# j j 0 , j 0 A k Y k,0 = Q k k A k+(m 1)d, λ k (n) 1 n Y k+(m 1)d j,j Q k+(m 1)d = 1 n Y k+(m 1)d,j, j 0, Ȳ k,j (n) 1 n j=0 j=0 Y k j,j = k

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1 L = λw L = λw L λ W 25

2 l 1 R l 1 l 1 k j j 0 X {X k,j : k 0; j 0} X k,j k j 0 Y k,j i=j X k,i k j j 0 A k Y k,0 = X k,j k Q k k Y k j,j = k, j 0 A k j Y k j,j A k j k 0 0/0 1 0 n λ k (n) 1 n Q k (n) 1 n Ȳ k,j (n) 1 n n A k+(m 1)d, m=1 n Q k+(m 1)d = 1 n m=1 n m=1 n Y k+(m 1)d,j, j 0, m=1 k+(m 1)d Y k+(m 1)d j,j F k,j(n) c Ȳk,j(n) n λ k (n) = m=1 Y k+(m 1)d,j n m=1 A, j 0, k+(m 1)d W k (n) F k,j(n), c 0 k d 1. λ k (n) k (m 1)d + k 0 k d 1 m 1 Qk (n) k,

3 Ȳk,j(n) k j F c k,j (n) k j n Wk (n) k n n n [k] k mod d k d ( ) (A1) λk (n) λ k, n, 0 k d 1, c (A2) F k,j (n) Fk,j, c n, 0 k d 1, j 0, (A3) Wk (n) W k Fk,j c n, 0 k d 1, k Z [k] k mod d λ k = λ [k], F c k,j = F c [k],j, j 0, W k = W [k], 0 k d 1 ( Q k (n), L k (n)) (L k, L k ) n, L k λ k j Fk j,j c < L k (n) = k λ k j (n) F k j,j(n) c + m=1 j=1 d λ d j (n) F d j,(m 1)d+j+k c (n) λ [k j] F c [k j],j, n 1, λ k (n) F k,j c (n) W k (n) F k,j c (n) d k Q k (n) L k (n)) L k L k

4 R d d R x = (x 0, x 1, ) l 1 R R l R R l 1 {x = (x 0, x 1, ) R : x 1 x i < }, i=0 l {x = (x 0, x 1, ) R : x sup x i < } i l 1 l 1 l 1 l l 1 l (l 1 ) d l 1 y (y 0, y 1,, y d 1 ) (l 1 ) d y i (y i,0, y i,1, ) l 1 y 1,d = max i=0,,d 1 { y i 1 } 0 k d 1 F c k(n) ( F c k,0(n), F c k,1(n), F c k,2(n), ) l 1, F c k (F c k,0, F c k,1, F c k,2, ) R. λ λ λ(n) ( λ0 (n), λ 1 (n),, λ d 1 (n)) R d, F c (n) ( F 0 c (n), F 1 c (n),, F d 1(n)) c (l 1 ) d, W W W (n) ( W 0 (n), W 1 (n),, W d 1 (n)) R d, Q Q Q(n) ( Q 0 (n), Q 1 (n),, Q d 1 (n)) R d, L L L(n) ( L 0 (n), L 1 (n),, L d 1 (n)) R d, R R R(n) L L L(n) Q Q Q(n) R d, ˆλˆλˆλ(n) n( λ λ λ(n) λ) R d, ˆF c (n) n( F c (n) F c ) (R ) d, ŴŴŴ (n) n( W W W (n) W ) R d, ˆQˆQˆQ(n) n( Q Q Q(n) L) R d, ˆLˆLˆL(n) n( L L L(n) L) R d, ˆRˆRˆR(n) ˆLˆLˆL(n) ˆQˆQˆQ(n) R d, λ (λ 0, λ 1,, λ d 1 ) R d, F c (F0 c, F1 c,, Fd 1) c (R ) d, W (W 0, W 1,, W d 1 ) R d, L (L 0, L 1,, L d 1 ) R d,

5 πk 1 : Rd R k = 0, 1,, d 1 R d (k + 1) πj 2 : l 1 R j 0 l 1 (j + 1) πk 1(λ) = λ k πj 2( F k c(n)) = F k,j c (n) k = 0, 1,, d 1 Π 1 k : Rd l Π 2 k : (l 1) d l 1 Π 1 k(x) (π 1 k(x), π 1 k 1(x),, π 1 0(x), π 1 d 1(x), π 1 d 2(x),, π 1 0(x), π 1 d 1(x), π 1 d 2(x), ) = (x k, x k 1,, x 0, x d 1, x d 2,, x 0, x d 1, x d 2, ), Π 2 k(y) (π 2 0 π 1 k(y), π 2 1 π 1 k 1(y),, π 2 k π 1 0(y), π 2 k+1 π 1 d 1(y), π 2 k+1 π 1 d 2(y), ) = (y k,0, y k 1,1,, y 0,k, y d 1,k+1, y d 2,k+2,, y 0,k+d, y d 1,k+d+1, y d 2,k+d+2, ). Π 1 k ( ) Π 2 k ( ) Π 1 k(x) Π 2 k(y) = x [k j] y [k j],j, Π 1 k ( ) Π 2 k ( ) : l l 1 R z (l 1 ) d f z : R d R d (l 1 ) d R d g : (l 1 ) d R d f z (x (1), x (2), y) (f z,0 (x (1), x (2), y), f z,1 (x (1), x (2), y),, f z,d 1 (x (1), x (2), y)), g(y) ( y 0,j, y 1,j,, y d 1,j ), f z,k : R d R d (l 1 ) d R f z,k (x (1), x (2), y) Π 1 k(x (1) ) Π 2 k(y) + Π 1 k(x (2) ) Π 2 k(z), 0 k d 1. f z (x (1), x (2), y) g(y) f g y (R ) d g 0 (y 0 ) y 0,j R R y (i) 0,j I {i=j} i, j 0 y (i) 0 (y (i) 0,0, y(i) 0,1, ) 0 i R y (i) 0 0 i lim g 0(y (i) 0 ) = 1 0 = g 0(0) i 0 R d 0 k = 0 k ( ) (C1) (ˆλˆλˆλ(n), ˆF c (n)) (Λ, Γ ) R d (l 1 ) d n, (C2) ˆRˆRˆR(n) 0 n, (C3) W = g(f F c ) L = f F c (0, λ, 0).

6 (C1) (C2) ˆF c (n) (l 1 ) d Γ (l 1 ) d (C3) f z g ( ( λ λ λ(n), F c (n), Q Q Q(n), ) L L L(n), W W W (n), (ˆλˆλˆλ(n), ˆF c (n), ˆQˆQˆQ(n), ˆLˆLˆL(n), ŴŴŴ (n), ˆRˆRˆR(n) )) (( λ,f F c, L, L, W ), ( Λ, Γ, Υ, Υ, Ω, 0 )) (R d (l 1 ) d R 3d ) (R d (l 1 ) d R 4d ), Ω = g(γ ) Υ = f F c (λ, Λ, Γ ) Ω k = Γ k,j Υ k = λ [k j] Γ [k j],j + Λ [k j] F c [k j],j k 0. (l 1 ) d (C1) ˆF c (n) Γ (l 1 ) d l 1 B B (l 1 ) d (l 1 ) d l 1 l 1 l 1 l 1 l 1 (l 1 ) d 1,d (l 1 ) = l (l 1 ) d (l ) d (l 1 ) d U (n) (U (n) 0,..., U (n) d 1 ) (l 1) d U (n) k (U (n) k,j : j 0) l 1 U {U (n) } U h(u (n) ) h(u) h ((l 1 ) d ) = (l ) d {U (n) } ϵ > 0 K (l 1 ) d P (U (n) K) 1 ϵ n (l 1 ) d n 1 O n {(y (R ) d : y k,j = 0, 0 k d 1, j n 1}, O = n=1o n (l ) d O (l ) d (l ) d y (l ) d {y (i) } O y { y (i) k,j = y k,j, j < i, 0,, 0 k d 1 i 1 y (i) O y ((l ) d ) = (l,s ) d l,s {x = (x 0, x 1, ) R : i=0 x i < } x ((l ) d ) = (l,s ) d x(y (i) ) = d 1 k=0 d 1 x k,j y (i) k,j = i 1 k=0 x k,j y k,j d 1 k=0 x k,j y k,j = x(y) i, y (i) y O (l ) d

7 (l 1 ) d ( (l 1 ) d ) K (l 1 ) d K ϵ > 0 J ϵ y K d 1 k=0 j=j ϵ y k,j < ϵ K (l 1 ) d K ϵ > 0 {y (i) } N K K N ϵ J ϵ d 1 k=0 j=j ϵ y (i) k,j < ϵ y(i) K y K y (i) y y (i) 1,d < ϵ d 1 d 1 y k,j y k,j y (i) k,j + d 1 y (i) k,j d 1 y k,j y (i) k,j + d 1 y (i) k,j k=0 j=j ϵ k=0 j=j ϵ k=0 j=j ϵ k=0 k=0 j=j ϵ d 1 d y y (i) 1,d + dϵ + ϵ. k=0 j=j ϵ y (i) k,j K (l 1 ) d ϵ > 0 J ϵ y K d 1 k=0 j=j ϵ y k,j < ϵ (l 1 ) d K K K C/d y 1,d C y K ϵ > 0 J ϵ K 0 {x (R Jϵ ) d : x 1 C} {x (i) } N ) d ϵ K (RJϵ 0 {y (i) } N (l 1) d {x (i) } N ) d (l (RJϵ 1 ) d y (i) k,j = x(i) k,j j J ϵ 0 y K x y (R Jϵ ) d x 1 d y 1,d C x K 0 x (i) x x (i) 1 ϵ d 1 y y (i) 1,d J ϵ 1 k=0 ϵ + ϵ, d 1 y k,j y (i) k,j + k=0 j=j ϵ y k,j y (i) k,j x d 1 x(i) 1 + y k,j j=j ϵ K ϵ (l 1 ) d {y (i) } N K (l 1 ) d m, C > 0 d 1 K m,c {x (l 1 ) d : x 1,d C, k=0 j=m n y k,j < n 1 k=0 n 1} ( (l 1 ) d ) U (n) U (l 1 ) d ( ) ϵ > 0 m C K m,c ( ) J 0 J < P (U (n) K m,c ) > 1 ϵ (U (n) (n) k,j : 0 k d 1; 0 j J) (U k,j : 0 k d 1; 0 j J) RdJ. (ii) ( ) J 0 J < {a k,j : 1 k d 1, 0 j J} d 1 J k=0 J=0 d 1 a k,j U (n) k,j k=0 J a k,j U k,j.

8 {U (n) } y(u (n) ) y(u) y O ((l 1 ) d ) U (n) U (l 1 ) d l 1 l 1 B q C q x i B ( x i q ) 1/q C q ϵ i x i, i i {ϵ i } P (ϵ i = 1) = P (ϵ i = 1) = 1/2 l 1 2 U B h B h(x) U B Eh(U) = 0 Eh 2 (U) < h B B U = (U 0, U 1, ) l 1 (E U k 2 ) 1/2 < ; k=0 U U n 1/2 n U (i) U (i) U U B h(u) h B l 1 U = (U 0, U 1, ) l 1 U U Y Fj c P (Y j) k = 0, 1, Y F j 1 Fj c Y (i) Y U (i) j I {Y (i) j} F c j. U (i) l 1 U Eh(U) = 0 h l h (l 1 ) U (E(I {Y (1) j} Fj c ) 2 ) 1/2 = (F j Fj c ) 1/2 <. k=0 k=0 F c j O(j (2+ϵ) ) ϵ > 0. Eh(U) = 0 Eh 2 (U) < h l U

9 (ˆλˆλˆλ(n), ˆF c (n)) (Λ, Γ ) C1 C2 (C1) R d (R J ) d J (C1) (C3) C2 (Λ, Γ ) (Λ, Λ) = Σ Λ (π 1 k (Γ ), π1 l (Γ )) = (Γ k, Γ l ) = Σ Γ :k,l (Λ, π 1 k (Γ )) = (Λ, Γ k ) = Σ Λ,Γ :k 1 k, l d 1 (Ω, Υ ) (Ω, Ω) k,l = (Υ, Υ ) k,l = (Ω, Υ ) k,l = j=1 i=0 + Σ Γ :k,l i,j, λ [k i] λ [l j] Σ Γ :[k i],[l j] i+1,j+1 + F[k i],i c λ Λ,Γ :[l j] [l j]σ i=0 λ [l j] Σ Γ :k,[l j] i,j+1 + i=0 [k i]+1,j+1 + i=0 λ [k i] F[l j],j c :[k j] ΣΛ,Γ [l j]+1,i+1 F[l j],j c :k ΣΛ,Γ [l j]+1,i, F c [k i],i F c [l j],j ΣΛ [k i]+1,[l j]+1, [x] x mod d ( ) L L L(n) Q Q Q(n) ˆLˆLˆL(n) ˆQˆQˆQ(n) Υ k Λ Γ Υ k k m = 1, 2, A m = (A 0+(m 1)d, A 1+(m 1)d,, A d 1+(m 1)d ) R d. {A m } m = 1, 2, EA m = λ > 0 (A m ) = Σ Λ k Fk,j c O(j (3+δ) ) k δ > 0

10 (Λ, Γ ) R d (l 1 ) d Λ N(0, Σ Λ ) (Γ k,j, Γ k,s ) = λ k F k,j Fk,s c 0 k d 1 0 j s Λ, Γ 0, Γ 1,, Γ d 1 (Ω, Υ ) 1 k l d 1 (Ω, Ω) k,l = (Υ, Υ ) k,l = (Ω, Υ ) k,l = { λk i=0 F k,min{i,j}fk,max{i,j} c, k = l, 0, k l, k λ 3 s( s=0 + + l s=k+1 d 1 s=l+1 m=0 n=0 λ 3 s( λ 3 s( F s,min{k s+md,l s+nd} Fs,max{k s+md,l s+nd} c ) m=1 n=0 m=1 n=1 d 1 d 1 + c k,i c k,j Σi,j, Λ i=0 m=0 c k,j = F s,min{k s+md,l s+nd} Fs,max{k s+md,l s+nd} c ) F s,min{k s+md,l s+nd} Fs,max{k s+md,l s+nd} c ) λ 2 kf k,min{i,l k+md} Fk,max{i,l k+md} c. { m=0 F c j,k j+md 0 j k, m=1 F c j,k j+md k + 1 j d 1. ( (I3)) (I3) 3 + δ (I3) 2 + δ (Ω, Υ ) N m, n Nd i, j Nd d = 1 d = 1 Υ = f F c (λ, Λ, Γ ) = λω + W Λ Λ Λ = λ 3/2 Ũ Ω Ñ( t) t 1/2 k=1 ( W k w) = t Ñ( t) Ñ( t) Ñ( t) t 1/2 ( k=1 = t Ñ( t) Ñ( t) t 1/2 ( k=1 = t Ñ( t) ( t 1/2 ( Ñ( t) k=1 W k Ñ( t) w) W k λ t w + λ t w Ñ( t) w) W k λ t w) t 1/2 w(ñ( t) λ t) ) λ 1 ( λ 1/2 ( W wũ) + w λ 3/2 Ũ) = Ω,

11 λ 1/2 W = λω + λ 1/2 wũ w λ 3/2 Ũ Υ ˆQˆQˆQ(n) Υ ˆLˆLˆL(n) λ 1/2 ( W wũ) = λω + λ 1/2 wũ w λ 3/2 Ũ λ 1/2 wũ = λω w λ 3/2 Ũ = λω + wλ. λ w λ W d = 1 t 1/2( Ñ( t) Ñ( t) λ t, k=1 W k Ñ( t) w ) (Λ, Ω), C3 C3 ˆLˆLˆL(n) ˆQˆQˆQ(n) f z (x (1), x (2), y) R d R d (l 1 ) d R d f z,k (x (1), x (2), y) R d R d (l 1 ) d R x (i) = (x (i) 0, x(i) 1,, x(i) d 1 ) Rd i = 1, 2 y = (y 0, y 1,, y d 1 ) (l 1 ) d z = (z 0, z 1,, z d 1 ) (l 1 ) d R d x max x k x = (x 0, x 1,, x d 1 ) R d 0 k d 1 R d l Π 1 k Π2 k Π1 k (x) = x < Π 2 k (y) 1 d 1 k=0 y k 1 < Π 1 k (x) l Π 2 k (y) l 1 Π 1 k (x) Π1 k ( x x x) = Π1 k (x x x x) Π k 2(y) Π2 k (ỹỹỹ) = Π2 k (y ỹỹỹ) x, x x x R d y,ỹỹỹ (l 1 ) d (x (1), x (2), y) ( x x x (1), x x x (2),ỹỹỹ) (x (1), x (2), y) ( x x x (1), x x x (2),ỹỹỹ) R d R d (l = d 1 1) d x(1) x x x (1) + x (2) x x x (2) + y k ỹỹỹ k 1 < δ, k=0

12 f z,k (x (1), x (2), y) f z,k ( x x x (1), x x x (2),ỹỹỹ) = Π 1 k(x (1) ) Π 2 k(y) + Π 1 k(x (2) ) Π 2 k(z) Π 1 k( x x x (1) ) Π 2 k(ỹỹỹ) Π 1 k( x x x (2) ) Π 2 k(z) Π 1 k(x (1) ) Π 2 k(y) Π 1 k( x x x (1) ) Π 2 k(ỹỹỹ) + (Π 1 k(x (2) x x x (2) )) Π 2 k(z) Π 1 k(x (1) ) (Π 2 k(y ỹỹỹ) + (Π 1 k(x (1) x x x (1) )) (Π 2 k(y ỹỹỹ)) + (Π 1 k(x (1) x x x (1) )) Π 2 k(y) + (Π 1 k(x (2) x x x (2) )) Π 2 k(z) Π 1 k(x (1) ) Π 2 k(y ỹỹỹ 1 + Π 1 k(x (1) x x x (1) ) Π 2 k(y ỹỹỹ) 1 + Π 1 k(x (1) x (1) ) Π 2 k(y) 1 + Π 1 k(x (2) x x x (2) ) Π 2 k(z) 1 d 1 d 1 x (1) δ + δ 2 + δ y k 1 + δ z k 1 0 δ 0, d=0 d=0 f z,k f z g(y) = f 0 (e, 0, y) e = (1, 1,, 1) R d g ˆF c (n) (l 1 ) d F c (l 1 ) d ( λ λ λ(n), F c (n)) (λ,f F c ) R d (l 1 ) d. W W W (n) = g( F c (n)) W = g(f F c ) ( λ λ λ(n), W W W (n), F c (n)) (λ, W,F F c ) R 2d (l 1 ) d. L L L(n) L L L(n) ( λ λ λ(n), F c (n)) L L L(n) = f 0 ( λ λ λ(n), 0, F c (n)) ( λ λ λ(n), W W W (n), L L L(n), F c (n)) (λ, W, L,F F c ) R 3d (l 1 ) d. ˆRˆRˆR(n) 0 R R R(n) = L L L(n) Q Q Q(n) 0 ρ(( λ λ λ(n), W W W (n), L L L(n), F c (n)), ( λ λ λ(n), W W W (n), Q Q Q(n), F c (n))) 0, ρ R 3d (l 1 ) d ( λ λ λ(n), W W W (n), Q Q Q(n), L L L(n), F c (n)) (λ, W, L, L,F F c ) R 4d (l 1 ) d. ŴŴŴ (n) = g( ˆF c (n)) (ˆλˆλˆλ(n), ŴŴŴ (n), ˆF c (n)) (Λ, Ω, Γ ) R 2d (l 1 ) d, Ω = g(γ ) ( λ λ λ(n), W W W (n), Q Q Q(n), L L L(n), ˆλˆλˆλ(n), ŴŴŴ (n), F c (n), ˆF c (n)) (λ, W, L, L, Λ, Ω,F F c, Γ ) R 6d (l 1 ) 2d. ˆLˆLˆL(n)

13 k = 0, 1,, d 1 k ˆLˆLˆL(n) n( Lk (n) L k ) = ( k n( λ k j (n) F k j,j(n) c + ( k λ k j Fk j,j c + d m=1 j=1 m=1 j=1 d λ d j (n) F d j,(m 1)d+j+k c (n)) λ d j Fd j,(m 1)d+j+k c ) ) = ( k n ( λk j (n) F k j,j(n) c λ k j (n)fk j,j c + λ k j (n)fk j,j c λ k j Fk j,j) c = + m=1 j=1 d ( λd j (n) F d j,(m 1)d+j+k c (n) λ d j (n)fd j,(m 1)d+j+k c + λ d j (n)fd j,(m 1)d+j+k c λ d jfd j,(m 1)d+j+k c ) ) k + ( πk j( λ λ λ(n) 1 ) π 2 j πk j 1 ( ˆF c (n) ) + πk j(ˆλˆλˆλ(n) 1 ) ) π 2 j πk j(f 1 F c ) d m=1 j=1 +π 1 d j ( πd j( λ λ λ(n) 1 ) π 2 (m 1)d+j+k πd j 1 ( ˆF c (n) ) ) ) (ˆλˆλˆλ(n) π 2 (m 1)d+j+k πd j(f 1 F c ) = Π 1 k( λ λ λ(n) ) Π 2 k ( ˆF c = f F c,k( λ λ λ(n), ˆλˆλˆλ(n), ˆF c (n)), ˆF c (n) ) + Π 1 ) k(ˆλˆλˆλ(n) Π 2 k(f F c ) ˆLˆLˆL(n) = f F ( λ λ λ(n), ˆλˆλˆλ(n), ˆF c c (n) ). f F c Γ (l 1 ) d ( λ λ λ(n), W W W (n), Q Q Q(n), L L L(n), ˆλˆλˆλ(n), ŴŴŴ (n), ˆLˆLˆL(n), F c (n), ˆF c (n)) (λ, W, L, L, Λ, Ω, Υ,F F c, Γ ) R 7d (l 1 ) 2d, Υ = f F c (λ, Λ, Γ ) ˆRˆRˆR(n) 0 ˆLˆLˆL(n) ˆQˆQˆQ(n) 0 ( λ λ λ(n), W W W (n), Q Q Q(n), L L L(n), ˆλˆλˆλ(n), ŴŴŴ (n), ˆQˆQˆQ(n), ˆLˆLˆL(n), F c (n), ˆF c (n)) (λ, W, L, L, Λ, Ω, Υ, Υ,F F c, Γ ) R 8d (l 1 ) 2d. ˆRˆRˆR(n) X i N(µ i, σi 2) i = 1, 2, X n X n X N(µ, σ 2 ) µ = lim µ i σ 2 = lim n n σ2 i X n X L 2 n

14 ϕ Xi (t) = Ee itxi = exp(iµ i t 2 1 σi 2t2 ) X i X n X lim n EeitXi = Ee itx t lim µ i = µ lim n n σ2 i = σ2 µ σ 0 lim mu i = lim n n σ2 i = Ee itx = exp(iµt 2 1 σ 2 t 2 ) X N(µ, σ 2 ) lim µ i = µ lim n n σ2 i = σ2 sup EXn 4 = sup(µ 4 n + 6µ 2 nσn 2 + 3σn) 4 < n n {Xi 2} X n X L 2 n X = a in i a T N Y = b in i b T N N (N, N) = Σ N (X, Y ) (X, X) = (X) = a i a j Σi,j N <, j=1 (Y, Y ) = (Y ) = b i b j Σi,j N < j=1 (X, Y ) = E(XY ) = a i b j Σi,j N <. j=1 α β R αx + βy X n = n a in i Y n = n b in i X Y αx n + βy n αx + βy αx n + βy n n n j=1 (αa i + βb i )(αa j + βb j )Σi,j N αx + βy j=1 (αa i + βb i )(αa j + βb j )Σi,j N (X, Y ) E(XY ) X n X Y n Y L 2 X n Y n XY L 1 E(XY ) = lim E(X ny n ) = n n n j=1 a ib j Σi,j N = j=1 a ib j Σi,j N < X Y (X, Y ) = E(XY ) (X, X) (Y, Y ) X = Y π 1 k (Λ) = Λ k Ga k,j = π 2 j π1 k (Γ ) k = 0, 2,, d 1 j = 0, 1, 2 Γ (l 1 ) d Ω = g(γ ) Υ = f F c F c F c (λ, Λ, Γ ) (Ω, Υ )

15 L 2 ( (Ω, Ω)) k,l = ( πj 2 πk(γ 1 ), πj 2 πl 1 (Γ )) = Σ Γ :k,l i,j, ( (Υ, Υ )) k,l = (f F c,k(λ, Λ, Γ ), f F c,l(λ, Λ, Γ )) = ( πj 2 (Π 1 k(λ))πj 2 (Π 2 k(γ )) + j=1 πj 2 (Π 2 k(f F c ))πj 2 (Π 1 k(λ)), πj 2 (Π 1 l (λ))πj 2 (Π 2 l (Γ )) + πj 2 (Π 2 l (F F c ))πj 2 (Π 1 l (Λ))) = ( λ [k j] Γ [k j],j + F[k j],j c Λ [k j], = i=0 + λ [k i] λ [l j] Σ Γ :[k i],[l j] i+1,j+1 + F[k i],i c λ Λ,Γ :[l j] [l j]σ i=0 i=0 [k i]+1,j+1 + i=0 ( (Ω, Υ )) k,l = ( πj 2 πk(γ 1 ), πj 2 (Π 1 l (λ))πj 2 (Π 2 l (Γ )) + = λ [l j] Σ Γ :k,[l j] i+1,j+1 + λ [l j] Γ [l j],j + λ [k i] F[l j],j c :[k j] ΣΛ,Γ [l j]+1,i+1 F c [l j],j Λ [l j]) F c [k i],i F c [l j],j ΣΛ [k i]+1,[l j]+1, F[l j],j c :k ΣΛ,Γ [l j]+1,i+1, πj 2 (Π 2 l (F F c ))πj 2 (Π 1 l (Λ))) [x] x mod d W k k = 0, 1,, d 1 F c k,j 1 F k,j P (W k j) O(j (3+δ) ) j 0 δ > 0 W (i) k (I W (i) 0, I k W (i) 1, ) F c k (F k,0 c, F k,1 c k R d EY (i) = µ Y I (i) k W (i) k, ) X(i) k µ Y I(i) k W k F c k Y (i) µ Y (µ Y,0, µ Y,1,, µ Y,d 1 ) > 0 (Y (i) ) = Σ Y Y (j) k = 0, 1,, d 1 i, j 1 S(n) (S 0 (n), S 1 (n),, S d 1 (n)) S 0(n) G(n) (G 0 (n), G 1 (n),, G d 1 (n)) ( n S1(n) 0, Y (i) Sd 1(n) 1,, d 1 ). n 1/2 (S(n) nµ µ Y, G(n)) (Λ, Γ (Γ 0, Γ 1,, Γ d 1 )) R d (l 1 ) d, (Λ, Γ ) R d (l 1 ) d Λ N(0, Σ Y ) (Γ k,j, Γ k,s ) = µ Y,k F k,j F c k,s 0 k d 1 0 j s Λ, Γ 0, Γ 1,, Γ d 1

16 n 1/2 (S(n) nµ µ Y ) Λ N(0, Σ Y ). Z k l 1 0 k d 1 Z k (Z k,j, Z k,l ) = F k,j Fk,l c S k (n) (S k (n)) 1/2 S k (n) n 1/2 k Z k l 1 0 k d 1, k Γ k = µ 1/2 Y,k Z k l 1 0 k d 1. Γ k (Γ k,j, Γ k,s ) = µ Y,k F k,j Fk,s c j s Y (i) {ỸY (i) } Y (i) S(n) ( S 0 (n), S 1 (n),, S d 1 (n)) n ỸY (i) n 1/2 (S(n) nµ µ Y ) n 1/2 G 0 (n) n 1/2 S 0(n) n 1/2 (S(n) nµ µ Y, G 0 (n)) (Λ, Γ 0 ) R d l 1, Λ Γ 0 n 1/2 G 0 (n) G 0 (n) 1 0 n ϵ > 0 0 S P (n 1/2 G 0 (n) G 0(n) 0 (n) 1 > ϵ) = P ( S 0(n) P ( 2P ( S0(n) 0 max I=1,2,, n 3/4 =2P ( max 2P ( I I=1,2,, n 3/4 max I=1,2,, n 3/4 S0(n) > n 1/2 ϵ) 0 1 > n 1/2 ϵ, S 0 (n) S 0 (n) n 3/4 ) + P ( S 0 (n) S 0 (n) > n 3/4 ) 0 1 > n 1/2 ϵ) + P ( S 0 (n) S 0 (n) > n 3/4 ) I I 0,j > n1/2 ϵ) + P ( S 0 (n) S 0 (n) > n 3/4 ) 0,j > n1/2 ϵ) + P ( S 0 (n) S 0 (n) > n 3/4 ). δ 1 δ/2 C ( j (1+δ1) ) 1 ( 0,j ) = F 0,jF c 0,j P ( 3 27 max I=1,2,, n 3/4 max I=1,2,, n 3/4 P ( I 0,j > n1/2 ϵ) I C 2 n 1 ϵ 2 j 2+2δ1 n 3/4 F 0,j F0,j c 27C 2 ϵ 2 n 1 n 3/4 P ( max I=1,2,, n 3/4 0,j > Cn1/2 ϵj (1+δ1) /3) 27 I 0,j > Cn1/2 ϵj (1+δ1) ) max I=1,2,, n 3/4 ( I X(i) 0,j ) C 2 nϵ 2 j (2+2δ1) j 2+2δ1 F0,j c 0 n,

17 Fk,j c O(j (3+δ) ) j2+2δ1 F0,j c P ( S 0 (n) S 0 (n) > n 3/4 ) (S 0(n) S 0 (n)) n 3/2 2nΣY 1,1 n 3/2 0 n. 0 n n 1/2 (S(n) nµ µ Y, G 0 (n)) (Λ, Γ 0 ) R d l 1. Y (i) G 1 (n), G 2 (n),, G d 1 (n) λ = E(A 0+(m 1)d, A 1+(m 1)d,, A d 1+(m 1)d ) F c W L W (C1) Y (i) A i W (i) k i k + (m 1)d m = 1, 2, EW (i) k = W k σw,k 2 (i) (W k ) < (C1) (C2) ˆRˆRˆR(n) 0 ˆRˆRˆR(n) = ˆLˆLˆL(n) ˆQˆQˆQ(n) = n 1/2 ( L L L(n) Q Q Q(n)) 0 k d 1 n 1/2 ( L k (n) Q k (n)) 0 n F c L k (n) Q k (n) = n 1 n d Y d j+(m 1)d,j+k+(s 1)d. m=1 j=1 s=n m+1 ϵ > 0 P (n 1/2 ( L k (n) Q k (n)) > ϵ) n d =P (n 1/2 Y d j+(m 1)d,j+k+(s 1)d > ϵ) m=1 j=1 s=n m+1 d n P ( Y d j+(m 1)d,j+k+(s 1)d > n 1/2 d 1 ϵ) j=1 m=1 s=n m+1 d (P ( j=1 + P ( n n 1/4 m=1 n Y d j+(m 1)d,j+k+(s 1)d > n 1/2 d 1 ϵ) s=n m+1 m=n n 1/4 +1 s=n m+1 Y d j+(m 1)d,j+k+(s 1)d > n 1/2 d 1 ϵ)). n

18 n n 1/4 P ( m=1 n n 1/4 P ( P ( m=1 n s=n m+1 s= n 1/4 +1 m=1 s= n 1/4 +1 n m=1 n 1/2 dλ d j ϵ 1 Y d j+(m 1)d,j+k+(s 1)d > n 1/2 d 1 ϵ) Y d j+(m 1)d,j+k+(s 1)d > n 1/2 d 1 ϵ) Y d j+(m 1)d,j+k+(s 1)d > n 1/2 d 1 ϵ) s= n 1/4 +1 EY d j+(m 1)d,j+k+(s 1)d n 1/2 dϵ 1 n 1/2 d 1 ϵ s= n 1/4 +1 F c d j,s n 1/2 Cdλ d j ϵ 1 n 1/4 s= n 1/4 +1 s (3+δ) ds λ d j F c d j,j+k+(s 1)d =n 1/2 Cdλ d j (2 + δ) 1 ϵ 1 n 1/4 (2+δ) 0 n. S k (n) = n m=1 A k+(m 1)d n P ( Y d j+(m 1)d,j+k+(s 1)d > n 1/2 d 1 ϵ) P ( m=n n 1/4 +1 s=n m+1 n m=n n 1/4 +1 s=0 S d j ( n 1/4 ) Y d j+(m 1)d,s > n 1/2 d 1 ϵ) = P ( W (i) d j > n1/2 d 1 ϵ) ( S d j( n 1/4 ) W (i) d j ) + (E( S d j ( n 1/4 ) W (i) d j ))2 nd 2 ϵ 2 =n 1 d 2 ϵ 2 ( n 1/4 Σd j+1,d j+1w Λ d j + n 1/4 λ d j σw,d j 2 + n 1/4 2 λ 2 d jwd j) 2 0 n. n 1/2 ( L k (n) Q k (n)) 0 n (C2) (Ω, Υ ) Σ Λ:k,l = 0 k l Σ Λ,Γ :k = 0 k Σ Γ :k,k i+1,j+1 = λ kf k,i Fk,j c 0 i j F k,j c (n) = 0 0 k d 1 n j (C1) Fk,j c = 0 k j ˆF c (n) (l 1 ) d Γ k,j = 0 k j Γ (l 1 ) d (C2) ˆRˆRˆR(n) = ˆLˆLˆL(n) ˆQˆQˆQ(n) = n( L L L(n) Q Q Q(n)) ne(n) 0 E(n) L L L(n) Q Q Q(n) 1 n > N Y k,j = 0 j > Nd ne(n) = n 1 n d j=1 n d m=1 j=1 s=(n m)d n 1/2 n m=n N+1 Y d j+(m 1)d,j+s = n 1/2 n d Nd m=n N+1 j=1 s=(n m)d NdA d j+(m 1)d n 1/2 d 2 N 2 M 0 n, M (C2) Y d j+(m 1)d,j+s

19 L = λw L = λw L = λw L = λw L = λw L = λw 50 th L = λw L = λw L = λw L = λw

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