 + E T ξ C 4) { E ζ (2 + T ξ C ) 4 3/4 E(2 + T ξc ) 4

24 24 L.H.Y Chen and Q.-M. Shao 8(16 + 3(E T ξ C 2 ) κ 3 j J E ξ j 4 ) 2 16( 1 + κ 2 j J E ξ j 3) 2 17 α. Ths proves (3.28). Case II. 0 < α < 1. Defne 0 for w a ζ α 1 2α (w a ζ + α) 2 h ζ (w) = w a ζ + α/2 1 2α (w b ζ α) 2 + b ζ a ζ + α b ζ a ζ + α for a ζ α < w a ζ for a ζ < w b ζ for b ζ < w b ζ + α for w > b ζ + α and f ζ (w) = (1 + w) 3 h ζ (w). Clearly, h ζ s a contnuous functon gven by 1, for a ζ w b ζ (3.32) h ζ(w) = 0, for w a ζ α or w b ζ + α, lnear, for a ζ α w a ζ or b ζ w b ζ + α and 0 h ζ (w) b ζ a ζ + α. Wth ths f ζ, and by the fact that for every j C c, ξ j and (ζ, T ξ Aj ) are ndependent, (3.33) E ζ {T f ζ (T ) = j C c = j C c + j C c := G 1 + G 2. Wrte G 1 = 3G 1,1 3G 1,2 + G 1,3, where G 1,1 = j C c G 1,2 = j C c E ζ ( ξ j {f ζ (T ) f ζ (T ξ Aj ) ) E ζ ( ξ j {(1 + T ) 3 (1 + T ξ Aj ) 3 h ζ (T ξ Aj ) ) E ζ {ξ j (1 + T ) 3 (h ζ (T ) h ζ (T ξ Aj )) E ζ ( ξ j ξ Aj (1 + T ) 2 h ζ (T ξ Aj ) ), ( ) E ζ ξ j ξa 2 j (1 + T )h ζ (T ξ Aj )

25 Normal Approxmaton under Local Dependence 25 G 1,3 = j C c ( ) E ζ ξ j ξa 3 j h ζ (T ξ Aj ). Then (ξ j, ξ Aj, T ξ C ) and ζ are ndependent for each j C c. Hence by (3.30), G 1,2 (b ζ a ζ + α) j C c (b ζ a ζ + α) j C c C(b ζ a ζ + α) j C c C(b ζ a ζ + α) j C c E ζ { ξ j ξ 2 A j (1 + T ) C(b ζ a ζ + α) j J E{ ξ j ξ 2 A j E{ ξ j ξ 2 A j (3 + T ξ C ) E{ ξ j ξ 2 A j (1 + T ξ C j ) E{ ξ j ξ 2 A j E(1 + T ξ C j ) C(b ζ a ζ + α)κ 2 j J C(b ζ a ζ + α). E ξ j 3 Smlarly by (3.30), G 1,3 C(b ζ a ζ + α). To bound G 1,1, wrte G 1,1 = j C c + j C c ( = E ( ) E(ξ j ξ Aj )E ζ (1 + T ξ C j ) 2 h ζ (T ξ C j ) j C c + j C c + j C c ( ) E ζ ξ j ξ Aj {(1 + T ) 2 h ζ (T ξ Aj ) (1 + T ξ C j ) 2 h ζ (T ξ C j ) ξ j ξ Aj ) E ζ ( (1 + T ) 2 h ζ (T ) ) ( ) E(ξ j ξ Aj )E ζ (1 + T ξ C j ) 2 h ζ (T ξ C j ) (1 + T ) 2 h ζ (T ) ( ) E ζ ξ j ξ Aj {(1 + T ) 2 h ζ (T ξ Aj ) (1 + T ξ C j ) 2 h ζ (T ξ C j ) := G 1,1,1 + G 1,1,2 + G 1,1,3.

26 26 L.H.Y Chen and Q.-M. Shao Frst we have G 1,1,1 = E(T ξ C ) 2 E ζ ( (1 + T ) 2 h ζ (T ) ) C(b ζ a ζ + α)e ζ (1 + T ) 2 C(b ζ a ζ + α)(1 + E(1 + T ξ C ) 2 ) C(b ζ a ζ + α). Next, as n boundng G 1,2, we obtan G 1,1,3 j C c + j C c j C c ( ) E ζ ξ j ξ Aj (1 + T ) 2 {h ζ (T ξ Aj ) h ζ (T ξ C j ) ( ) E ζ ξ j ξ Aj {(1 + T ) 2 (1 + T ξ C j ) 2 h ζ (T ξ C j ) ( ) E ζ ξ j ξ Aj (1 + T ) 2 ξ Aj ξ C j +C(b ζ a ζ + α) C j J j C c ( E ξ j ξ 2 A j + E ξ j ξ Aj ξ C j C(b ζ a ζ + α). ( ) E ζ ξ j ξ Aj ξ C j (1 + T ) ) + C(b ζ a ζ + α) E ξ j ξ Aj ξ C j j J Fnally, n a smlar way, G 1,1,2 C(b ζ a ζ + α). Combnng the above nequaltes yelds (3.34) G 1 C(b ζ a ζ + α). Now we bound G 2. Usng the defnton of ˆK j,ξ (t), we wrte G 2 = j C c = j C c E ζ( 0 ) ξ j (1 + T ) 3 h ζ(t + t)dt ξ Aj E ζ( (1 + T ) 3 h ζ(t + t) ˆK ) j,ξ (t)dt

27 Normal Approxmaton under Local Dependence 27 = E ζ( (1 + T ) 3 h ζ(t + t) ˆM ) ξ (t)dt = E ζ( ) (1 + T ) 3 h ζ(t )M ξ (t)dt +E ζ( ) (1 + T ) 3 (h ζ(t + t) h ζ(t ))M ξ (t)dt +E ζ( (1 + T ) 3 h ζ(t + t)( ˆM ) ξ (t) M ξ (t))dt := G 2,1 + G 2,2 + G 2,3. Note that ET 2 = E(T ξ C ) 2 (1/2)ET 2 Eξ 2 C 1/4 1/8=1/8 and hence G 2,1 = ET 2 E ζ{ (3.35) (1 + T ) 3 h ζ(t ) (1/8)E ζ{ (1 + T ) 3 I(a ζ T b ζ ). By the Cauchy nequalty, (3.26) and (3.31), G 2,3 E ζ( (1 + T ) 4 ) [h ζ(t + t)] 2 dt + E ζ( + T ) (1 2 ( ˆM ) ξ (t) M ξ (t)) 2 dt (b ζ a ζ + 2α)E ζ (1 + T ) 4 + CE ζ{ (1 + (T ξ C ) 2 ) C(b ζ a ζ + α)(1 + E(T ξ C ) 4 ) + G 2,3,1 ( ˆM ξ (t) M ξ (t)) 2 dt C(b ζ a ζ + α) + G 2,3,1, where G 2,3,1 = { E (1 + (T ξ C ) 2 )( ˆM ξ (t) M ξ (t)) 2 dt. By (3.22) and (3.27), we have E {(1 + (T ξ C ) 2 )( ˆM ξ (t) M ξ (t)) 2 Cκ j J E ˆK j,ξ (t) 2 + Cκ 3 j J E ξ j 4 + Cκ 3 j J E ˆK j,ξ (t) 4

28 28 L.H.Y Chen and Q.-M. Shao and hence G 2,3,1 Cκ j J E ξ j 2 ξ Aj + Cκ 3 j J E ξ j 4 + Cκ 3 j J E ξ j 4 ξ Aj Cκ E ξ j 2 ξ Aj + Cκ 2 E ξ j 3 j J j J Cκ E ξ j 2 ξ l + Cκ 2 E ξ j 3 j J l A j j J Cκ 2 E ξ j 3 Cα. j J To bound G 2,2, defne L ζ (α) = lm sup k x 0,x Q E ζ {(1 + T ) 3 I(x 1/k T x + 1/k + α), where Q s the set of ratonal numbers and E ζ s regarded as a regular condtonal expectaton gven ζ. Then for a ζ > 1, so that a ζ α > 0, we have G 2,2 = E ζ( (1 + T ) 3 1 and = α α 1 0 t (3.36) G 2,2 α 1 0 t 0 h ζ (T + s)dsm ξ (t)dt ) E ζ( 0 (1 + T ) t ) h ζ (T + s)dsm ξ (t)dt E ζ{ (1 + T ) 3 (I(a ζ α T + s a ζ ) I(b ζ T + s b ζ + α)) dsm ξ (t)dt t 0 E ζ{ (1 + T ) 3 (I(a ζ α T + s a ζ ) I(b ζ T + s b ζ + α)) dsm ξ (t)dt t 0 = α 1 L ζ (α) L ζ (α)ds M ξ (t) dt + α 1 0 (1/2)κ 2 α 1 L ζ (α) j J (1/16)L ζ (α). 1 0 t L ζ (α)ds M ξ (t) dt tm ξ (t) dt (1/2)α 1 L ζ (α) j J E ξ j ξ 2 A j E ξ j 3 Therefore (3.37) G 2 (1/8)E ζ{ (1 + T ) 3 I(a ζ T b ζ ) C(b ζ a ζ + α) (1/16)L ζ (α).

29 Normal Approxmaton under Local Dependence 29 Now by (3.31) (3.38) E ζ {T f ζ (T ) (b ζ a ζ + α)e ζ T (1 + T ) 3 C(b ζ a ζ + α)e( T (1 + T 3 )) C(b ζ a ζ + α). So combnng (3.33), (3.34), (3.37) and (3.38), we have for a ζ > 1, (3.39) E ζ{ (1 + T ) 3 I(a ζ T b ζ ) C(b ζ a ζ + α) + (1/2)L ζ (α). For 0 < a ζ 1, t suffces to consder b ζ a ζ 1. Applyng Proposton 3.2 to {ξ, J, we obtan (3.40) E ζ {(1 + T ) 3 I(a ζ < T < b ζ + α) CP ζ (a ζ < T < b ζ + α) C(b ζ a ζ + α). Now take a ζ = x 1/k and b ζ = x + 1/k + α. By takng supremum over x Q and then lettng k, (3.39) and (3.40) mply L ζ (α) Cα + (1/2)L ζ (α). Hence (3.41) L ζ (α) Cα. Ths together wth (3.39) and (3.40) proves (3.28) and hence Proposton Proofs of Theorems We frst derve a Sten dentty for W. Let f be a bounded absolutely contnuous functon. Then (4.1) E{W f(w ) = J E{X (f(w ) f(w Y )) = 0 E {X f (W + t)dt J Y = { E f (W + t) ˆK (t)dt J

30 30 L.H.Y Chen and Q.-M. Shao = E f (W + t) ˆK(t)dt and hence by the fact that K(t)dt = EW 2 = 1, (4.2) Ef (W ) EW f(w ) = E f (W )K(t)dt E = E +E +E +E f (W )(K(t) ˆK(t))dt t >1 := R 1 + R 2 + R 3 + R 4. (f (W ) f (W + t)) ˆK(t)dt f (W + t) ˆK(t)dt (f (W ) f (W + t))( ˆK(t) K(t))dt (f (W ) f (W + t))k(t)dt Now choose f to be f z,α, the unque bounded soluton of the dfferental equaton (4.3) f (w) wf(w) = h z,α (w) Nh z,α where α > 0 s to be determned later, 1, for w z (4.4) h z,α (w) = 1 + (z w)/α, for z w z + α 0, for w z + α and Nh zα = (2π) 1/2 h z,α (x)e x2 /2 dx. The soluton of (4.3) s gven by f z,α (w) = e w2 /2 = e w2 /2 w w [h z,α (x) Nh z,α ]e x2 /2 dx [h z,α (x) Nh z,α ]e x2 /2 dx. From Lemma 2 and arguments on pages 23 and 24 n Sten (1986), we have for all w and v, (4.5) 0 f z,α (w) 1,

31 Normal Approxmaton under Local Dependence 31 (4.6) f z,α(w) 1, f z,α(w) f z,α(v) 1 and f z,α(w + s) f z,α(w + t) (4.7) (4.8) ( w + 1) mn( s + t, 1) + α 1 t s I(z w + u z + α)du ( w + 1) mn( s + t, 1) + I(z s t w z s t + α). Proof of Theorem 2.1. By (4.6) (4.9) R 1 = Ef (W ) J (X Y EX Y ) r 1 and (4.10) R 2 J E X Y I( Y > 1) = r 2. By (4.8), (4.11) R 3 E ( W + 1) t ˆK(t) K(t) dt +E +E I(z t W z + α) ˆK(t) K(t) dt I(z W z t + α) ˆK(t) K(t) dt r 3 + r 4 + R 3,1 + R 3,2, where R 3,1 = E R 3,2 = E I(z t W z + α) ˆK(t) K(t) dt, I(z W z t + α) ˆK(t) K(t) dt. Let δ = 0.625α + 4r r 3 + 4r 5. Then by Proposton 3.1, P (z t W z + α) δ t

32 32 L.H.Y Chen and Q.-M. Shao for t > 0. Hence by the Cauchy nequalty, { 1 R 3,1 E 0 0.5α + 0.5α 1 δ ( 0.5α(δ t) 1 I(z t W z + α) + 0.5α 1 (δ t) ˆK(t) K(t) 2) dt Var( ˆK(t))dt α 1 tvar( ˆK(t))dt. A smlar nequalty holds for R 3,2. Thus we arrve at 0 (4.12) R 3 α + 0.5α 1 δr α 1 r r 3 + r 4. By (4.7) and Proposton 3.1 agan, we have (4.13) R 4 E ( W + 1) tk(t) dt t +α 1 P (z W + u z + α)du K(t) dt 2r 3 + α 1 0 Combnng the above nequaltes yelds tδ K(t) dt 2r 3 + α 1 δr 3. Eh z,α (W ) Nh z,α r 1 + r 2 + 3r 3 + r 4 + α + α 1{ δ(0.5r 5 + r 3 ) r 2 6 r 1 + r r 3 + r r 5 + α +α 1{ (4r r 3 + 4r 5 )(0.5r 5 + r 3 ) r 2 6 Usng the fact that Eh z α,α (W ) P (W z) Eh z,α (W ) and that Φ(z + α) Φ(z) (2π) 1/2 α, we have. (4.14) Lettng sup z P (W z) Φ(z) sup Eh z,α (W ) Nh z,α + 0.5α. z α = 2/3 ( ) 1/2 (4r r 3 + 4r 5 )(0.5r 5 + r 3 ) r6 2 yelds (4.15) sup P (W z) Φ(z) z

33 Normal Approxmaton under Local Dependence 33 r 1 + r r 3 + r r ( ) 1/2 (4r r 3 + 4r 5 )(0.5r 5 + r 3 ) r6 2 ( ) r 1 + r r 3 + r r r (4r r 3 + 4r 5 ) + 4(0.5r 5 + r 3 ) r 1 + 4r 2 + 8r 3 + r r r 6. Ths proves Theorem 2.1. Proof of Theorem 2.2. Let p 3 = 3 p. Usng the followng well-known nequalty: x 0, α 0 wth α = 1 (4.16) x r α x, we have r 2 J J E X Y p3 1 { 1 p 3 E X p3 + p 3 1 p 3 E Y p3 and r 3 J J E X Y p3 1 { 1 p 3 E X p3 + p 3 1 p 3 E Y p3. Smlarly, we have and r 5 2,j J,B B j,j J,B B j 2κ J r 2 6 (1/2) {E X X j Y p3 2 + E X X j Y p3 2 { 1 E X p3 + 1 E X j p3 + (p 3 2) E Y p3 p 3 p 3 p 3 { 2 p 3 E X p3 + (p 3 2) p 3 E Y p3,j J,B B j {E X X j Y p 2 + E X X j Y p 2

34 34 L.H.Y Chen and Q.-M. Shao κ { 2 p E X p + J (p 2) E Y p. p Now we estmate r 4. Recall that Z = j B X j and that (X, Y ) and W Z are ndependent. We have r 4 J {E W Z E X (Y 2 1) + E Z X (Y 2 1) {(1 + E Z )E X (Y 2 1) + E Z X Y p3 2 J E X Y p3 1 + E X j E X Y p3 2 J J j B + E X j X Y p3 2 J j B J { 1 E X p3 + (p 3 1) E Y p3 + 2κ { 2 E X p3 + (p 3 2) E Y p3. p 3 p 3 p 3 p 3 J To estmate r 1, let ξ = X Y I( X Y 1). We have r 1 E J (ξ Eξ ) + 2 J E X Y I( X Y > 1) Var( J ξ ) 1/2 + 2 J E X Y p3/2 Var( J ξ ) 1/2 + J {E X p3 + E Y p3. Smlarly to boundng r 6 Var( ξ ) J (E ξ ξ j + E ξ E ξ j ),j J,B B j,j J,B B j (E X Y p/4 X j Y j p/4 + E X Y p/4 E X j Y j p/4 ) κ (E X p + E Y p ). J Combnng the nequaltes above yelds (2.5). Proof of Theorem 2.3. The dea of the proof s smlar to that of Theorem 2.1. Notng that ˆK and W Z are ndependent, we rewrte (4.2) as (4.17) Ef (W ) EW f(w )

35 where = J = J E Normal Approxmaton under Local Dependence 35 { E f (W )K (t)dt E f (W + t) ˆK (t)dt + J E (f (W ) f (W Z + t))k (t)dt = Q 1 + Q 2 + Q 3 + Q 4, (f (W Z + t) f (W + t)) ˆK (t)dt Q 1 = E (f (W ) f (W Z + t))k (t)dt, J Q 2 = E (f (W ) f (W Z + t))k (t)dt, J Q 3 = E J Q 4 = E J By (4.6), smlar to (4.10), t >1 t >1 (f (W Z + t) f (W + t)) ˆK (t)dt, (f (W Z + t) f (W + t)) ˆK (t)dt. (4.18) Q 2 + Q 3 2 J E X Y I( Y > 1) = 2r 2. To bound Q 4, wrte Q 4 = Q 4,1 + Q 4,2, where Q 4,1 = EI( X > 1) (f (W Z + t) f (W + t)) ˆK (t)dt, J Q 4,2 = EI( X 1) J Then by (4.6) (f (W Z + t) f (W + t)) ˆK (t)dt. (4.19) Q 4,1 J E X Y I( X > 1) = r 7. From (4.7), we obtan (4.20) Q 4,2 EI( X 1) ( W + t + 1)( Z 1) ˆK (t) dt J

36 36 L.H.Y Chen and Q.-M. Shao +α 0 1 EI( X 1) I(Z 0) I(z W + t + u z + α)du ˆK (t) dt J Z +α Z 1 EI( X 1) I(Z < 0) I(z W + t + u z + α)du ˆK (t) dt J 0 J E( W + 1) X I( X 1)( Z 1)( Y 1) J E X ( Y 2 1) + Q 4,3 r 8 + r r 3 + Q 4,3, where Q 4,3 = α { 1 E I( X 1) J +α { 1 E I( X 1) J and ξ = (X, Y, Z ). By Proposton 3.2, 0 I(Z 0) P ξ (z W + t + u z + α)du ˆK (t) dt Z Z I(Z < 0) P ξ (z W + t + u z + α)du ˆK (t) dt 0 Q 4,3 (4.21) α { 1 E I( X 1) J (0.625σ 1 α + 4σ 2 r σ 3 r 3 + 4σ 3 r 10 ) Z ˆK (t) dt 0.625λ J E{ X I( X 1)( Y 1) Z + α 1{ 4λ 2 r λ 3 r 3 + 4λ 3 r 10 r 8 = 0.625λr 8 + α 1{ 4λ 2 r λ 3 r 3 + 4λ 3 r 10 r 8. Combnng (4.19), (4.20) and (4.21) yelds (4.22) Q 4 r 7 + r λr r 3 + α 1{ 4λ 2 r λ 3 r 3 + 4λ 3 r 10 r 8. Smlarly we have, (4.23) Q 1 J P ( X > 1)E X ( Y 1) + J E{( W + 1)( Z 1)E X ( Y 1) + r 3 +α 1{ 0.625λα + 4λ 2 r λ 3 r 3 + 4λ 3 r 10 { 0.5r 3 + E{( W + 1)( Z 1)E X ( Y 1) J

37 Normal Approxmaton under Local Dependence 37 r 11 + r 12 + r 3 +α 1{ 0.625λα + 4λ 2 r λ 3 r 3 + 4λ 3 r 10 (0.5r 3 + r 12 ). Combnng (4.17), (4.18), (4.22), (4.23) and (4.14), we obtan (4.24) sup F (z) Φ(z) z 0.5α + 2r 2 + 2λr 3 + r λr 8 + r 9 + r λr 12 +α 1{ 4λ 2 r λ 3 r 3 + 4λ 3 r 10 (r r 3 + r 12 ). Let α = ( 2(+4λ 2 r λ 3 r 3 + 4λ 3 r 10 )(r r 3 + r 12 )) 1/2. Then the rght hand sde of (4.24) s = 2r 2 + 2λr 3 + r λr 8 + r 9 + r λr 12 { 1/2 + 2(4λ 2 r λ 3 r 3 + 4λ 3 r 10 )(r r 3 + r 12 ) 2r 2 + 2λr 3 + r λr 8 + r 9 + r λr λ 3/2 (4λ 2 r λ 3 r 3 + 4λ 3 r 10 ) + λ 3/2 (r r 3 + r 12 ) 4λ 3/2 (r 2 + r 3 + r 7 + r 8 + r 9 + r 10 + r 11 + r 12 ). Ths proves Theorem 2.3. Proof of Theorem 2.4. We can assume that (4.25) κ p 1 J E X p 1/75. Otherwse (2.9) s trval. Let ξ = j N(C ) X j. Then by (4.25) (4.26) Eξ 2 (E ξ p ) 1/p ( N(C ) p 1 j N(C ) E X j p ) 1/p

38 38 L.H.Y Chen and Q.-M. Shao (κ p 1 j J E X j p ) 1/p (1/75) 1/p (1/75) 1/3 < and σ EW 2 Eξ = Thus 4λ 3/ By (4.16) and the Mnkowsk nequalty r 8 J E X Z Y p 2 = J E(κ (p 1)/p X ( Z /κ 1/p )( Y /κ 1/p ) p 2 ) J κp 1 p 1 p κp 1 E X p + ( 1 pκ E Z p (p 2) + pκ E Y p ) J J E X p + J κ p 1 J E X p. C p 1 (p 1) pκ j C E X j p Smlarly, we have r 2 + r 3 + r 7 + r 11 2κ p 1 J E X p. Note that N(B ) N(C ). Followng the proof for r 2 6 yelds r 10 2,j J,B B j {E(κ (p 2)/p X κ (p 2)/p X j ( Y /κ 2/p ) p 2 ) +E(κ (p 2)/p X κ (p 2)/p X ( Y /κ 2/p ) p 2 ),j J,B B j { κp 2 2κ p 1 J E X p. p (E X p + E X j p ) + (p 2) pκ 2 E Y p The r 9 can be bounded as r 8 and r 4. By (4.26) r 9 E W ξ E X ( Y 1)( Z 1) + E ξ X Y p 2 J J J E X Y Z p 2 + J E X ξ Y p 2

39 Normal Approxmaton under Local Dependence κ p 1 J E X p. Smlarly, r κ p 1 J E X p. Theorem 2.4 follows from (2.7) and the above nequaltes. 5. Proof of Theorem 2.5 The basc dea of the proof of Theorem 2.5 s smlar to that of Theorem 2.3. We use the same notaton as n Secton 2.2 and remnd the reader that {X, J satsfes (LD4 ) and that E X p < for 2 < p 3. Frst we need a few prelmnary lemmas. Let (5.1) τ = 1/(8κ), X = X I( X τ), ˆX = X I( X > τ), W = J X, Y = j A X j and (5.2) β 1 = J E X Y I( X > τ), β 2 = J EX 2 I( X > τ), β 3 = J E X 3 I( X τ). Our frst lemma shows that W s close to W. Lemma 5.1. Assume β 2 τ/16. Then there exsts an absolute constant C such that for z 0 (5.3) P (W > z) P ( W > z) J P ( Y > (z + 1)/4) + C(z + 1) 3 κ 3 β 3 +64κ 2 β 2 {(1 + z) 3 + P (W > (z 1)/2).

40 40 L.H.Y Chen and Q.-M. Shao Proof. Observe that (5.4) P (W > z) = P (W > z, max J X τ) + P (W > z, max J X > τ) P ( W > z) + J P (W > z, X > τ) P ( W > z) + J { P (W Y > 3(z 1/3)/4, X > τ) + P (Y > (z + 1)/4, X > τ) P ( W > z) + J P ( W > z) + J { P (W Y > 3(z 1/3)/4)P ( X > τ) + P (Y > (z + 1)/4) { P (W > (z 1)/2)P ( X > τ) + P ( Y > (z + 1)/4) +P (Y > (z + 1)/4) P ( W > z) + P (W > (z 1)/2)τ 2 J E X 2 I( X > τ) + J P ( Y > (z + 1)/4) = P ( W > z) + 64κ 2 β 2 P (W > (z 1)/2) + J P ( Y > (z + 1)/4). Smlarly, notng that j A Xj 1, we have (5.5) P ( W > z) P (W > z) + κ 2 β 2 P ( W > z 2). Note that β 2 τ/16 mples E W 1/16 and by (3.22), (5.6) Var( W ) 1 = Var( W ) Var(W ) 2(EW 2 ) 1/2 (Var( J ˆX )) 1/2 + Var( J ˆX ) 2(κβ 2 ) 1/2 + κβ 2 2/3. Applyng (3.23) to W E W yelds (5.7) P ( W > z 2) E W E W (z + 2 E W ) 4

41 Normal Approxmaton under Local Dependence 41 C(z + 1) 4( 1 + κ ) 3 E X 4 I( X τ) J C(z + 1) 4( 1 + κ 3 τ ) E X 3 I( X τ) J C(z + 1) 4 (1 + κ 2 β 3 ). By (5.4) - (5.7) and assumpton that κβ 2 1., (5.3) s proved and hence the lemma. Lemma 5.2. Assume E X p < for some 2 < p 3. Then there exsts an absolute constant C such that for z 0 (5.8) P (W > (z 1)/2) C(1 + z) p (1 + κ p 1 γ) and P ( Y > (z + 1)/4) + κ 3 (1 + z) 3 β 3 + (1 + z) 3 β 1 J + κ 2 β 2 {(1 + z) 3 + P (W (z 1)/2) (5.9) Cκ p (1 + z) p γ + C(1 + z) 3 κ 2p 1 γ 2, where γ = J E X p. Proof. If κ p 1 γ > (1 + z) p 2, then (5.8) s trval because P (W > (z 1)/2) 4(z + 1) 2 E(W + 1) 2 = 8(z + 1) 2. To prove (5.8) for κ p 1 γ (1 + z) p 2, let X = X I( X (1 + z)τ) EX I( X (1 + z)τ), X = X I( X > (1 + z)τ) EX I( X > (1 + z)τ), W = J X, W = J X. Then P (W > (z 1)/2) P ( W > (z 3)/4) + P ( W > (z + 1)/4)

42 42 L.H.Y Chen and Q.-M. Shao C(1 + z) 4 E W (1 + z) 1 E W C(1 + z) 4 (1 + E W 4 ) + C(1 + z) 1 ((1 + z)τ) p+1 γ. Smlar to (5.6), κ p 1 γ (1 + z) p 2 mples Var( W ) 4. Hence by (3.23) E W 4 C(1 + κ 3 J E X 4 I( X (1 + z)τ)) C(1 + κ p 1 (1 + z) 4 p γ). By combnng the above nequaltes, (5.8) s proved. We now prove (5.9). From (5.8), the Chebyshev nequalty and the Hölder nequalty, the left hand sde of (5.9) s bounded by ((z + 1)/4) p J +C(1 + z) 3 κ 2 τ p+2 γ(1 + κ p 1 γ) E Y p + κ 3 (1 + z) 3 τ 3 p γ + (1 + z) 3 τ p+2 J E X p 1 Y C(1 + z) p κ p γ + C(1 + z) 3 κ 2p 1 γ 2. Ths completes the proof of the lemma. Proof of Theorem 2.5. Wthout loss of generalty, assume z 0. When κ p 1 γ > 1, (2.10) follows drectly from (5.8). When κ p 1 γ 1, then (2.10) s a consequence of (5.9) and the followng nequalty (5.10) F (z) Φ(z) J P ( Y > (z + 1)/4) + C(1 + z) 3 β 1 +Cκ 2 β 2 {(1 + z) 3 + P (W (z 1)/2) +Cκ 2 (1 + z) 3 β 3. So t suffces to prove (5.10). It s clear that for z 0, P (W > z) (1 Φ(z)) P (W > (z 1)/2) + 16(1 + z) 3.

43 Normal Approxmaton under Local Dependence 43 So (5.10) holds f κ 2 β 2 > 1/16. It remans to consder the case that (5.11) κ 2 β 2 1/16. Let W be defned as n (5.2). By Lemma 5.1, t suffces to show that (5.12) P ( W z) Φ(z) C(1 + z) 3 β 1 + Cκ(1 + z) 3 β 2 + Cκ 2 (1 + z) 3 β 3. Let σ 2 = Var( W ), T = ( W E W )/σ. Then (5.13) P ( W z) Φ(z) = P (T (z E W )/σ) Φ(z) P (T (z E W )/σ) Φ((z E W )/σ) + Φ((z E W )/σ) Φ(z). By the Chebyshev nequalty, (5.14) E W τ 1 J EX 2 I( X > τ) 1/16. By (5.6), we have 1/3 < σ 2 < 2, and moreover, smlar to (5.6), (5.15) σ 2 1 = 2 J E( ˆX W ) + Var( J ˆX ) 2 J = 2β 1 + κβ 2. E X Y I( X > τ) + κ J EX 2 I( X > τ) Thus by (5.14) and (5.15), (5.16) Φ((z E W )/σ) Φ(z) Φ((z E W )/σ) Φ(z/σ) + Φ(z/σ) Φ(z) C(1 + z) 3 E W + C(1 + z) 3 σ 2 1 C(1 + z) 3 κβ 2 + C(1 + z) 3 β 1.

44 44 L.H.Y Chen and Q.-M. Shao Wth x = (z E W )/σ (> 1/2), we only need to show that (5.17) P (T x) Φ(x) Cκ 2 (1 + x) 3 β 3. Put ξ = X /σ, ξ A = ξ j, ξ B = ξ j, j A j B ˆK (t) = ξ (I( ξ A < t 0) I(0 < t < ξ A )), K (t) = E ˆK (t). By the defnton of τ, we have ξ A 1/4 and ξ B 1/4. If (1 + x)κ 2 β 3 > 1, then by (3.23), P (T > x) (1 Φ(x)) (1 + x) 4 E 1 + T 4 + (1 + x) 4 C(1 + x) 4( 1 + κ 3 E ξ 4) J C(1 + x) 4( 1 + κ 3 τ E ξ 3) J C(1 + x) 4 (1 + κ 2 β 3 ) C(1 + x) 3 κ 2 β 3, whch proves (5.17). If (1 + x)κ 2 β 3 1, let α = 64κ 2 β 3. Also let h x,α (w) be as n (4.4) and let f(w) = f x,α (w) be the unque bounded soluton of the Sten equaton (4.3) wth x replacng z. Then by (4.1) and smlar to (4.2), Ef (T ) ET f(t ) = (5.18) E (f (T ξ B + t) f (T + t)) ˆK (t)dt J + E (f (T ) f (T ξ B + t))k (t)dt J := R 1 + R 2. Let g(w) = (wf(w)) and let R 1,1 = ξb E g(t + u)du ˆK (t)dt, J 0

45 Notng that Normal Approxmaton under Local Dependence 45 R 1,2 = α ξb 1 E I(x T + u x + α)du ˆK (t)dt. J 0 (5.19) f x,α(w + t) f x,α(w + s) α 1 t s t I(x w + u x + α)du, s g(w + u)du we have R 1 R 1,1 + R 1,2. Let ζ = (ξ, ξ A, ξ B ). By Lemma 5.3 below, R 1,1 ξb E E ζ g(t + u)du ˆK (t) dt J 0 C(1 + x) 3 E ξ B ˆK (t) dt J C(1 + x) 3 J E ξ ξ A ξ B Cκ 2 (1 + x) 3 β 3, and by Proposton 3.3 R 1,2 α ξb 1 E P ζ (x T + u x + α)du ˆK (t) dt J 0 Cα 1 (1 + x) 3 E (κ 2 β 3 + α) ξ B ˆK (t) dt J Cα 1 (1 + x) 3 (κ 2 β 3 + α) J E ξ ξ A ξ B Cκ 2 (1 + x) 3 β 3. Ths proves R 1 Cκ 2 (1 + x) 3 β 3.

46 46 L.H.Y Chen and Q.-M. Shao Smlarly we have R 2 Cκ 2 (1 + x) 3 β 3. Hence we have (5.20) Eh x,α (T ) Nh x,α Cκ 2 (1 + x) 3 β 3. Fnally usng the fact that Eh x, α (T ) P (T x) Eh x,α (T ) and that Φ(x + α) Φ(x) C(1 + x) 3 α for x > 1/2, we have (5.17). Ths completes the proof of Theorem 2.5. It remans to prove Lemma 5.3 whch was used above. Lemma 5.3. Let ζ = ζ = (ξ, ξ A, ξ B ) and (5.21) g(w) = g x,α (w) = (wf x,α (w)). Then for x > 1/2 and u 4, we have (5.22) E ζ g(t + u) C(1 + x) 3 (1 + κ 2 β 3 ). Proof. From the defntons of Nh x,α, f x,α and g, we have Nh x,α = Φ(x) + α 1 se (x+α αs)2 /2 ds, 2π 0 2πe w 2 /2 Φ(w)(1 Nh x,α ), w x 2πe w 2 /2 (1 Φ(w))Nh x,α f x,α (w) = αe w2 /2 1+(z w)/α se (z+α αs)2 /2 ds, x < w x + α 0 2πe w 2 /2 (1 Φ(w))Nh x,α, w > x + α

47 Normal Approxmaton under Local Dependence 47 ( 2π(1 + w 2 )e w2 /2 Φ(w) + w) (1 Nh x,α ), w < x, ( 2π(1 g(w) = + w 2 )e w2 /2 (1 Φ(w)) w) Nh x,α + r x,α (w), x w x + α ( 2π(1 + w 2 )e w2 /2 (1 Φ(w)) w) Nh x,α, w > x + α, where r x,α (w) = we w2 /2 x+α w < x + α, we have by ntegraton by parts, r x,α e w2 /2 = e w2 /2 w x+α w x+α w (1 + x s 2 α )e s /2 ds + (1 + (x w)/α). For x < s(1 + x s 2 α )e s /2 ds + (1 + (x w)/α) e s2 /2 ds 0. So 0 r x,α 1 for x < w < x + α. It can be verfed that 0 < g(w) C(1 + x ) for all x. Therefore (5.22) holds when 1 < x 6. When x > 6, the proof of (5.22) s very smlar to that of Lemma 5.2 of Chen and Shao (2001). A drect calculaton shows that for w 0, (5.23) 0 2π(1 + w 2 )e w2 /2 (1 Φ(w)) w w 3. Ths mples that g 0, g(w) 2(1 Φ(x)) for w 0 and 2 1+w 3 +I(x < w < x+α) for w x; furthermore, g s clearly ncreasng for 0 w < x. Therefore E ζ g(t + u) = E ζ g(t + u)i(t + u x 1) + E ζ g(t + u)i(t + u x) +E ζ g(t + u)i(x 1 < T + u < x) 2(1 Φ(x)) + g(x 1) + 2(1 + x 3 ) 1 + P ζ (x T + u x + α) +E ζ g(t + u)i(x 1 < T + u < x) ( ) C (1 + x) 3 + x 2 e (x 1)2/2 (1 Φ(x)) + C(1 + x) 3 α

48 48 L.H.Y Chen and Q.-M. Shao +E ζ g(t + u)i(x 1 < T + u < x) (5.24) C(1 + x) 3 (1 + α) + E ζ g(t + u)i(x 1 < T + u < x). By Proposton 3.3 and the fact that w u x 5 > (x + 1)/7 for w x 1 5 and u 4, E ζ g(t + u)i(x 1 < T + u < x) = x x 1 g(w)dp ζ (w < T + u < x) = g(x 1)P ζ (x 1 < T + u < x) + C(1 + x) 3 + C(1 + x) 3 x C(1 + x) 3{ 1 + αg(x ) + C(1 + x) 3{ x 1 + α x + x 1 C(1 + x) 3{ 1 + xf x,α (x) x 1 x x 1 x x 1 g (w)p ζ (w < T + u < x)dw g (w){α + (x w)dw (x w)dg(w) g(w)dw (5.25) C(1 + x) 3. Combnng (5.24) and (5.25) yelds (5.22). Ths completes the proof of Lemma 5.3. Acknowledgments. The authors thank the referee for hs valuable comments whch have led to substantal mprovement on the presentaton of the paper. REFERENCES [1] Bald, P. and Rnott, Y. (1989). On normal approxmaton of dstrbutons n terms of dependency graph. Ann. Probab. 17,

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