Irregular sets and Central Limit Theorems for dependent triangular arrays

Irregular sets and Cen tral Limit Theorems for dep enden t triangular arra ys Beatri z Marr´ on ∗ , Ana T ablar 1 , ∗ Ma y 1 0 , 2 001 Abstract In previous pap ers, w e studied the asymptotic b eha viour of S N ( A, X ) = (2 N + 1) − d/ 2 P n ∈ A N X n , where X is a cen tered, stationary and w eakly de- p enden t random field, a nd A N = A ∩ [ − N , N ] d , A ⊂ Z d . This leads to the definition o f asymptotically measurable sets, which enjoy the prop erty that S N ( A ; X ) has a G aussian w eak limit f o r any X b elonging to a cer- tain class. Here we extend this t yp e o f results to the case of weakly de- p enden t triangular a r ra ys and presen t an applicatio n of this tec hnique to regression mo dels. Indeed, w e prov e that CL T and related results hold for X N n = ϕ ( ξ N n , Y N n ) , n ∈ Z d , where ϕ satisfies certain regular ity conditions, ξ and Y are inde p enden t random fields, ξ is w eakly dependen t a nd Y satisfies some Strong La w of Large Num b ers. Keywor ds: Cen tral Limit Theorems, weakly dep enden t random fields, trian- gular arr a ys, regression mo dels, asymptotically me asurable sets . AMS subje ct classific ations: 60 F05, 60G6 0. 1 In tro duc tion The notion of an “ a symptotically measurable set” was in tro duced in [7] and [8], and it w as motiv ated b y some statistical problems concerning random fields. ∗ Departamento de Matem´ atica. Univ ersidad Nacional del Sur. 1 Corresp o nding author : actablar@ uns.edu.ar 1 Let us denote Z d the la t tice of p oints of R d with integer co ordinates. A subset A of Z d is said to b e asymptotic al ly m e asur able (AM), if for eac h n ∈ Z d , the limit, as N tends to infinity , of H N ( n ; A ) = car d { A N ∩ ( n + A N ) } (2 N +1) d exists, where A N = A ∩ [ − N , N ] d ; furthermore, if w e denote H ( n ; A ) this limit, it satisfies 0 < H ( n ; A ) < 1. W e denote M ( Z d ) the class o f asymptotically measurable se ts. Sets with regular b o rders (in the sense that their b orders are negligible), p erio dic sets and certain random sets are examples of elemen ts of M ( Z d ). The class o f cen t ered, stationar y , with finite second momen t random fields whic h satisfy certain w eak dep endence conditions is denoted b y F , then S N ( A, X ) = 1 √ (2 N +1) d P n ∈ A N X n has a non- trivial weak limit for any X ∈ F if and only if A ∈ M ( Z d ), this is the main prop ert y of this class of sets. F or statistical purp oses, a generalization of the notion of AM set is needed. W e say that a collection { A i : i = 1 , ..., r } of subsets of Z d is an asymptotic al ly me asur able c ol l e ction ( AMC) if lim N H N ( n ; A i , A j ) = H ( n ; A i , A j ) ∀ n ∈ Z d , i, j = 1 , ..., r , where H N ( n ; A i , A j ) = car d { A i N ∩ ( n + A j N ) } (2 N +1) d . No w consider X = ( X 1 , ..., X r ) , a R r -v alued, cen tered, stationary and w eakly dependen t random field, and define M N ( A 1 , ..., A r ; X 1 , ..., X r ) =  S N ( A 1 , X 1 ) , ..., S N ( A r , X r )  , then M N ( A 1 , ..., A r ; X 1 , ..., X r ) con ve rges we akly for an y X in a suitable class, if and only if A 1 , ..., A r is an AMC. F or instance, t o b e more precise, ta k e k x k = max 1 ≤ i ≤ r | x i | , x = ( x 1 , ..., x r ) ∈ R r , and let X = ( X n ) n ∈ Z d ∈ F b e a random fields suc h that the followin g conditions hold: ( C 1) L et us call r X ( k ) = E { X 0 X k } , then P k ∈ Z d | r X ( k ) | < ∞ . ( C 2) L et us define X J , the truncation b y J of the ra ndom field X , that is X J n = X n 1 I {k X n k≤ J } − E { X n 1 I {k X n k≤ J } } . 2 ( i ) There exists no negative num b ers ρ (1) , ρ (2) , · · · suc h that, for all k ∈ Z d , J ≥ 0, P k ∈ Z d ρ ( k ) < ∞ , and | r X J ( k ) | < ρ ( k ) . ( ii ) There exists a sequence b ( J ) suc h tha t lim J → ∞ b ( J ) = 0 and for eac h A ⊂ Z d , w e ha v e E  ( S N ( A, X ) − S N ( A, X J )) 2  ≤ b ( J ) car d ( A N ) (2 N + 1) d . ( C 3) F or each J > 0 , there exists a real n umber C ( X, J ) suc h that f o r all N ∈ N , A ⊂ Z d w e ha v e E  ( S N ( A, X J )) 4  ≤ C ( X , J )  car d ( A N ) (2 N +1) d  2 . ( C 4) There exis ts a b ounded real function g a nd a sequence d ( J ) with lim J → ∞ d ( J ) = 0 suc h that   E  exp[ it S N ( A ∪ B , X J )]  − E  exp[ it S N ( A, X J )]  · E  exp[ it S N ( B , X J )]    ≤ d ( J ) g ( t ) , t ∈ R , holds fo r an y A, B ⊂ Z d that satisfy d ist ( A, B ) ≥ J . Theorem 1.1 If A is a n AM set and c onditions (C1)-( C 4) hold, t hen S N ( A, X ) w − → N (0 , σ 2 ( A, X )) , wher e σ 2 ( A, X ) = P n ∈ Z d r X ( n ) H ( n ; A ) . The pro of of this theorem is obta ined b y Bernsh tein “big and small blo c ks” metho d [1], and it is similar to the proo f of Proposition 2.2 in [9]. Some final remarks o n general notation w e use all along t his pap er: • The w eak con v ergence of proba bility measures is denoted b y “ w − → ”. • The sym b ol “0”represen ts b o th, the real zero and the zero elemen t of R d ; t he con text will make its me aning cle ar. • N ( µ , σ 2 ) denotes a Gaussian distribution with mean µ and v ariance σ 2 . • car d ( A ) is the cardinal of A . • [ x ] is the inte ger pa r t of the n um b er x . • X ≈ Y means t ha t X and Y has the same distribution. 3 2 The cen tral limit theore m for triang ular ar- ra ys Here w e shall deal with a sp ecial case of random fields, the triangular a rra y . A triangular array is a double sequence of random v ariables X N n , n ∈ Z d , N ∈ N , and the ra ndo m v ariables in eac h row are indep enden t. The purp ose of this pap er is to establish a CL T for w eakly dep endent triangular array s. Theorem 2.1 L et X N =  X N n  n ∈ Z d b e a triangular arr ay that satisfies ( H 1) F or a l l N ∈ N , X N is a stationary, c enter e d, with finite se c ond momen t r andom fields, that satisfie s P k ∈ Z d    r X N ( k )    < ∞ , wh e r e r X N ( k ) = E  X N 0 X N k  . ( H 2) F or al l J > 0 we de fi ne X N ,J n = X N n 1 I {k X N n k≤ J } − E  X N n 1 I {k X N n k≤ J }  and let us supp ose that ( i ) Ther e exists ρ ( k ) ≥ 0 such that P k ∈ Z d ρ ( k ) < ∞ and for any k ∈ Z d , N ∈ N    r X N,J ( k )    ≤ ρ ( k ) . ( ii ) T her e e xists a se quenc e b ( J ) such that lim J → + ∞ b ( J ) = 0 and for any B ⊂ Z d , N ∈ N E n  S N  B , X N  − S N  B , X N ,J  2 o ≤ b ( J ) car d ( B N ) (2 N + 1) d . ( H 3) F or al l J > 0 , ther e exis ts C ( J ) < ∞ such that for al l B ⊂ Z d , and for al l N ∈ N E n S N  B , X N ,J  4 o ≤ C ( J ) car d B N (2 N + 1) d ! 2 . ( H 4) T her e exists a de cr e asing r e al function h : R + → R + such that lim x → + ∞ h ( x ) = 0 and a r e al function g ( J, t ) such that for al l fixe d J > 0 , g is b ounde d on the se c ond variab l e , sup t ∈ R g ( J, t ) = g J < ∞ , such that   E  exp[ it S N ( A ∪ B , X N ,J )]  − E  exp[ it S N ( A, X N ,J )]  E  exp[ it S N ( B , X N ,J )]    4 ≤ g ( J , t ) h ( d ist ( A, B )) , for any disjoint sets A, B ⊂ Z d , for al l N ∈ N , t ∈ R . ( H 5) F or al l k ∈ Z d , J > 0 , ther e exists γ J ( k ) , γ ( k ) such that lim N → + ∞ r X N,J ( k ) = γ J ( k ) and lim J → + ∞ γ J ( k ) = γ ( k ) . If A is an AM set, then S N  A, X N  ω → N N (0 , σ 2 ( A )) , with σ 2 ( A ) = X k ∈ Z d γ ( k ) H ( k ; A ) . The pro of is based on the follow ing steps: First, constrain t the pro blem to w ork with a b ounded cen tered field, with the truncation pro p osed in (H2). Then, follow Bernsh tein “big and small blo c ks”methods, so that t he sum of v ariables o v er small blo ck s is negligible, and the sum of v ariables in tw o differen t large blo c ks is asymptotically indep enden t. Pro of: W e consider t wo nondecreasing sequences of p o sitiv e integer p N and q N suc h that: lim N →∞ p N = lim N →∞ q N = 0 lim N →∞ q N p N = 0 , lim N →∞ p N N = 0 and lim N →∞ ( k N ) d .h ( q N ) = 0, w here k N = h 2 N +1 p N + q N i . W e call I N ( j ) = [ − N + j ( p N + q N ) ; − N + j ( p N + q N ) + p N ], 0 ≤ j ≤ k N ; I N = k N S j =0 I N ( j ) and ∆ N = I d N is the union of ( k N ) d disjoin ts d -cubes of side p N ∆ N = ( k N ) d [ ℓ =1 ∆ N ( ℓ ) , (1) car d (∆ N ( ℓ )) = ( p N + 1) d , hence car d (∆ N ) = ( k N ) d ( p N + 1) d . Ev en more, if ℓ 6 = ℓ ′ , dist (∆ N ( ℓ ) , ∆ N ( ℓ ′ )) ≥ q N . W e can decompo se S N  A, X N ,J  as fo llo ws S N  A, X N ,J  = S N  A ∩ ∆ N , X N ,J  + S N  A ∩ ∆ C N , X N ,J  . (2) 5 As X N ,J is a stationary pro cess, by Lemma 3.1 and H 2 ( i ) E n  S N  A ∩ ∆ C N , X N ,J  2 o ≤ C car d  A ∩ ∆ C N  N  (2 N + 1) d ≤ C car d  ∆ C N  N  (2 N + 1) d = C (2 N + 1) d − ( k N ) d · ( p N + 1) d (2 N + 1) d = C " 1 −  p N + 1 p N + q N  d # . (3) By ( 3 ), the second term in (2) conv erges in L 2 to 0 , therefore it is enough to prov e that S N  A ∩ ∆ N , X N ,J  con ve rges w eakly to a Gaussian la w. Let Y N 1 , . . . , Y N ( k N ) d b e a sequence of random independen t v ariables such that Y N ℓ ≈ S N  A ∩ ∆ N ( ℓ ) , X N ,J  for each J > 0 . In order to show that S N  A ∩ ∆ N , X N ,J  ≈ P ( k N ) d ℓ =1 Y N ℓ , it is sufficien t to pro v e       E    exp   it ( k N ) d X ℓ =1 S N  A ∩ ∆ N ( ℓ ) , X N ,J       − E    exp   it ( k N ) d X ℓ =1 Y N ℓ            − → N →∞ 0 . Applying Lemma 3.2 with Z ℓ = exp  it S N  A ∩ ∆ N ( ℓ ) , X N ,J  w e ha v e       E    ( k N ) d Y ℓ =1 exp  it S N  A ∩ ∆ N ( ℓ ) , X N ,J     − ( k N ) d Y ℓ =1 E  exp  it S N  A ∩ ∆ N ( ℓ ) , X N ,J        ≤ ( k N ) d − 1 X j =1       E    ( k N ) d Y ℓ = j exp  it S N  A ∩ ∆ N ( ℓ ) , X N ,J     − E  exp  it S N  A ∩ ∆ N ( j ) , X N ,J  E    ( k N ) d Y ℓ = j +1 exp  it S N  A ∩ ∆ N ( ℓ ) , X N ,J           ≤ ( k N ) d − 1 X j =1 g ( J, t ) h   dist   A ∩ ∆ N ( j ) , ( k N ) d [ ℓ = j +1 A ∩ ∆ N ( ℓ )     ≤ ( k N ) d g J h ( q N ) − → 0 as N − → ∞ , (4) 6 b y (H4) and as the distance b etw een ∆ N ( ℓ ) is larger than q N , hence S N  A ∩ ∆ N , X N ,J  has the same asymptotic distribution of P ( k N ) d ℓ =1 Y N ℓ . These rando m v ariables are a tringular array of indep enden ts copies of S N  A ∩ ∆ N ( ℓ ) , X N ,J  , cen- tered and with finite v ariance σ 2 N = E  Y N ℓ  2 = E  S N  A ∩ ∆ N ( ℓ ) , X N ,J  2 . By Lyapuno v’s ce n tral limit theorem if, ( k N ) d P ℓ =1 E h ( Y N ℓ ) 2+ δ i σ 2+ δ → N →∞ 0 for some p os- itiv e δ , then ( k N ) d X ℓ =1 Y N ℓ ω → N N (0 , σ 2 N ) . (5) F rom ( H 3) ( k N ) d X ℓ =1 E  S N  A ∩ ∆ N ( ℓ ) , X N ,J  4 ≤ ( k N ) d X ℓ =1 C ( J )  car d (( A ∩ ∆ N ( ℓ )) N ) (2 N + 1) d  2 ≤ ( k N ) d X ℓ =1 C ( J )  car d ((∆ N ( ℓ )) N ) (2 N + 1) d  2 = C ( J )( k N ) d  p N + 1 2 N + 1  2 d = C ( J )  ( p N + 1) 2 ( p N + q N ) (2 N + 1)  d . The last equation tends to 0 as N → ∞ , so the L yapuno v’s condition ho lds, with δ = 2, so (5) f ollo ws. By Lemma 3.1 w e can compute σ 2 N as ( k N ) d X ℓ =1 E n  S N  A ∩ ∆ N ( ℓ ) , X N ,J  2 o = ( k N ) d X ℓ =1 X k ∈ Z d r X N,J ( k ) H N ( k ; A ∩ ∆ N ( ℓ )) = X k ∈ Z d r X N,J ( k ) · ( k N ) d X ℓ =1 H N ( k ; A ∩ ∆ N ( ℓ )) . (6) By the other hand H N ( k ; A ∩ ∆ N ( ℓ )) ≤ car d ( A ∩ ∆ N ( ℓ )) (2 N + 1) d ≤ car d (∆ N ( ℓ )) (2 N + 1) d , 7 and 0 ≤ ( k N ) d X ℓ =1 H N ( k ; A ∩ ∆ N ( ℓ )) ≤ ( k N ) d X ℓ =1 car d (∆ N ( ℓ )) (2 N + 1) d ≤ 1 . F or each set A we can decompose A N lik e A N = ( A ∩ ∆ N ) ∪ ( A ∩ ∆ C N ). F or short, set D N = A ∩ ∆ N and C N = A ∩ ∆ C N , then H N ( k ; A ) = car d ( A N ∩ ( k + A N )) (2 N + 1) d = car d ( D N ∩ ( k + D N )) (2 N + 1) d + car d ( D N ∩ ( k + C N )) (2 N + 1) d + car d ( C N ∩ ( k + C N )) (2 N + 1) d + car d ( C N ∩ ( k + D N )) (2 N + 1) d . (7) As the last three terms ab ov e are b o unded b y car d (∆ N ( ℓ )), H N ( k ; A ) ≤ car d ( D N ∩ ( k + D N )) (2 N + 1) d + 3 car d (∆ C N ) (2 N + 1) d . (8) The second term in (8) con v erges to 0 if N → ∞ , then asymptotically H N ( k ; A ) ≃ car d ( D N ∩ ( k + D N )) (2 N +1) d , a nd car d ( D N ∩ ( k + D N )) (2 N + 1) d = ( k N ) d X ℓ,ℓ ′ =1 car d { ( A ∩ ∆ N ( ℓ )) ∩ ( k + ( A ∩ ∆ N ( ℓ ′ ))) } (2 N + 1) d = ( k N ) d X ℓ =1 car d { ( A ∩ ∆ N ( ℓ )) ∩ ( k + ( A ∩ ∆ N ( ℓ ))) } (2 N + 1) d + X ℓ 6 = ℓ ′ car d { ( A ∩ ∆ N ( ℓ )) ∩ ( k + ( A ∩ ∆ N ( ℓ ′ ))) } (2 N + 1) d . F or N large enough suc h that q N > k k k , the second term in (9) is equal to 0, then H N ( k ; A ) ≃ ( k N ) d X ℓ =1 H N ( k ; A ∩ ∆ N ( ℓ )) . (9) Since A is measurable, H N ( k ; A ) − → N →∞ H ( k ; A ) , this holds ( k N ) d X ℓ =1 H N ( k ; A ∩ ∆ N ( ℓ )) − → N →∞ H ( k ; A ) . (10) 8 Applying ( H 5) a nd (10) in (6), w e hav e σ 2 N − → N →∞ σ 2 J ( A ) = P k ∈ Z d γ J ( k ) H ( k ; A ) . In summary w e pro v ed that S N  A, X N ,J  ω → N N  0 , σ 2 J ( A )  . (11) F rom h yp otesis ( H 2) i ),    r X N,J ( k )    ≤ ρ ( k ) , for an y k , N , J , therefore, lim N →∞    r X N,J ( k )    ≤ ρ ( k ). So, b y ( H 5) for a ny k , J   γ J ( k )   ≤ ρ ( k ) . As P k ∈ Z d ρ ( k ) < ∞ and since 0 ≤ H ( k , A ) ≤ 1, a pplying the theorem of Dominated Conv ergence, w e get that σ 2 J ( A ) is finite. Hence, lim J → + ∞ σ 2 J ( A ) = σ 2 ( A ) . (12) F or a rbitrat y ǫ > 0, b y Tc heb yshev inequalit y and ( H 2) ii ) P    S N  A, X N ,J  − S N  A, X N    > ǫ  ≤ E  S N  A, X N ,J  − S N  A, X N  2 ǫ 2 ≤ b ( J ) car d ( A N ) (2 N + 1) d ǫ 2 ≤ b ( J ) ǫ 2 , then lim J → + ∞ lim sup N →∞ P    S N  A, X N ,J  − S N  A, X N    ≥ ǫ  = 0 , and the theorem is pro v ed. ✷ Corollary 2.1 L et X N =  X N n  n ∈ Z d b e a stationary, c enter e d and m -dep endent triangular arr ay that satisfies the fol lowing c onditions: C1) F or e ach N ∈ N , E  X N 0  4 < C , C > 0 , a c onstant indep endent of N . C2) The c ovarianc e function is uniformely b ounde d, t hat is    r X N ( k )    ≤ ρ ( k ) 0 ≤ k k k ≤ m, ∀ N ∈ N . 9 C3) F or every k ∈ Z d , J > 0 , ther e exists γ J ( k ) < ∞ and γ ( k ) < ∞ such that lim N → + ∞ r X N,J ( k ) = γ J ( k ) and lim J → + ∞ γ J ( k ) = γ ( k ) . If A is AM, then S N  A, X N  ω → N N (0 , σ 2 ( A )) , with σ 2 ( A ) = P k ∈ Z d k k k ≤ m γ ( k ) H ( k ; A ) . Pro of: W e set X N ,J n as in the theorem ab ov e, acc ording to this theorem it is enough to sho w that conditions ( H 1), ( H 2), ( H 3), and ( H 4) hold. The h yp o t hesis ( H 1) and ( H 2 ) i ) are direct from the fact that    r X N ( k )    = 0 if k k k ≤ m . Let us sho w ( H 2) ii ) E n  S N  B , X N  − S N  B , X N ,J  2 o = E      X n ∈ B N X N n − X N ,J n q (2 N + 1) d   2    . Let us call Y N ,J n =  X N n − X N ,J n  , so P k ∈ Z d    r Y N,J ( k )    = P k ∈ Z d k k k ≤ m    r Y N,J ( k )    < ∞ . Applying Lemma 3.1 E n  S N  B , X N  − S N  B , X N ,J  2 o = E n S N  B , Y N ,J  2 o ≤ car d ( B N ) (2 N + 1) d X k ∈ Z d k k k ≤ m    r Y N,J ( k )    , taking b ( J ) = P k ∈ Z d || k | | ≤ m    r Y N,J    to prov e the h yp othesis ( H 2) ii ) it is enough to show that    r Y N,J    − → 0 as J → ∞ , ∀ N ∈ N . 10    r Y N,J ( k )    =    E n Y N ,J 0 , Y N ,J 0 o    =    E  X N 0 X N k  − E n X N 0 X N k 1 I {k X N k k ≤ J } o − E n X N 0 1 I {k X N 0 k ≤ J } X N k o + r X N,J ( k )    . As applying Cauc hy -Sc hw artz inequalit y , we hav e E   X N 0 X N k   < ∞ , then by Dominate Conv ergence Theorem E n X N 0 X N k 1 I {k X N k k ≤ J } o − → J → ∞ E  X N 0 X N k  , E n X N 0 1 I {k X N 0 k ≤ J } X N k o − → J → ∞ E  X N 0 X N k  , and r X N,J ( k ) − → J → ∞ r X N ( k ), so b ( J ) = P k ∈ Z d k k k ≤ m    r Y N,J ( k )    − → J → ∞ 0 , ∀ N ∈ N . T o s ho w ( H 3), let B ⊆ Z d b e an arbitrary s et, E n S N  B , X N ,J  4 o = 1 (2 N + 1) 2 d      X i ∈ B N E  X N ,J i  4 + 4 X i, j ∈ B N i 6 = j E   X N ,J i  3 X N ,J j  +6 X i, j ∈ B N i 6 = j E   X N ,J i  2  X N ,J j  2  + 12 X i, j, k ∈ B N i 6 = j 6 = k E   X N ,J i  2 X N ,J j X N ,J k  +24 X i, j, k , l ∈ B N i 6 = j 6 = k 6 = l E n X N ,J i X N ,J j X N ,J k X N ,J l o      . As  X N ,J i  4 ≤  X N i  4 , f rom C 1) w e get X i ∈ B N E  X N ,J i  4 ≤ car d ( B N ) C . 11 Applying Cauc hy-Sc h w artz inequalit y and b y the m -dep endence, w e get b oundes fo r eac h term X i, j ∈ B N i 6 = j E   X N ,J i  3 X N ,J j  ≤ car d ( B N ) ( car d ( B N ) − 1) C , X i, j ∈ B N i 6 = j E   X N ,J i  2  X N ,J j  2  ≤ car d ( B N ) ( car d ( B N ) − 1) 2 C , X i, j, k ∈ B N i 6 = j 6 = k E   X N ,J i  2 X N ,J j X N ,J k  ≤ ( car d ( B N )) 2 2(2 m + 1) d C , X i, j, k , l ∈ B N i 6 = j 6 = k 6 = l E n X N ,J i X N ,J j X N ,J k X N ,J l o ≤ ( car d ( B N )) 2 3(2 m + 1) d C . W e conclude that E n S N  B , X N ,J  4 o ≤ C ⋆  car d ( B N ) (2 N + 1) d  2 ∀ J > 0 . T o prov e ( H 4) let us no t e that the characteristic function of S N ( A, X N ,J ) and the c ha r acteristic function of S N ( B , X N ,J ) are indep enden t random v ari- ables if ( A, B ) > m , then it is enough to consider h ( x ) = 0 if x > m . ✷ 3 App endix Lemma 3.1 If X = ( X n ) n ∈ Z d is a we akly stationary r andom field such that for al l n, E ( X n ) = 0 , E ( X 2 n ) < ∞ , then for any B ⊂ Z d , E  S N ( B , X ) 2  = X k ∈ Z d r X ( k ) · H N ( k ; B ) , with H N ( k ; B ) = car d ( B N ∩ ( k + B N )) (2 N +1) d . In p articular, if P k ∈ Z d | r X ( k ) | < ∞ then E  S N ( B , X ) 2  ≤ C car d ( B N ) (2 N +1) d . 12 Lemma 3.2 L et Z 1 , Z 2 , · · · , Z n b e a se quenc e of c omplex-v a lue d r andom vari- ables such that | Z i | ≤ 1 , for al l i , then      E ( n Y i =1 Z i ) − n Y i =1 E { Z i }      ≤ n − 1 X j =1      E ( n Y i = j Z i ) − E { Z j } E ( n Y i = j +1 Z i )      . Pro of:      E ( n Y i =1 Z i ) − n Y i =1 E { Z i }      ≤      E ( n Y i =1 Z i ) − E { Z 1 } E ( n Y i =2 Z i )      +      E { Z 1 } E ( n Y i =2 Z i ) − n Y i =1 E { Z i }      ≤      E ( n Y i =1 Z i ) − E { Z 1 } E ( n Y i =2 Z i )      +      E ( n Y i =2 Z i ) − n Y i =2 E { Z i }      . In the same w ay , w e can b ound the second term and finally w e obtain      E ( n Y i =1 Z i ) − n Y i =1 E { Z i }      ≤ ≤      E ( n Y i =1 Z i ) − E { Z 1 } E ( n Y i =2 Z i )      +      E ( n Y i =2 Z i ) − E { Z 2 } n Y i =3 E { Z i }      + · · · +      E ( n Y i = n − 1 Z i ) − n Y n − 1 E { Z i }      ≤ n − 1 X j =1      E ( n Y i = j Z i ) − E { Z j } E ( n Y i = j +1 Z i )      . ✷ 13 References [1] Bernsh tein (19 44). Extension of the cen tra l limit theorem of probability theory to sums o f dep enden t random v ariables. Usp ehi Mat. Nauk 10, 65- 114 (in Russian). 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