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ψ₅(w;μ₁, μ₂, σ₁, σ₂, ρ) :=_2π·σ ¹

1σ₂√

1−ρ²3uU

−^σ¹²(^w1−μ₁)²^−2σ1σ₂ρ(^w1−μ₁)(^w2−μ₂)^+σ²2(^w2−μ₂)²

σ₁²σ²₂(1−ρ²)

, VlYSW ψ₁(w;ρ) :=_2π·√¹

1−ρ²3uU

−^w²¹^−2ρw_1−ρ¹^w2²^+w²²

. VlYS$W

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Yt =

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Z_k,t

t∈ZN0 Z_,t

t∈Z$w Z_k,t:= Φ⁻¹

G_k Y_k,t

t∈Z,

Z_,t:= Φ⁻¹ G

Y_,t

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$CqaCj3 03NcCjw R8Zk:h:tj(v₁, v₂)Y V,W r@3N

h∈Zρ_k:v|p(h)<∞. j@3 IR,I ;nccCN ,aRccAcU3,janL j j@3 URCNjvCc 03A

~N30 c

f_k:v|p(ω) := ^∞

h=−∞

ρ_k:v|p(h)·e^−2πiωh, ω∈

−¹₂,¹₂

. VlYkW

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V$W i@3 8RIIRsCN< @RI0c s@3Nv˘:= (v₂, v₁)Cc j@3 0C<RNI a33,jCRN R8v= (v₁, v₂)d

f_k:v|p(ω) =f_k:˘_v|p(ω), VlY:W

f_k:v|p(ω) =ρ_k:v|p(0) +^∞

h=1

ρ_k:˘_v|p(h)·e^+2πiωh+^∞

h=1

ρ_k:v|p(h)·e^−2πiωh. VlY:$W

:

Slf TU3ac

(5)

TaRR8Y Bj3LVW8RIIRsc cCN,3 j@3 IR,I ;nccCN ,aRccA,Raa3IjCRNcρ_k:v|p(h)$w ,RNcjan,jCRN ,RCN,C03c sCj@ j@3 Ra0CNaw V<IR$IW ,aRccA,Raa3IjCRNcρ(h)CN j@3 ;nccCN ,c3Y 7Ra j@3 UaRR8 R8 Cj3LV$W. j@3 G3w R$c3aqjCRN Cc j@j j@3 0C<RNI 8RI0CN< UaRU3ajw j@j sc R$c3aq30 8Ra j@3 IR,I ;nccCN njRAcU3,janL. c33 VDRa0N<3a N0 iEǹ[email protected]. I3LL+YSW. 3uj3N0c 0Ca3,jIw jR j@3 Ua3c3Nj ,c3. CY3Yρ_k|v(−h) =ρ_k|˘_v(h). s@3a3v˘= (v₂, v₁)Cc j@3 0C<RNIIw a3A

3,j30 URCNj ,Raa3cURN0CN< jRvY i@Cc CLUIC3c j@jf_k:v|p(ω) =f_k:v|p(−ω) =f_k:˘_v|p(ω). N0 Cj IcR 8RIIRsc j@j 3\Y VlYkW ,N $3 a33uUa3cc30 c 3\Y VlY:$WY

lYk `3Ij30 IR,I ;nccCN 3NjCjC3c

7aRL j@3 03~NCjCRN R8 j@3 IR,I ;nccCN ,aRccAcU3,janL. Cj Cc URccC$I3 jR 03~N3 a3Ij30 cU3,ja CN j@3 cL3 LNN3a c j@Rc3 L3NjCRN30 8Ra j@3 Ra0CNaw cU3,janL CN 3\Y VSYlWY

/3~NCjCRN lYkY i@3 IR,I ;nccCN q3acCRNc R8 j@3 ,RcU3,janLc_k(ω). j@3 \n0ajna3 cU3,A janLq_k(ω). j@3 LUICjn03 cU3,janLα_k(ω)N0 j@3 U@c3 cU3,janLφ_k(ω). a3 <Cq3N $w

c_k:v|p(ω) :=`3

f_k:v|p(ω)

=ρ_k:v|p(0) +^∞

h=1

,Rc(2πωh)

ρ_k:v|p(h) +ρ_k:˘_v|p(h)

, VlY9W q_k:v|p(ω) :=−BL

f_k:v|p(ω)

= ∞ h=1

cCN(2πωh)

ρ_k:v|p(h)−ρ_k:˘_v|p(h)

, VlY9$W

α_k:v|p(ω) :=KR0

f_k:v|p(ω)

=

c_k:v|p² (ω) +q_k:v|p² (ω), VlY9,W φ_k:v|p(ω) :=a<

f_k:v|p(ω)

∈(−π, π]. VlY90W

−φ_k|˘_v(ω)Y

`3LaG lY:Y 7Ra ;nccCN 0CcjaC$njCRNc. j@3 IR,I ;nccCN ,Raa3IjCRNc sCII Iswc $3 3\nI jR j@3 Ra0CNaw V<IR$IW ,Raa3IjCRNc.^lN0 j@3 IR,I ;nccCN ,RNcjan,jCRNc CN 03~NCjCRNclYS N0lYksCII j@nc ,RCN,C03 sCj@ j@3 Ra0CNaw V<IR$IW q3acCRNc 8Ra LnIjCqaCj3 ;nccCN jCL3 c3aC3cY ,RLUaCcRN R8 j@3 IR,I N0 <IR$I 3cjCLj3c CN j@3 cL3 UIRj Cc j@nc R8 CNj3a3cj s@3N <Cq3N cLUI3 Cc ,RNcC03a30. cCN,3 j@Cc ,RnI0 03j3,j NRNICN3a CNj3a,jCRNc R8 j@3 jCL3 c3aC3c nN03a CNq3cjC<jCRNY

`3LaG lY9Y Bj Cc URccC$I3 jR 03~N3 IR,I ;nccCN NRIR<n3 R8 j@3 c\na30 ,R@3a3N,3 L3NA jCRN30 CN c3,jCRNS$w a3UI,CN< j@3 Ra0CNaw ,aRccA N0 njRAcU3,ja sCj@ j@3 ,Raa3cURN0CN< IRA

@3a3 cCN,3Qk:v|p(ω)CN <3N3aI I,G30 j@3 NC,3 UaRU3ajC3c GNRsN 8aRL j@3 Ra0CNaw <IR$I ,c3Y BN UajC,nIa. j@3 IR,I ;nccCN njRAcU3,jaf_kk:v|p(ω)N0f_:v|p(ω)sCII CN <3N3aI $3 ,RLA UI3u qIn30 8nN,jCRNc. cR N CNcU3,jCRN R8Qk:v|p(ω)Lncj j@nc $3 $c30 RN UIRjc R8 Cjc a3I N0 CL<CNaw Uajc VRa Cjc LUICjn03 N0 U@c3WY KRa3Rq3a. j@3c3 UIRjc 0C0 LRa3 R8j3N j@N NRj jnaN Rnj jR $3 aj@3a @a0 jR CNq3cjC<j3. cCN,3 j@3 3cjCLj3c R8f_kk:v|p(ω)N0f_:v|p(ω)V8Ra cRL3 0CcjaC$njCRNc N0 cRL3 8a3\n3N,C3cωW <q3 qIn3c q3aw ,IRc3 jR y3aR CN j@3 03NRLCNjRaY

li@Cc Cc 0n3 jR j@3 sw j@3 IR,I ;nccCN ,Raa3IjCRN Cc 03~N30. c33iEǹcj@3CL N0 ?n8j@LL3aVlzSkW 8Ra 03jCIcY

9

lYl MRNICN3a ,aRccAcU3,janL NIwcCc qC j@3 IR,I ;nccCN ,Raa3IjCRN Sle

(6)

lY: 2cjCLjCRN

i@3 3cjCLjCRN R8 j@3 IR,I ;nccCN ,aRccAcU3,janL f_k:v|p(h)8aRL c3,jCRN lYl8RIIRsc j@3 cL3 c3jnU j@j sc nc30 CN VDRa0N<3a N0 iEǹ[email protected]. c3,jCRNlYkW 8Ra j@3 3cjCLjCRN R8 j@3 IR,I ;nccCN njRAcU3,janL. sCj@ j@3 R$qCRnc 0C{3a3N,3 j@j cRL3 3uja CN0C,3c a3 N33030 CN j@3 Ua3c3Nj ,c3Y i@3 3cjCLjCRN R8 j@3 a3Ij30 cU3,jac_k:v|p(h).q_k:v|p(h).α_k:v|p(h) N0φ_k:v|p(h)8aRL c3,jCRNlYkCc j@3N R$jCN30 8aRL j@3 3cjCLj3 R8f_k:v|p(h)CN N R$qCRnc LNN3aY

/3~NCjCRN lY:Y 7Ra cLUI3 yt=

y_1,t, . . . , y_d,tn

t=1R8 cCy3n8aRL LnIjCqaCj3 jCL3 c3A aC3c. NmAjanN,j30 3cjCLj3f_k:v|p^m (ω)R8f_k:v|p(ω)Cc ,RNcjan,j30 $w L3Nc R8 j@3 8RIIRsCN<

UaR,30na3Y

VW mc3 j@3 nNCqaCj3 La<CNIc y_k,tn

t=1 N0 y_,tn

t=1 jR ~N0 3cjCLj3cG_k:nN0G_:nR8 j@3 ,Raa3cURN0CN< La<CNI ,nLnIjCq3 0CcjaC$njCRN 8nN,jCRNc. N0 ,RLUnj3 8aRL j@Cc j@3 Uc3n0RANRaLICc30 R$c3aqjCRNc

z_k,t:= Φ⁻¹ G_k:n

y_k,tⁿ

t=1N0

z_,t:= Φ⁻¹ G_:n

y_,tⁿ

t=1Y V$W +a3j3 j@3 I<hUc3n0RANRaLICc30 UCac

z_k,t+h,z_,tn−h

t=1 8Rah= 0, . . . , m. N0 3cjCA Lj3 8Ra j@3 URCNjv= (v₁, v₂)j@3 IR,I ;nccCN ,aRccA,Raa3IjCRNc

ρ_k:v|p(h|^b^h)m h=0. s@3a3 j@3{bh}^mh=0Cc j@3 $N0sC0j@c j@j a3 nc30 8Ra j@3 0C{3a3Nj I<cY

V,W +a3j3 j@3 I<hUc3n0RANRaLICc30 UCac

z_,t+h,z_k,tn−h

t=1 8Rah= 1, . . . , m. N0 3cjCA Lj3 8Ra j@3 0C<RNIIw a33,j30 URCNjv˘= (v₂, v₁)j@3 IR,I ;nccCN ,aRccA,Raa3IjCRNc ρ_k:˘_v|p(h|^b^h)m

h=0Y

V0W 0Encj 3\Y VlY:$W 8aRL I3LLlYlV$WsCj@ cRL3 I<AsCN0Rs 8nN,jCRNλ_m(h)jR <3j j@3 3cjCLj3

f_k:v|p^m (ω) :=ρ_k:v|p(0) + m h=1

λ_m(h)·ρ_k:˘_v|p(h)·e^+2πiωh+ m h=1

λ_m(h)·ρ_k:v|p(h)·e^−2πiωh,

VlYfW s@3a3 j@3{bh}^mh=0@c $33N cnUUa3cc30 8aRL j@3 NRjjCRN CN Ra03a jR <3j LRa3 ,RLU,j 8RaLnIY

/3~NCjCRN lY9Y 7Ra LnIjCqaCj3 cLUI3{yt}ⁿt=1R8 cCy3n. c 03c,aC$30 CN 03~NCjCRNlY:. j@3 mAjanN,j30 3cjCLj3c R8 j@3 IR,I ;nccCN q3acCRNc R8 j@3 ,RcU3,janL. \n0ajna3 cU3,janL.

LUICjn03 cU3,janL N0 U@c3 cU3,janL Cc <Cq3N $w

c^m_k:v|p(ω) :=`3

f_k:v|p^m (ω)

=ρ_k:v|p(0) +^∞

h=1

,Rc(2πωh)

ρ_k:v|p(h) +ρ_k:˘_v|p(h)

, VlYeW q_k:v|p^m (ω) :=−BL

f_k:v|p^m (ω)

=^∞

h=1

cCN(2πωh)

ρ_k:v|p(h)−ρ_k:˘_v|p(h)

, VlYe$W

α^m_k:v|p(ω) :=KR0

f_k:v|p^m (ω)

=

c^m_k:v|p(ω)2

+

q_k:v|p^m (ω)2

, VlYe,W

φ^m_k:v|p(ω) :=a<

f_k:v|p^m (ω)

∈(−π, π]. VlYe0W

`3LaG lYfY i@3 ,RLL3Njc CN VDRa0N<3a N0 iEǹcj@3CL.lzSe. a3LaGclY9jRlY4W @RI0c 8Ra j@3 Ua3c3Nj ,c3 jRRY BN UajC,nIa. j@3 3cjCLj30 La<CNI ,nLnIjCq3 0CcjaC$njCRNcG_k:nN0G_:n 8aRL 03~NCjCRNlY:VW,N 3Cj@3a $3 $c30 RN j@3 Va3c,I30W 3LUCaC,I ,nLnIjCq3 0CcjaC$njCRN

f

Sl4 TU3ac

(7)

8nN,jCRNc Ra j@3w ,RnI0 $3 $nCIj nURN IR<cUICN3 j3,@NC\n3 ICG3 j@3 RN3 CLUI3L3Nj30 CNQjN3CL N0 iEǹcj@3CLVlzSfWY 7naj@3aLRa3. 8Ra j@3 cwLUjRjC, CNq3cjC<jCRN. j@3 a<nL3Njc CN VQjN3CL N0 iEǹ[email protected]. b3,jCRN kW a3q3Ic j@j j@3 Uc3n0RANRaLICcjCRN R8 j@3 La<CNIc 0R3c NRj {3,j j@3 ~NI ,RNq3a<3N,3 aj3c. s@C,@ Vc sc 0RN3 CN VDRa0N<3a N0 iEǹ[email protected] CLUIC3c j@j j@3 Ua3c3Nj j@3Ra3jC,I NIwcCc ,N C<NRa3 j@3 0CcjCN,jCRN $3js33N j@3 RaC<CNI R$c3aqjCRNc N0 j@3 Uc3n0RANRaLICc30 R$c3aqjCRNcY

lY9 cwLUjRjC, j@3Raw 8Ra f

_k:v|p^m

( ω )

i@3 cwLUjRjC, j@3Raw 8Ra j@3 IR,I ;nccCN ,aRccAcU3,janLf_k:v|p^m (ω)8RIIRsc 8aRL 83s LCNRa 0EncjL3Njc R8 j@3 cwLUjRjC, j@3Raw j@j sc 03q3IRU30 8Ra j@3 IR,I ;nccCN njRAcU3,jaY c CN VDRa0N<3a N0 iEǹ[email protected]. c3,jCRNlY:W. j@3 ccnLUjCRNc N0 a3cnIjc sCII $3 cjj30 8Ra j@3 RaC<CNI R$c3aqjCRNc CNcj30 R8 j@3 Uc3n0RANRaLICc30 R$c3aqjCRNc. cCN,3 j@Cc LG3c j@3 NIwcCc 3cC3a N0 cCN,3 j@3 ~NI ,RNq3a<3N,3 aj3c a3 nN{3,j30 $w j@Cc 0CcjCN,jCRN. c33 a3LaGlYf8Ra 03jCIcY

lY9YS bRL3 03~NCjCRNc N0 N ccnLUjCRN 8RaYt

c 8Ra j@3 nNCqaCj3 ,c3 CN VDRa0N<3a N0 iEǹ[email protected]. j@3 ccnLUjCRNc jR $3 CLURc30 RN j@3kN0,RLURN3Njc R8 j@3 LnIjCqaCj3 jCL3c c3aC3c{Yt}t∈ZN330 jR $3 U@ac30 a3IjCq3 jR j@3 $CqaCj3 UCac j@j ,N $3 ,a3j30 c 0C{3a3Nj ,RL$CNjCRNc R8 3I3L3Njc 8aRL j@3 nNCqaCj3 La<CNIc

Y_k,t

t∈ZN0 Y_,t

t∈ZY MRj3 j@j j@3 8RI0CN< UaRU3ajw 8aRL Cj3LV0WR8 03~NCjCRNlY:

CLUIC3c j@j Cj Cc cn|,C3Nj jR 8RaLnIj3 j@3 ccnLUjCRN $c30 RN NRNAN3<jCq3 qIn3c R8 j@3 I<hY

/3~NCjCRN lYfY 7Ra cjaC,jIw cjjCRNaw LnIjCqaCj3 jCL3 c3aC3c{Yt}t∈Z. sCj@Yt =

Y_1,t, . . . , Y_d,t . N0 8Ra c3I3,j30 UCa R8 CN0C,3ckN0. 03~N3 j@3 8RIIRsCN< $CqaCj3 UCac 8aRL j@3 nNCqaCj3 La<CNIc

Y_k,t

t∈ZN0 Y_,t

t∈ZY Yk:h:t:=

Y_k,t+h, Y_,t

, h≥0, VlY4W

Yk:h:t:=

Y_,t+h, Y_k,t

, h≥1, VlY4$W

N0 I3jg_k:h(yk:h)N0g_k:h(yk:h)03NRj3 j@3 a3cU3,jCq3 UaR$$CICjw 03NcCjw 8nN,jCRNcY

i@3 $cC, C03 8Ra j@3 ,RNcjan,jCRN R8f_k:v|p(ω)Cc j@j URCNjv= (v₁, v₂)c@RnI0 $3 c3A I3,j30 j s@C,@ 8Ra IIhj@3 03NcCjw 8nN,jCRNcg_k:h(yk:h)R8Yk:h:t sCII $3 UUaRuCLj30 $w ψ_p

yk:h;θv|k:h|p

. s@3a3ψ_pCc RN3 R8 j@3 $CqaCj3 ;nccCN 03NcCjw 8nN,jCRNc 8aRL 3\Y VlYSWY i@3 ,Raa3IjCRNAUaL3j3a 8aRL j@3 UUaRuCLjCN< ;nccCN 03NcCjw 8nN,jCRN sCII $3 03NRj30 ρ_k:v|p(h). N0 Cj sCII $3 a383aa30 jR c j@3 IR,I ;nccCN I<h,aRccA,Raa3IjCRN R8Y_k,t+hN0 Y_,tVCN j@j Ra03aW j j@3 URCNjvY

i@Cc IR,I CNq3cjC<jCRN a3\nCa3c $N0sC0j@ q3,jRab= (b₁, b₂)N0 G3aN3I 8nN,jCRNK(w).

s@C,@ Cc nc30 jR 03~N3Kk:h:b(yk:h−v) :=_b₁¹_b₂K_y

k,h−v₁ b₁ ,^y^,0^−v²

b₂

. s@C,@ CN jnaN Cc nc30 CN q_v|k:h:b|p:=

R²

Kk:h:b(yk:h−v) ψ_p

yk:h;θv|k:h|p

−IR<ψ_p

yk:h;θv|k:h|p

g_k:h(yk:h) 0yk:h,

VlYOW

e

lYl MRNICN3a ,aRccAcU3,janL NIwcCc qC j@3 IR,I ;nccCN ,Raa3IjCRN SlO

(8)

LCNCLCc3a R8 s@C,@ c@RnI0 cjCc8w j@3 q3,jRa 3\njCRN

R²

Kk:h:b(yk:h−v)uk:h|p

yk:h;θv|k:h|p g_k:h(yk:h)−ψ_p

yk:h;θv|k:h|p

0yk:h=0, VlYSzW s@3a3uk:h|p

yk:h;θv|k:h|p

:=∇k:h|pIR<ψ_p

yk:h;θv|k:h|p

Cc j@3 c,Ra3 8nN,jCRN R8ψ_p

yk:h;θv|k:h|p

VsCj@ ∇k:h|p := ∂/∂θv|k:h|pWY mN03a j@3 ccnLUjCRN j@j j@3a3 Cc $N0sC0j@ bk:h:0 cn,@

j@j j@3a3 3uCcjc LCNCLCc3a θv|k:h:b|p R8 3\Y VlYOW s@C,@ cjCc~3c 3\Y VlYSzW 8Ra Nw bsCj@

0<b<bk:h:0. j@Ccθv|k:h:b|psCII $3 a383aa30 jR c j@3 URUnIjCRN qIn3 8Ra j@3 <Cq3N $N0A sC0j@bY

`3LaG lYeY i@Cc UUaR,@ sc CNjaR0n,30 CN LRa3 <3N3aI ,RNj3uj CN?ERaj N0 DRN3cVSOOfW.

s@3a3 Cj sc nc30 jR 03~N3 IR,I UUaR,@ jR 03NcCjw 3cjCLjCRN. N0 j@3 N3s C03 CN ViEǹcA j@3CL N0 [email protected] sc jR 8R,nc nURN j@3 3cjCLj30 IR,I ;nccCN UaL3j3ac θv|k:h|pVCNcj30 R8 j@3 3cjCLj30 03NcCjC3cWY i@3 cwLUjRjC, UaRU3ajC3c R8 j@3 3cjCLj30 UaLA 3j3ac sc CNq3cjC<j30 CN ViEǹcj@3CL N0 [email protected] $w j@3 @3IU R8 j@3 FICLGRAM3IcRN UUaR,@^k N0 cnCj$Iw 03~N30 IR,I U3NIjw 8nN,jCRN Q_v|k:h:n|p

θv|k:h|p

Vc33 3\Y V#YSW CN UU3N0Cu#YSWY

i@3 ccnLUjCRNc jR $3 CLURc30 RNY_t Cc a3Ij30 jR j@3 3cjCLjCRN R8 3\Y VlYSzW. N0 j@nc a3\nCa3c 83s 00CjCRNI 03~NCjCRNcY

/3~NCjCRN lYeY 7Raψ_p yk:h;θp

j@3 IR,I ;nccCN 03NcCjw nc30 s@3N UUaRuCLjCN<g_k:h(yk:h) j j@3 URCNjv= (v₁, v₂). N0 8Raθv|k:h:b|pj@3 URUnIjCRN qIn3 j@j LCNCLCc3c j@3 U3NIjw 8nN,A jCRNq_v|k:h:b|p8aRL 3\Y VlYOW. 03~N3 8Ra IIh∈NN0 IIq∈ {1, . . . , p}

u_k:h:q:b(w) := ∂

∂θ_p:qIR<

ψ_p

yk:h;θp

(^yk:h;θp)⁼^w;^θv|k:h:b|p

, VlYSSW

s@3a3∂/∂θ_p:qCc j@3q^j@UajCI 03aCqjCq3 VsCj@ a3cU3,j jRθ_pWY

i@3 8RIIRsCN< a3\nCa3L3Njc RN j@3 G3aN3I 8nN,jCRN a3 C03NjC,I jR j@Rc3 <Cq3N CN VDRa0N<3a N0 iEǹ[email protected]. 03~NCjCRN#YOWY

ki@3 FICLGRAM3IcRN UUaR,@ Vc33FICLGR N0 M3IcRNVSOe4WW c@Rsc @Rs j@3 cwLUjRjC, UaRU3ajC3c R8 N 3cjCLj3 R8 j@3 UaL3j3ac R8 U3NIjw 8nN,jCRNQ,N $3 3uUa3cc30 a3IjCq3 jR j@3 cwLUjRjC, UaRU3ajC3c R8 V3NjCjC3c a3Ij30 jRW j@3 U3NIjw 8nN,jCRN Cjc3I8Y i@3 CNj3a3cj30 a303a ,N ,RNcnIj VDRa0N<3a N0 iEǹcj@3CL.

lzSe. UU3N0Cu#YSW 8Ra LRa3 03jCI30 Ua3c3NjjCRN R8 j@3 FICLGRAM3IcRN UUaR,@ s@3N IR,I U3NIjwA 8nN,jCRN Cc nc30Y

4

Skz TU3ac

(9)

/3~NCjCRN lY4Y 7aRL $CqaCj3. NRNAN3<jCq3. N0 $RnN030 G3aN3I 8nN,jCRNK(w). j@j cjA Cc~3c

R²

K(w₁, w₂)0w10w₂= 1, VlYSlW

K1:k(w₂) :=

R¹

K(w₁, w₂)w^k₁0w₁ Cc $RnN030 8Rak∈ {0,1,2}, VlYSl$W K2:(w₁) :=

R¹

K(w₁, w₂)w₂0w₂ Cc $RnN030 8Ra∈ {0,1,2}, VlYSl,W

R²

K(w₁, w₂)|w^k₁w₂|0w₁0w₂<∞, k, ≥0N0k+≤2· ν, VlYSl0W s@3a3ν >2Cc 8aRL ccnLUjCRNlYSV$WVN0·Cc j@3 ,3CICN< 8nN,jCRNW. 03~N3

Kh:b(yk:h−v) := 1 b₁b₂K

y_h−v₁

b₁ ,y₀−v₂ b₂

. VlYSkW

/3~NCjCRN lYOY #c30 RNYk:h:t.u_k:h:q:b(w)N0Kh:b(yk:h−v). 03~N3 j@3 N3s $CqaCj3 aNA 0RL qaC$I3cX_k:h:q:t^n|v c 8RIIRsc.

X_k:h:q:t^n|v :=

b₁b₂Kh:b(Yk:h:t−v)u_k:h:q:b(Yk:h:t). VlYS:W MRj3 j@j UaR0n,j R8 j@3 aN0RL qaC$I3cX_k:h:q:t^n|v N0X_k:i:r:s^n|v sCII $3 8nN,jCRN R8Y_k,t+h. Y_,t.Y_k,s+iN0Y_,s. s@C,@ 03U3N0CN< RN j@3 ,RN~<najCRN R8 j@3 CN0C,3ch, i, s, tsCII $3 3Cj@3a $CqaCj3. jaCqaCj3 Ra j3jaqaCj3 8nN,jCRNY i@3 3uU3,jjCRN R8 j@3 UaR0n,jX_k:h:q:t^n|v ·X_k:i:r:s^n|v sCII j@nc V03U3N0CN< RN j@3c3 CN0C,3cW 3Cj@3a a3\nCa3 $CqaCj3. jaCqaCj3 Ra j3jaqaCj3 03NcCjw 8nN,jCRNY

ccnLUjCRN lYSY i@3 LnIjCqaCj3 UaR,3cc{Yt}t∈ZsCII $3 ccnL30 jR cjCc8w j@3 8RIIRsCN<

UaRU3ajC3c. sCj@v= (v₁, v₂)VCN Cj3LV0WW j@3 URCNj j s@C,@f_k:v|p^m (ω). j@3 3cjCLj3 R8f_k:v|p(ω).

Cc jR $3 ,RLUnj30Y

VW {Yt}t∈ZCc cjaC,jIw cjjCRNawY

V$W {Y_t}_t∈ZCc cjaRN<Iw LCuCN<. sCj@ LCuCN< ,R3|,C3Njα(j)cjCc8wCN<

∞ j=1

j^a[α(j)]^1−2/ν<∞ 8Ra cRL3ν >2N0a >1−2/ν. VlYS9W V,W pa(||Yt||²)<∞. s@3a3|| · ||Cc j@3 2n,IC03N NRaLY

i@3 $CqaCj3 03NcCjw 8nN,jCRNc g_k:h(yk:h)N0g_k:h(yk:h). ,Raa3cURN0CN< jR j@3 I< hUCac CNjaR0n,30 CN 3\Y VlY4W. Lncj cjCc8w j@3 8RIIRsCN< a3\nCa3L3Njc 8Ra <Cq3N URCNjv= (v₁, v₂)Y

O

lYl MRNICN3a ,aRccAcU3,janL NIwcCc qC j@3 IR,I ;nccCN ,Raa3IjCRN SkS

(10)

V0W g_k:h(yk:h)Cc 0C{3a3NjC$I3 jv. cn,@ j@j iwIRaȕc j@3Ra3L ,N $3 nc30 jR saCj3g_k:h(yk:h)c g_k:h(yk:h) =g_h(v) +g_h(v)[yk:h−v] +R_h(yk:h)[yk:h−v], VlYSfW

s@3a3g_h(v) =

∂

∂y_hg_k:h(yk:h)

y_k:h=v, ^∂

∂y₀g_k:h(yk:h)

y_k:h=v

N0 ICL

y_k:h−→vR_h(yk:h),

sCj@ j@3 cL3 a3\nCa3L3Nj 8Rag_k:h(yk:h)j j@3 0C<RNIIw a33,j30 URCNjv˘= (v₂, v₁)Y V3W i@3a3 3uCcjc $N0sC0j@bk:h:0cn,@ j@j j@3a3 8Ra 3q3aw0<b<bk:h:0Cc nNC\n3 LCNA

CLCc3aθv|k:h:b|pR8 j@3 U3NIjw 8nN,jCRNq_v|k:h:b|p8aRL 3\Y VlYOWY

V8W i@3 ,RII3,jCRN R8 $N0sC0j@c{bk:h:0}h∈Z@c URcCjCq3 CN~LnL. CY3Y j@3a3 3uCcjc bk:0

cn,@ j@j^:

0<bk:0:=CN8

h∈Zbk:h:0, VlYSeW

s@C,@ CLUIC3c j@j j@Ccbk:0,N $3 nc30 cCLnIjN3RncIw 8Ra II j@3 I<cY

V<W 7RaX_k:h:q:t^n|v 8aRL 03~NCjCRNlYO. j@3 a3Ij30 $CqaCj3. jaCqaCj3 N0 j3jaqaCj3 03NcCjw 8nN,jCRNc Lncj $3 cn,@ j@j j@3 3uU3,jjCRNc 2

X_k:h:q:t^n|v

. 2X_k:h:q:t^n|v ^ν N0 2

X_k:h:q:t^n|v ·X_k:i:r:s^n|v II a3 ~NCj3Y

`3LaG lY4Y i@3 Ua3c3Nj ccnLUjCRNlYSCc CN 3cc3N,3 C03NjC,I jR VDRa0N<3a N0 iEǹcj@3CL.

lzSe. ccnLUjCRNlYSW sCj@ cRL3 3uja CN0C,3c. cR j@3 a3LaGc 8aRL VDRa0N<3a N0 iEǹcj@3CL.

lzSeW Cc R8 CNj3a3c3j @3a3 jRRY BN UajC,nIa. j@3αALCuCN< a3\nCa3L3Nj CN Cj3LV$WCLUIC3c j@j Y_k,t+hN0Y_,tsCII $3 cwLUjRjC,IIw CN03U3N03Nj ch→ ∞. CY3Y j@3 $CqaCj3 03NcCjw 8nN,jCRNc g_k:h(yk:h)sCII 8Ra Ia<3 I<chUUaR,@ j@3 UaR0n,j R8 j@3 La<CNI 03NcCjC3c. N0 j@3 cCjnjCRN sCII j@nc cj$CICc3 s@3NhCc Ia<3 3NRn<@Y i@Cc Cc CN UajC,nIa R8 CLURajN,3 8Ra Cj3LV8W. cCN,3 Cj CLUIC3c j@j Cj sCII $3 URccC$I3 jR ~N0 NRNy3aRbk:0j@j sRaGc 8Ra IIhY KRa3Rq3a. j@3

~NCj3N3cc ccnLUjCRNc CN Cj3LV<Wa3 Iswc jaCqCIIw cjCc~30 C8 j@3 a3\nCa30 03NcCjwA8nN,jCRNc a3 ~NCj3Y

lY9Yl N ccnLUjCRN 8Ra

Y_k,t, Y_,t

N0 j@3 c,Ra3 8nN,jCRNup(w;θp)R8ψ_p(w;θp) i@3 8RIIRsCN< ccnLUjCRN Cc CN 3cc3N,3 C03NjC,I jR VDRa0N<3a N0 iEǹ[email protected]. ccnLUA jCRNlYlW. s@C,@ sc CN,In030 0n3 jR j@3 N330 8Ra j@3 cwLUjRjC, a3cnIjc 8aRLiEǹcj@3CL N0

?n8j@LL3aVlzSkW jR $3 UUIC30 8Ra II j@3 0C{3a3Nj I<chY ccnLUjCRN lYlY i@3 ,RII3,jCRN R8 IR,I ;nccCN UaL3j3ac

θv|k:h|p

j j@3 URCNjv8Ra j@3

$CqaCj3 UaR$$CICjw 03NcCjw 8nN,jCRNcg_k:h(yk:h). Lncj II $3 cn,@ j@j VW up(v;θv|k:h|p)=08Ra II ~NCj3hY

V$W ICLu_p(v;θv|k:h|p)=0Y

MRj3 j@j N CNcU3,jCRN R8 j@3p3\njCRNc CNup(w;θp) =0,N $3 nc30 jR C03NjC8w s@3N Cj3LcVWN0V$WR8 ccnLUjCRNlYlLC<@j 8CI. ,8Y j@3 0Cc,nccCRN CN VDRa0N<3a N0 iEǹcj@3CL.

lzSe. c3,jCRNlY:YlW 8Ra 8naj@3a 03jCIcY

:BN3\nICjC3c CNqRIqCN< q3,jRac a3 jR $3 CNj3aUa3j30 CN ,RLURN3NjAsCc3 LNN3aY

Sz

Skl TU3ac

(11)

lY9Yk ccnLUjCRNc 8Ran.mN0b= (b₁, b₂)

i@3 8RIIRsCN< ccnLUjCRN Cc C03NjC,I jR VDRa0N<3a N0 iEǹ[email protected]. ccnLUjCRNlYkWY i@3 CNj3aNI ,RNcCcj3N,w R8 Cj sc q3aC~30 CN VDRa0N<3a N0 iEǹ[email protected]. I3LL+YkWY ccnLUjCRN lYkY H3jm:=m_n→ ∞$3 c3\n3N,3 R8 CNj3<3ac 03NRjCN< j@3 NnL$3a R8 I<c jR CN,In03. N0 I3jb:=bn→0⁺$3 j@3 $N0sC0j@c nc30 s@3N 3cjCLjCN< j@3 IR,I ;nccCN ,Raa3IjCRNc 8Ra j@3 I<ch= 1, . . . , mV$c30 RNnR$c3aqjCRNcWY H3jb₁N0b₂a383a jR j@3 jsR ,RLURN3Njc R8b. N0 I3jα.νN0a$3 c CNjaR0n,30 CN ccnLUjCRNlYSV$WY H3js:=s_n→ ∞

$3 c3\n3N,3 R8 CNj3<3ac cn,@ j@js=o

nb₁b₂/m

. N0 I3jτ $3 URcCjCq3 ,RNcjNjY i@3 8RIIRsCN< a3\nCa3L3Njc Lncj $3 cjCc~30 8Ra j@3c3 3NjCjC3cY⁹

VW IR<n/n(b₁b₂)⁵−→0. VRNIw a3\nCa30 8Ra j@3 ,c3p= 5WY V$W nb₁b₂/m−→ ∞Y

V,W m^δ(b₁∨b₂)−→0, s@3a3δ= 2∨_{ν(a−1)−2}^ν(a+1) Y V0W

nm/b₁b₂·s^τ·α(s−m+ 1)−→ ∞Y V3W m=o

(nb₁b₂)^{τ/(2+5τ)−λ}

, 8Ra cRL3λ∈(0, τ/(2 + 5τ))Y V8W m=o(s)Y

lY9Y: +RNq3a<3N,3 j@3Ra3Lc 8Raf_k:v|p^m (ω).α_k:v|p^m (ω)N0φ_k:v|p^m (ω) b33 UU3N0Cu8Ra j@3 UaRR8c R8 j@3 j@3Ra3Lc cjj30 $3IRsY

n(b₁b₂)^(p+1)/2/m·

c_k:v|p^m (ω) q_k:v|p^m (ω)

−

c_k:v|p(ω) q_k:v|p(ω)

−→d M 0

0

,

σ_c|k:v|p² (ω) 0 0 σ_q|k:v|p² (ω)

,

VlYS4W s@3Nω∈ ¹₂·Z=

. . . ,−1,−¹₂,0,¹₂,1, . . .

. s@3a3 j@3 qaCN,3cσ_c|k:v|p² (ω)N0σ_q|k:v|p² (ω) a3 <Cq3N $w

σ_c|k:v|k|p² (ω) = ICL

m→∞

1 m

-

˜

σ_v|k|p² (0) + m h=1

λ²_m(h)·,Rc²(2πωh)·

˜

σ_v|k|p² (h) + ˜σ_v|k|p_˘² (h).

VlYSOW σ_q|k:v|k|p² (ω) = ICL

m→∞

1 m

- _m

h=1

λ²_m(h)·cCN²(2πωh)·

˜

σ_v|k|p² (h) + ˜σ_v|k|p_˘² (h).

, VlYSO$W sCj@ ˜σ_v|k|p² (h)N0σ˜_v|k|p_˘² (h)j@3 cwLUjRjC, qaCN,3c a3Ij30 jR j@3 3cjCLj3c ρ_k:v|p(h)N0

ρ_k:˘_v|p(h). c33 j@3Ra3L#YS8Ra j@3 03jCIcY

i@3 IR,I ;nccCN \n0ajna3 cU3,janL Cc C03NjC,I jR y3aR s@3Nω∈¹₂·Z. N0 8Ra j@Rc3 8a3\n3N,C3c j@3 8RIIRsCN< cwLUjRjC, a3cnIj @RI0c nN03a j@3 <Cq3N ccnLUjCRNc

n(b₁b₂)^(p+1)/2/m·

f_k:v|p^m (ω)−f_k:v|p(ω)

−→d M

0, σ_c|k:v|p² (ω)

, ω∈¹₂·Z. VlYlzW

9MRjjCRNI ,RNq3NjCRN- Ȕ∨ȕ 03NRj3c j@3 LuCLnL R8 jsR NnL$3ac. s@3a3c Ȕ∧ȕ 03NRj3c j@3 LCNCLnLY

SS

lYl MRNICN3a ,aRccAcU3,janL NIwcCc qC j@3 IR,I ;nccCN ,Raa3IjCRN Skk

(12)

i@3 cwLUjRjC, a3cnIjc 8Ra j@3 IR,I ;nccCN LUICjn03A N0 U@c3AcU3,ja Cc 0Ca3,j ,RNc3A

\n3N,3 R8 j@3Ra3LlYSzN0 V#aR,Gs3II N0 /qCc.SO4f. UaRURcCjCRN fY:Yk. UY lSSWY

i@3Ra3L lYSSY mN03a ccnLUjCRNclYSjRlYk. s@3Nα_k:v|p(ω)>0N0ω∈¹₂·Z. j@3 3cjCLj3

α_k:v|p^m (ω) =

c_k:v|p^m (ω)2

+

q_k:v|p^m (ω)2

cjCc~3c

n(b₁b₂)^(p+1)/2/m·

α_k:v|p^m (ω)−α_k:v|p(ω) _d

−→M(0, σ_α²(ω)), VlYlSW s@3a3σ_α²(ω)Cc <Cq3N a3IjCq3 jRσ_c|k:v|p² (ω)N0σ_q|k:v|p² (ω)V8aRL 3\Y VYOW CN j@3Ra3LlYSzW c

σ_α²=

c²_k:v|p(ω)·σ_c|k:v|p² (ω) +q_k:v|p² (ω)·σ_q|k:v|p² (ω)

/α²_k:v|p(ω). VlYllW i@3Ra3L lYSlY mN03a ccnLUjCRNclYSjRlYk. s@3Nα_k:v|p(ω)>0N0ω∈¹₂·Z. j@3 3cjCLj3 φ_k:v|p^m (ω) =a<c

c_k:v|p^m (ω)−i·q_k:v|p^m (ω) cjCc~3c

n(b₁b₂)^(p+1)/2/m·

φ_k:v|p^m (ω)−φ_k:v|p(ω)

−→d M

0, σ_φ²(ω)

, VlYlkW

s@3a3σ_φ²(ω)Cc <Cq3N a3IjCq3 jRσ_c|k:v|p² (ω)N0σ_q|k:v|p² (ω)V8aRL 3\Y VYOW CN j@3Ra3LlYSzW c σ_φ²(ω) =

q_k:v|p² (ω)·σ_c|k:v|p² (ω) +c²_k:v|p(ω)·σ_q|k:v|p² (ω)

/α⁴_k:v|p(ω). VlYl:W

`3LaG lYOY i@3 cwLUjRjC, NRaLICjw a3cnIjc CN j@3Ra3Lc lYSz jR lYSl0R NRj N3,3ccaCIw

@3IU Ln,@ C8 ,RLUnjjCRNc R8 URCNjsCc3 ,RN~03N,3 CNj3aqIc 8Ra j@3 3cjCLj30 IR,I ;nccCN 3cjCLj3c a3 R8 CNj3a3cj. cCN,3 Cj CN Ua,jC,3 Lw $3 nN83cC$I3 jR ~N0 03,3Nj 3cjCLj3c R8 j@3 qaCN,3cσ_c|k:v|p² (ω)N0σ_q|k:v|p² (ω)j@j R,,nac CN j@3Ra3LlYSzY i@3 URCNjsCc3 ,RN~03N,3 CNj3aqIc sCII j@nc Ij3a RN 3Cj@3a $3 3cjCLj30 $c30 RN cnCj$I3 \nNjCI3c R$jCN30 $w a3U3j30 cLUICN< 8aRL GNRsN 0CcjaC$njCRN. Ra j@3w sCII $3 $c30 RN $RRjcjaUUCN< j3,@NC\n3c 8Ra j@Rc3 ,c3c s@3a3 a3I 0j @c $33N CNq3cjC<j30Y +RN83ai3a¡cqCaj 3j IYVlzSz. ,@Y eYlY9 N0 eYlYfW 8Ra 8naj@3a 03jCIc sCj@ a3<a0 jR j@3 N330 8Ra $RRjcjaUUCN< CN cn,@ cCjnjCRNcY

k 2uLUI3c

i@Cc c3,jCRN sCII CNq3cjC<j3 C8 j@3mAjanN,j30 3cjCLj3c R8 j@3 IR,I ;nccCN ,aRccAcU3,janL f_k:v|p(ω)LC<@j $3 R8 CNj3a3cj jR ,RNcC03a s@3N LnIjCqCaj3 jCL3 c3aC3c a3 3N,RnNj3a30Y bCN,3 f_k:v|p^m (ω)Cc ,RLUI3uAqIn30. j@3 ,jnI CNq3cjC<jCRN sCII $3 $c30 RN UIRjc R8 j@3 ,Raa3cURN0A CN< IR,I ;nccCN q3acCRNc R8 j@3 ,RcU3,janL. \n0ajna3 cU3,janL. U@c3 cU3,janL N0 j@3 LUICjn03 cU3,janLY

i@3 Ua3c3Nj c3jnU Cc cCLCIa jR j@3 RN3 nc30 8Ra j@3 CNq3cjC<jCRN R8 j@3 IR,I ;nccCN njRA cU3,janL. c33 VDRa0N<3a N0 iEǹ[email protected]. c3,jCRNkW. CY3Y Cj sCII ~acj $3 ,@3,G30 j@j j@3 3uU3,j30 a3cnIj Cc R$jCN30 s@3N j@3 RaC<CN R8 j@3 0j Cc $CqaCj3 ;nccCN jCL3 c3aC3cY 8j3a j@Cc cCLnIjCRN 8aRL $CqaCj3 3uj3NcCRN R8 j@3 IR,I jaC<RNRL3jaC, jCL3 c3aC3c 8aRL VDRa0N<3a N0 iEǹ[email protected]. c3,jCRNkYkYlW sCII $3 ,RNcC03a30. N0 j@3a3 sCII LRa3Rq3a

$3 N 3uLUI3 $c30 RN a3I LnIjCqaCj3 0j jR<3j@3a sCj@ cCLnIjCRN 8aRL LR03I $c30 RN j@3c3 0jY c CN VDRa0N<3a N0 iEǹ[email protected]. Cj sCII $3 c33N j@j UIRjc a3Ij30 jR j@3 3cjCLj30 f_k:v|p^m (ω)LC<@j $3 nc38nI c N 3uUIRajRaw jRRI. CY3Y j@j j@Cc UUaR,@ ,N 03j3,j NRNICN3a 03U3N03N,C3c N0 U3aCR0C,CjC3c $3js33N j@3 qaC$I3cY

Sl

Sk: TU3ac

(13)

b3q3aI UaL3j3ac Lncj $3 cU3,C~30 CN Ra03a 8Ra NmAjanN,j30 3cjCLj3 R8f_k:v|p(ω)jR $3 ,RLUnj30. N0 8Ra j@3 3uLUI3c CN j@3 Ua3c3Nj UU3a j@3 8RIIRsCN< qIn3c sCII $3 nc30-

SY p. j@3 NnL$3a R8 UaL3j3ac CN j@3 IR,I ;nccCN UUaRuCLjCRNY QNIw j@3 qIn3p= 5 sCII $3 nc30 CN j@3 Ua3c3Nj 3uLUI3c. cCN,3 j@3 a3cnIjc $c30 RNp= 1CN <3N3aI 8CIc jR ,Ujna3 j@3 IR,I cjan,jna3 CN UaRU3a LNN3a Vc33 j@3 0Cc,nccCRN CN VDRa0N<3a N0 iEǹ[email protected]. c3,jCRNkYeW 8Ra 8naj@3a 03jCIcWY

lY v= (v₁, v₂). j@3 URCNjc jR CNq3cjC<j3Y i@3 Ua3c3Nj CNq3cjC<jCRN sCII ,RNcC03a URCNjc s@Rc3 ~acj N0 c3,RN0 ,RRa0CNj3c ,Raa3cURN0 jR j@3 SzX. 9zX N0 OzX U3a,3NjCI3c R8 j@3 cjN0a0 NRaLI 0CcjaC$njCRN. CY3Y j@3 qIn3c a3−1.28.0N01.28Y BN8RaLjCRN

$Rnj j@3 URCNj R8 CNq3cjC<jCRN Cc ,RNjCN30 CN j@3 nUU3a aC<@j ,RaN3a R8 j@3 a3I3qNj UIRjc.

s@3a3 Cj sCII $3 LaG30 cRyW,,8yWN0 cR RNY

kY ω. j@3 8a3\n3N,C3c jR CNq3cjC<j3Y pIn3c $3js33N0N0¹₂Y

:Y b= (b₁, b₂). j@3 $N0sC0j@Aq3,jRa jR $3 nc30 s@3N ,RLUnjCN< j@3 IR,I ;nccCN njRA ,Raa3IjCRNcY i@3 qIn3b= (.6, .6)@c $33N nc30 8Ra II R8 j@3 ,c3c CN j@Cc UU3aY 9Y m. j@3 janN,jCRN I3q3I. CY3Y j@3 NnL$3a R8 I<c jR CN,In03 CN j@3 3cjCLj3f_v|p^m(ω)Y i@3

qIn3m= 10@c $33N nc30 CN j@Cc CNq3cjC<jCRN. N0 j@Cc NnL$3a Cc $w 038nIj <Cq3N CN j@3 nUU3a I38j ,RaN3a R8 j@3 a3I3qNj UIRjcY

fY λ_m(h). j@3 s3C<@jCN< 8nN,jCRN jR $3 nc30 8Ra j@3 cLRRj@CN< R8 j@3 0C{3a3Nj I<cY i@3 inG3wA?NNCN< I<AsCN0Rs G3aN3I @c $33N nc30 8Ra II j@3 Ua3c3Nj 3uLUI3c. CY3Y

λ_m(h) = ₁

2·

1 +,Rc π·_m^h

|h| ≤m,

0 |h|> m.

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