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(1)

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f(ω) :=

h∈Z

γ(h)·e^−2πiωh. VSYSW

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(2)

i@3 CNq3ac3 7RnaC3a jaNc8RaL <Cq3c j@3 a3IjCRNγ(h) =_1/2

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f(ω) :=

h∈Z

ρ(h)e^−2πiω. VSYlW

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Y_t+h, Y_t

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(3)

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f_ξ(ω) =

h∈Z

ξ_h·e^−2πiωh. VSYkW

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3uU(iuY_t),3uU

ivY_t−|j|

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h(ω, u, v) :=

h∈Z

σ_j(u, v)·e^−2πijω, VSY:W

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γ_h(x₁, x₂) :=+Rq

Y_t+h≤x₁

,{Y_t≤x₂}

, (x₁, x₂)∈R², VSY9W γ_h^U(τ₁, τ₂) :=+Rq

U_t+h≤τ₁

,{U_t≤τ₂}

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(4)

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j@3 qaCjCRN j@jξ_hsCII $3 IR,I L3cna3 R8 Y_t+h, Y_t

j@j 03U3N0c RN URCNjv= (v₁, v₂).

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(5)

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sCII j@3N $3 03~N30 jR $3 j@3 ,Raa3IjCRN R8 j@Cc IR,I ;nccCN UUaRuCLjCRNY

7Ra j@3 UnaURc3 R8 j@Cc CNq3cjC<jCRN. j@3 q3,jRa ,RNjCNCN< j@3 ~q3 UaL3j3acμ₁.μ₂.σ₁.σ₂ N0ρsCII $3 03NRj30 $wθ.^:N0 j@3 $CqaCj3 ;nccCN 03NcCjw 8nN,jCRN sCII $3 03NRj30ψ(w;θ).

CY3Y

ψ(w;θ) :=_2π·σ ¹

1σ₂√

1−ρ²3uU

−^σ¹²(^w¹^−μ¹)²^−2σ¹^σ²^ρ(^w¹^−μ¹)(^w²^−μ²)^+σ²²(^w²^−μ²)²

2σ²₁σ₂²(1−ρ²)

. VlYSW Bj Cc NjnaI jR a3\nCa3 j@j j@3 8RIIRsCN< 3\njCRNc a3 cjCc~30 CN Ra03a 8Raψ(w;θ)jR $3 ,RNcC03a30 <RR0 UUaRuCLjCRN R8g(w)CN N3C<@$Rna@RR0 R8 j@3 URCNjv.

g(v) =ψ(w;θ)|w=v, ∂

∂w₁g(v) = ∂

∂w₁ψ(w;θ)

_w=v, ∂

∂w₂g(v) = ∂

∂w₂ψ(w;θ) _w=v, VlYlW CY3YgN0ψc@RnI0 ,RCN,C03 jvN0 j@3w c@RnI0 @q3 ,RCN,C0CN< jN<3Nj UIN3c j@3a3Y

Bj Cc 3cw jR q3aC8w j@j cRInjCRNθ,N $3 8RnN0 8Ra Nw URCNjv s@3a3g(w)Cc cLRRj@ Ģ

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iR UaRU3aIw ,,RnNj 8Ra j@3 @C<@3a Ra03a j3aLc R8ψ(w;θ)j j@3 URCNjv. j@3 UUaRuCLjCRN L3j@R0 N330c jR CN,In03 N3C<@$Rna@RR0 aRnN0vY UUIwCN< j@3 UUaR,@ nc30 s@3N 3cjCA LjCN< 03NcCjC3c CN?ERaj N0 DRN3cVSOOfW. RN3 ,N ,RNcC03a b→0⁺ICLCj R8 UaL3j3acθb

j@j LCNCLCc3 j@3 U3NIjw 8nN,jCRN q=

K_b(w−v) [ψ(w;θ)−g(w)IR<(ψ(w;θ))]0w, VlYkW s@3a3K_b(w−v)Cc G3aN3I 8nN,jCRN sCj@ $N0sC0j@bY c 3uUICN30 CN V?ERaj N0 DRN3c.

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:i@3 q3,jRaθCc 8nN,jCRN R8 j@3 URCNjv. $nj j@Cc sCII $3 cnUUa3cc30 CN j@3 NRjjCRNY

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(6)

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K_b(w−v)u(w;θ) [ψ(w;θ)−g(w)]0w=0, VlY:W s@3a3u(w;θ) :=_∂θ^∂ IR<(ψ(w;θ))Cc j@3 c,Ra3 8nN,jCRN R8 j@3 UUaRuCLjCN< 03NcCjwψ(w;θ)Y i@3a3 sCII. nN03a cnCj$I3 ccnLUjCRNc. $3 nNC\n3 ICLCjCN< cRInjCRN R8 3\Y VlY:W. CY3Y

θ0= ICL

b→0⁺θb VlY9W

sCII $3 s3IIA03~N30. N0 j@3ρAUaj R8 j@3θ0Aq3,jRa ,N $3 nc30 jR 03~N3 H;+ j j@3 URCNjvY

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μ₁ μ₁

,

σ₁² σ₁σ₂ρ σ₁σ₂ρ σ²₂

, VlYfW

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lYSYl HR,I ;nccCN ,Raa3IjCRN. NRaLICc30 q3acCRN

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i@3 G3w R$c3aqjCRN Cc j@j j@3 NnL3aC,I 3cjCLjCRN UaR$I3L 0R3c NRj R,,na s@3N j@3 La<CNI 0CcjaC$njCRNc a3 cjN0a0 NRaLI A s@C,@ LRjCqj3c N 0Encj30 cjaj3<w cCLCIa jR j@3 ,RUnIA ,RN,3Uj 8aRLbGIaVSO9OWY bGIaȕc j@3Ra3L <Cq3c j@3 3uCcj3N,3 R8 ,RUnIC(u₁, u₂)cn,@ j@j j@3 ERCNj ,08G(w),N $3 3uUa3cc30 cC(G₁(w₁), G₂(w₂)). sCj@G_i(w_i)j@3 La<CNI ,08 ,RaA a3cURN0CN< jRW_iY i@Cc ,RUnIC,RNjCNc II j@3 CNj3a03U3N03N,3 CN8RaLjCRN $3js33N j@3 jsR La<CNI aN0RL qaC$I3cW₁N0W₂. Cj sCII $3 nNC\n3 s@3N j@3 jsR La<CNc a3 ,RNjCNnRnc.

N0 Cj sCII j@3N $3 CNqaCNj nN03a cjaC,jIw CN,a3cCN< jaNc8RaLjCRNc R8 j@3 La<CNcY⁹ mN03a j@Cc ,RNjCNnCjw ccnLUjCRN. j@3 aN0RL qaC$I3W = (W₁, W₂)sCII @q3 j@3 cL3 ,RUnI c j@3 jaNc8RaL30 aN0RL qaC$I3Z:= (Φ⁻¹(G₁(W₁)),Φ⁻¹(G₂(W₂))). s@3a3ΦCc j@3 ,08 R8 j@3 cjN0a0 NRaLI 0CcjaC$njCRN Ģ s@Rc3 ,Raa3cURN0CN< U08 c ncnI sCII $3 03NRj30 $wφY^fi@Cc jaNc8RaL30 q3acCRN R8W@c cjN0a0 NRaLI La<CNc. cR j@3 H;+A3cjCLjCRN I<RaCj@L sCII NRj anN CNjR NnL3aC,I UaR$I3Lc 8Ra j@Cc ,c3 Ģ s@C,@ LRjCqj3c j@3 8RIIRsCN< Ij3aNjCq3 UA UaR,@ jR j@3 03~NCjCRN R8 H;+- BNcj30 R8 ~N0CN< ;nccCN UUaRuCLjCN< R8 j@3 U08g(w) VR8 j@3 RaC<CNI aN0RL qaC$I3 WW j URCNjv. ~N0 ;nccCN UUaRuCLjCRN R8 j@3 U08

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(7)

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g(w)=c(G₁(w₁), G₂(w₂))g₁(w₁)g₂(w₂) UUaRuCLj3 jv= (v₁, v₂), VlYeW g_Z(z) =c(Φ(z₁),Φ(z₂))φ(z₁)φ(z₂) UUaRuCLj3 jvZ:= (Φ⁻¹(G₁(v₁)),Φ⁻¹(G₂(v₂))).

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jR j@3W8aRL 3\Y VlYfW sCII j@3N $3Z= ((W₁−μ₁) /σ₁,(W₂−μ₂) /σ₂). s@C,@ CLUIC3c Z ∼M

0 0

, 1 ρ

ρ 1

, VlY4W

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nb₁b₂WY BNcUCa30 $w j@3 8,j j@j j@3 jaNc8RaL30 aN0RL qaC$I3Z@q3 cjN0a0 NRaLI La<CNc. Cj

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V,W r@3N

h∈Zρ_v|p(h)<∞. j@3 IR,I ;nccCN cU3,jaI 03NcCjw j j@3 URCNj v Cc 03A

~N30 c

f_v|p(ω) := ^∞

h=−∞

ρ_v|p(h)·e^−2πiωh. VlYSzW

`3LaG lYlY i@3 a3\nCa3L3Nj

h∈Zρ_v|p(h)<∞CN 03~NCjCRNlYSV,WCLUIC3c j@j j@3 ,RN,3Uj R8 IR,I ;nccCN cU3,jaI 03NcCjw CN <3N3aI LC<@j NRj $3 s3II 03~N30 8Ra II cjjCRNaw jCL3 c3aC3c{Y_t}t∈ZN0 II URCNjcv∈R²Y

i@3 8RIIRsCN< 03~NCjCRN R8 jCL3 a3q3acC$I3 jCL3 c3aC3c. 8aRLiRN<VSOOz. 038Y :YfW. Cc N33030 CN I3LLlYkV,WY

/3~NCjCRN lYlY cjjCRNaw jCL3 c3aC3c{Y_t}_t∈ZCc jCL3 a3q3acC$I3 C8 8Ra 3q3aw URcCjCq3 CNj3<3an N0 3q3awt₁, t₂, . . . , t_n∈Z. j@3 q3,jRac

Y_t

1, Y_t

2. . . , Y_tn

N0

Y_−t

1, Y_−t

2. . . , Y₋_tn

@q3 j@3 cL3 ERCNj 0CcjaC$njCRNcY

H3LL lYkY i@3 8RIIRsCN< UaRU3ajC3c @RI0c 8Raf_v|p(ω)Y

VW f_v|p(ω),RCN,C03c sCj@f(ω)8Ra IIv∈R²s@3N{Y_t}t∈ZCc ;nccCN jCL3 c3aC3c. N0 s@3N{Y_t}t∈Z,RNcCcjc R8 CYCY0Y R$c3aqjCRNcY

4

k4 TU3ac

(9)

V$W i@3 8RIIRsCN< @RI0c s@3Nv˘:= (v₂, v₁)Cc j@3 0C<RNI a33,jCRN R8v= (v₁, v₂)d f_v|p(ω) = 1 +

∞ h=1

ρ_v|p_˘ (h)·e^+2πiωh+ ∞ h=1

ρ_v|p(h)·e^−2πiωh, VlYSSW

f_v|p(ω) =f_v|p_˘ (ω). VlYSS$W

V,W r@3N{Y_t}t∈ZCc jCL3 a3q3acC$I3. j@3Nf_v|p(ω)Cc a3I qIn30 8Ra IIv∈R². CY3Y f_v|p(ω) = 1 + 2·^∞

h=1

ρ_v|p(h)·,Rc(2πωh). VlYSlW V0W f_v|p(ω)sCII CN <3N3aI $3 ,RLUI3uAqIn30. $nj Cj sCII Iswc $3 a3I qIn30 s@3N j@3 URCNj vIC3c RN j@3 0C<RNI. CY3Y s@3Nv₁=v₂Y 2\njCRN VlYSlW sCII @RI0 CN j@Cc 0C<RNI ,c3 jRRY

TaRR8Y Bj3LVW8RIIRsc 8Ra j@3 ;nccCN ,c3 cCN,3 j@3 IR,I ;nccCN njR,Raa3IjCRNcρ_v|p(h)

$w ,RNcjan,jCRN ,RCN,C03c sCj@ j@3 Ra0CNaw V<IR$IW njR,Raa3IjCRNcρ(h)CN j@3 ;nccCN ,c3Y bCLCIaIw. s@3N{Y_t}t∈Z,RNcCcjc R8 CYCY0Y R$c3aqjCRNc. j@3N $Rj@ IR,I N0 <IR$I njR,Raa3IA jCRNc sCII $3 z s@3Nh= 0. N0 j@3 IR,I N0 <IR$I cU3,ja $Rj@ $3,RL3c SY Bj3LcV$WjRV0Wa3 jaCqCI ,RNc3\n3N,3c R8 j@3 0C<RNI 8RI0CN< UaRU3ajw 8aRL I3LL+YS. CY3Yρ_v|p(−h) =ρ_v|p_˘ (h).

N0 j@3 03~NCjCRN R8 jCL3 a3q3acC$CICjw. c33 UU3N0C,3c+YSN0+Yl8Ra 03jCIcY

`3LaG lYkY 7Ra <3N3aI URCNjcv= (v₁, v₂). j@3 ,RLUI3u qIn30 a3cnIj R8f_v|p(ω)LC<@j $3 @a0 jR CNq3cjC<j3 N0 CNj3aUa3j Ģ $nj. 0n3 jR I3LLlYkV0W. j@3 CNq3cjC<jCRN $3,RL3c cCLUI3a 8Ra URCNjc RN j@3 0C<RNIY^Oi@3 a3I qIn30 a3cnIjcf_v|p(ω)8RavIRN< j@3 0C<RNI ,N $3 ,RLUa30 sCj@ j@3 a3cnIj R8 j@3 Ra0CNaw V<IR$IW cU3,jaI 03NcCjwf(ω). c <Cq3N CN 3\Y VSYlW. N0 j@Cc LC<@j 03j3,j ,c3c s@3a3 j@3 jCL3c c3aC3c{Y_t}t∈Z03qCj3c 8aRL $3CN< ;nccCNY 7naj@3aLRa3. C8 j@3

<IR$I cU3,janLf(ω)Cc j. j@3N Nw U3Gc N0 jaRn<@c R8f_v|p(ω)LC<@j $3 CNj3aUa3j30 c CN0C,jRac R8 3Y<Y U3aCR0C,CjC3c j IR,I I3q3IY i@Cc CLUIC3c j@j 3cjCLj3c R8f_v|p(ω)LC<@j $3 nc38nI c N 3uUIRajRaw jRRI. N C03 j@j sCII $3 Unacn30 CN c3,jCRNkY

`3LaG lY:Y MRj3 j@j j@3 ,RII3,jCRN R8 IR,I ;nccCN njR,Raa3IjCRNc ρ_v|p(h)

h∈ZLC<@j NRj

$3 NRNAN3<jCq3 03~NCj3Y +njCRN Cc j@nc 0qCc30 C8 U3Gc N0 jaRn<@c R8f_v|p(ω)a3 jj3LUj30 CNj3aUa3j30 c j@3w sRnI0 @q3 $33N C8 j@3w @0 R,,naa30 8Ra N Ra0CNaw V<IR$IW cU3,jaI 03NA cCjwf(ω)Y b33 j@3 0Cc,nccCRN CN c3,jCRN:8Ra 8naj@3a 03jCIcY

lYk 2cjCLjCRN

i@3Ra3jC,I N0 NnL3aC,I 3cjCLj3c R8 j@3 Ra0CNaw cU3,jaI 03NcCjwf(ω)Cc jwUC,IIw CNq3cjCA

<j30 $w L3Nc R8 j@3 8cj 7RnaC3a jaNc8RaL V77iW N0 j3,@NC\n3c a3Ij30 jR j@3 U3aCR0R<aLY i@Cc 77iAUUaR,@ ,N NRj $3 nc30 CN j@3 IR,I ,c3 cCN,3 j@3a3 Cc NR NjnaI 8,jRaCcjCRN R8 j3aLc LGCN< nU IR,I 3cjCLj30 ,RqaCN,3. $nj j@3a3 0R3c 3uCcj Ua3A77i UUaR,@ 8Ra j@3 3cjCLjCRN R8f(ω). s@3a3 7RnaC3a jaNc8RaL Cc jG3N R8 j@3 3cjCLj30 njR,Raa3IjCRNc 8j3a

O 0C<RNI URCNj ,Raa3cURN0c jR cCjnjCRN s@3a3 R$c3aqjCRNc R8 j@3 cL3 Ȕc,I3 R8 L<NCjn03ȕ a3 ,RLUa30Y i@Cc ,N CN UajC,nIa $3 R8 CNj3a3cj 8Ra jCL3 c3aC3c 83jnaCN< N cwLL3jaC, $3@qCRna. cCN,3 ,RLUaCcRN R8 j@3 IR,I ;nccCN cU3,ja j URCNjc ,Raa3cURN0CN< jR 3Y<Y j@3 SzX N0 OzX \nNjCI3c LC<@j Vc c33N CN ~<YSSW a3q3I NRNICN3a cjan,jna3c s@C,@ j@3 Ra0CNaw cU3,jaI 03NcCjwf(ω)8CIc jR 03j3,jY

O

lYS MRNICN3a cU3,jaI NIwcCc qC j@3 IR,I ;nccCN ,Raa3IjCRN kO

(10)

j@3w @q3 $33N cLRRj@30 N0 janN,j30 $w L3Nc R8 cRL3 I<AsCN0Rs 8nN,jCRN Ģ N0 j@3 Ua3A 77i UUaR,@ ,N $3 0Uj30 jR 03I sCj@ j@3 3cjCLj3c R8 j@3 IR,I ;nccCN cU3,jaI 03NcCjC3cY /3~NCjCRN lY:Y 7Ra cLUI3{y_t}ⁿ_t=1R8 cCy3n. NmAjanN,j30 3cjCLj3f_v|p^m(ω)R8f_v|p(ω)a3 ,RNcjan,j30 $w L3Nc R8 j@3 8RIIRsCN< UaR,30na3Y

VW 7CN0 N 3cjCLj3G_nR8 j@3 La<CNI ,nLnIjCq3 0CcjaC$njCRN 8nN,jCRN. N0 ,RLUnj3 j@3 Uc3n0RANRaLICc30 R$c3aqjCRNc

z_t:= Φ⁻¹

G_n(y_t)n

t=1j@j ,Raa3cURN0c jR{y_t}ⁿ_t=1Y V$W +a3j3 j@3 I<hUc3n0RANRaLICc30 UCac

z_t+h,z_tn−h

t=1 8Rah= 1, . . . , m. N0 3cjCA Lj3. $Rj@ 8Ra j@3 URCNjv= (v₁, v₂)N0 Cjc 0C<RNI a33,jCRN v˘= (v₂, v₁). j@3 IR,I

;nccCN njR,Raa3IjCRNc

ρ_v|p(h|^b^h)m h=1N0

ρ_v|p_˘ (h|^b^h)m

h=1. s@3a3 j@3{bh}^mh=1Cc j@3

$N0sC0j@c nc30 0naCN< j@3 3cjCLjCRN R8 j@3 IR,I ;nccCN njR,Raa3IjCRN 8Ra j@3 0C8A 83a3Nj I<cY

V,W 0Encj 3\Y VlYSSW 8aRL I3LLlYkV$WsCj@ cRL3 I<AsCN0Rs 8nN,jCRNλ_m(h)jR <3j j@3 3cjCLj3

f_v|p^m(ω) := 1 + m h=1

λ_m(h)·ρ_˘_v|p(h|^b^h)·e^+2πiωh+ m h=1

λ_m(h)·ρ_v|p(h|^b^h)·e^−2πiωh. VlYSkW i@3 c3I3,jCRN R8 $N0sC0j@ N0 janN,jCRN I3q3I Cc 0Cc,ncc30 CN c3,jCRNc:YSN0:YlY i@3 8RIIRsCN< a3cnIj Cc N NIR<n3 jR 3\Y VlYSlW R8 I3LLlYkV,W

H3LL lY9Y r@3N Cj Cc ccnL30 j@j j@3 cLUI3{y_t}ⁿt=1,RL3c 8aRL jCL3 a3q3acC$I3 cjR,@cjC, UaR,3cc{Y_t}t∈Z. j@3mAjanN,j30 3cjCLj3f_v|p^m(ω),N 8Ra II URCNjcv∈R²$3 saCjj3N c

f_v|p^m(ω) = 1 + 2· m h=1

λ_m(h)·ρ_v|p(h|^bh)·,Rc(2πωh). VlYS:W KRa3Rq3a. 3\Y VlYS:W sCII Iswc @RI0 s@3N j@3 URCNjvIC3c RN j@3 0C<RNI. CY3Yv₁=v₂Y TaRR8Y i@Cc 8RIIRsc 8aRL Cj3LcV,WN0V0WR8 I3LLlYkY

`3LaG lY9Y i@3 3cjCLj30G_nCN 03~NCjCRNlY:V$W,N 3Y<Y $3 j@3 Va3c,I30W 3LUCaC,I ,nA LnIjCq3 0CcjaC$njCRN 8nN,jCRN ,a3j30 8aRL j@3 cLUI3{y_t}ⁿt=1. Ra Cj ,RnI0 $3 $c30 RN cRL3 IR<cUICN3 j3,@NC\n3 ICG3 j@3 RN3 CLUI3L3Nj30 CNQjN3CL N0 iEǹcj@3CLVlzSfWY

`3LaG lYfY i@3 $N0sC0j@cbh= (b_h1, b_h2)CN 03~NCjCRNlY:V$W0R3c NRj N330 jR $3 3\nI 8Ra II j@3 I<chs@3N N 3cjCLj3f_v|p^m(ω)Cc ,RLUnj30Y 7Ra j@3 cwLUjRjC, CNq3cjC<jCRN Cj Cc cn|,C3Nj jR a3\nCa3 j@jb_h1N0b_h2UUaR,@ y3aR j j@3 cL3 aj3. CY3Y j@j j@3a3 3uCcjcb= (b₁, b₂)cn,@

j@jb_hib_i8Rai= 1,2N0 8Ra IIhVj@j Cc jR cw. ICLb_hi/b_i= 1WY

`3LaG lYeY i@3 cwLUjRjC, j@3Raw 8Raρ_v|p(h|^bh). <Cq3N j@j j@3 a3\nCa30 a3<nIaCjw ,RN0CjCRNc a3 cjCc~30. 8RIIRsc 8aRLQjN3CL N0 iEǹcj@3CLVlzSfWdiEǹcj@3CL N0 ?n8j@LL3aVlzSkWY i@3 NIwcCc CN ViEǹcj@3CL N0 [email protected] ,RNcC03a30 j@3 <3N3aI ,c3 sCj@ p= 5 UaL3j3a IR,I ;nccCN UUaRuCLjCRN j j@3 URCNjv8Ra j@3 I<hUCac

y_t+h, y_tn−h t=1. CY3Y j@3 RaC<CNI R$c3aqjCRNc{y_t}ⁿt=1s3a3 nc30 CNcj30 R8 j@3 NRaLICc30 R$c3aqjCRNc{z_t:= Φ⁻¹(G(y_t))}ⁿt=1Y bCN,3 j@3 ,nLnIjCq3 03NcCjw 8nN,jCRNGCN <3N3aI sCII $3 nNGNRsN. j@3 Ua3c3Nj cwLUjRjC,

Sz

:z TU3ac

(11)

NIwcCc Lncj sRaG sCj@ j@3 Uc3n0RANRaLICc30 R$c3aqjCRNc{z_t}ⁿt=1. s@C,@ LG3c Cj N3,3cA caw jR jG3 CNjR ,,RnNj j@3 0C{3a3N,3 $3js33N j@3 jan3 NRaLICc30 qIn3cz_t N0 j@3 3cjCA Lj30 Uc3n0RANRaLICc30 qIn3cz_tY i@3 NIwcCc CN VQjN3CL N0 iEǹ[email protected] a3q3I30 j@jG_n(y_t)UUaR,@3cG(y_t)j 8cj3a aj3 j@N j@3 aj3 R8 ,RNq3a<3N,3 8Ra j@3 3cjCLj30 IR,I

;nccCN ,Raa3IjCRN. cR j@3 ,RNq3a<3N,3 aj3 R8ρ_v|p(h|^bh)sCII j@nc NRj $3 {3,j30 $w j@3 0CcA jCN,jCRN $3js33Nz_tN0z_tY i@3 Ua3c3Nj NIwcCc sCII NRj 0nUIC,j3 j@3 a<nL3Njc a3Ij30 jR j@Cc 0CcjCN,jCRN. N0 j@3 CNj3a3cj30 a303a c@RnI0 ,RNcnIj VQjN3CL N0 iEǹ[email protected]. b3,jCRN kW 8Ra j@3 03jCIcY

`3LaG lY4Y i@3 $CcAqaCN,3 $IN,3 R8 j@3mAjanN,j30 3cjCLj3cf^m(ω)R8 j@3 Ra0CNaw cU3,jaI 03NcCjwf(ω)03U3N0c RN j@3 cCy3 R8ma3IjCq3 jRnVj@3 cCy3 R8 j@3 cLUI3WY i@3 $CcA qaCN,3 $IN,3 8Ra j@3 3cjCLj3cf_v|p^m(ω)Lncj CN 00CjCRN ,RNcC03a j@3 cCy3 R8ma3IjCq3 jR

$Rj@nN0 j@3 $N0sC0j@c{bh}^mh=1. CY3Y j@3 G3aN3I 8nN,jCRN a30n,3c j@3 NnL$3a R8 R$c3aqjCRNc j@j 3{3,jCq3Iw ,RNjaC$nj3c jR j@3 ,RLUnjjCRNc R8 j@3 3cjCLj3c Ģ N0 j@j NnL$3a R8 3{3,jCq3 ,RNjaC$njRac ,N IcR 03U3N0 RN j@3 IR,jCRN R8 j@3 URCNjv. CY3Y s@3j@3a j@3 URCNjv IC3c j j@3 ,3Nj3a Ra CN j@3 U3aCU@3aw R8 j@3 Uc3n0RANRaLICc30 R$c3aqjCRNc

z_t+h,z_tn−h

t=1Y +RN83a c3,jCRNkYl8Ra 8naj@3a 03jCIcY

7C<na3Sc@Rsc j@3 3{3,j R8 j@3 Uc3n0RANRaLICcjCRN RN j@3/K#T3uLUI3^Szj@j sCII $3 0Cc,ncc30 CN c3,jCRNkY:Y i@3 nUU3aLRcj Uaj c@Rsc j@3 RaC<CNI/K#TAc3aC3c VR8 I3N<j@ SOe:W s@3a3c j@3 IRs3aLRcj Uaj c@Rsc j@3 Uc3n0RANRaLICc30 jaNc8RaLjCRN R8 Cj. N0 Cj Cc ,I3a j@j j@3 c@U3 R8 j@3 Uc3n0RANRaLICc30 q3acCRN a3c3L$I3c j@3 c@U3 R8 j@3 RaC<CNI q3acCRNY i@3 3{3,j R8 j@3 jaNc8RaLjCRN Cc jsR8RI0d Cj a3LRq3c j@3 3uja3L3 RnjIC3ac j j@3 cL3 jCL3 c Cj cUa30c Rnj j@3 ,3Nj3a R8 j@3 0CcjaC$njCRNY

original dmbp

−2

−10 1 2 3

0 500 1000 1500 2000

pseduo−normal dmbp

−2 0 2

0 500 1000 1500 2000

7C<na3 S-/K#T. RaC<CNI q3acCRN N0 Uc3n0RANRaLICc30 q3acCRNY

lY: cwLUjRjC, j@3Raw 8Ra f

_v|p^m

( ω )

i@Cc c3,jCRN Ua3c3Njc cwLUjRjC, a3cnIjc 8Ra j@3 ,c3c s@3a3f_v|p^m(ω)a3 a3IAqIn30 8nN,jCRNcY MRj3 j@j $Rj@ ccnLUjCRNc N0 a3cnIjc a3 cjj30 a3IjCq3 jR j@3 RaC<CNI R$c3aqjCRNc CNcj30 R8

Szi@Cc Cc j@3 /3njc,@3LaGg#aCjCc@ URnN0 2u,@N<3 `j3 V/K#TW 0j 8aRL#RII3acI3q N0 ;@wc3IcVSOOfW. s@C,@

Cc ,RLLRN $3N,@LaG 0j c3j 8Ra ;`+?AjwU3 LR03Ic. N0 c cn,@ LR03Ic a3 LRN< j@3 LRjCqjCN<

8,jRac 8Ra j@3 cjn0w R8 j@3 IR,I ;nccCN cU3,jaI 03NcCjw. Cj c33Lc NjnaI jR j3cj j@3 L3j@R0 RN/K#TY i@3 0j UIRjj30 @3a3 sc 8RnN0 CN j@3_AU,G<3`m;`+?. c33;@INRcVlzS9$W. s@3a3 j@3 8RIIRsCN< 03c,aCUjCRN sc <Cq3N- Ȕi@3 0CIw U3a,3Nj<3 NRLCNI a3jnaNc ,RLUnj30 c100 [IN(P_t)−IN(P_t−1)]. s@3a3P_tCc j@3

$CIj3aI /3njc,@3LaGg#aCjCc@ URnN0 aj3 ,RNcjan,j30 8aRL j@3 ,Raa3cURN0CN< mYbY 0RIIa aj3cYȕ

SS

lYS MRNICN3a cU3,jaI NIwcCc qC j@3 IR,I ;nccCN ,Raa3IjCRN :S

(12)

j@3 Uc3n0RANRaLICc30 R$c3aqjCRNcY i@Cc cCLUIC~,jCRN 0R3c NRj {3,j j@3 ~NI ,RNq3a<3N,3 aj3c Vc33 a3LaG lYe8Ra 03jCIcW N0 Cj LG3c j@3 NIwcCc 3cC3aY i@3 a3\nCa3L3Nj j@j j@3 H;b/ c@RnI0 $3 03~N30 a3IjCq3 jR j@3 NRaLICc30 R$c3aqjCRNc Cc 0n3 jR ,RLUnjjCRNI Cccn3c.

N0 j@3 j@3Ra3jC,I CNq3cjC<jCRN c@Rsc j@j Cj ,RnI0 Encj c s3II @q3 $33N U@ac30 CN j3aLc R8 j@3 RaC<CNI R$c3aqjCRNcY

lY:YS 03~NCjCRN N0 N ccnLUjCRN 8RaY_t

i@3 ccnLUjCRN jR $3 CLURc30 RN j@3 nNCqaCj3 jCL3 c3aC3c{Y_t}t∈Za3\nCa3c ,RLURN3Njc a3A Ij30 jR j@3 $CqaCj3 I<hUCac j@j ,N $3 ,RNcjan,j30 8aRL Cj. s@3a3c j@3 j@3Ra3jC,I NIwcCc R8f_v|p^m(ω)IcR a3\nCa3c j@j(m+ 1)AqaCj3 UCac a3 ,RNcC03a30Y MRj3 j@j Cj3LV,WR8 03~NCA jCRNlY:CLUIC3c j@j Cj Cc cn|,C3Nj jR RNIw ,RNcC03a URcCjCq3 qIn3c 8RahY

/3~NCjCRN lYfY 7Ra cjaC,jIw cjjCRNaw nNCqaCj3 jCL3 c3aC3c{Y_t}t∈Z. sCj@h≥1N0m≥2.

03~N3 $CqaCj3 N0(m+ 1)AqaCj3 jCL3 c3aC3c c 8RIIRsc.

Yh:t:=

Y_t+h, Y_t

, Ym:t:=

Y_t+m, . . . , Y_t

, VlYS9W

N0 I3jg_h(yh)N0g_m(ym)03NRj3 j@3 a3cU3,jCq3 UaR$$CICjw 03NcCjw 8nN,jCRNcY

`3LaG lYOY i@3 03NcCjC3cg_ha3 j@Rc3 N33030 s@3N CNq3cjC<jCN< j@3 IR,I ;nccCN 3cjCLj3c 8Ra j@3 0C{3a3Nj I<chY i@3 $CqaCj3 03NcCjC3cg_h,N II $3 R$jCN30 8aRL j@3(m+ 1)qaCj3 03NcCjwg_m$w CNj3<ajCN< Rnj j@3m−1a30nN0Nj La<CNIc. s@C,@ CN UajC,nIa CLUIC3c j@j C8 N (m+ 1)AqaCj3 8nN,jCRNη_h(ym) :R^m+1→R¹ Cc j@3 R$qCRnc 3uj3NcCRN^SSR8 $CqaCj3 8nN,jCRNη_h(yh) :R²→R¹. j@3N

2[η_h(Yh:t)] =2[η_h(Ym:t)], 8Rah∈ {1, . . . , m}. VlYSfW rCj@ j@3 NRjjCRN 8aRL 03~NCjCRNlYfj@3 8RIIRsCN< ccnLUjCRN ,N NRs $3 CLURc30 nURNY_tY MRj3 j@j Cj3LcV3WjRV<W,RNjCNc a383a3N,3c jR 03~NCjCRNc j@j ~acj a3 <Cq3N CN UU3N0Cu#d j@3c3 03~NCjCRNc a3 a3Ij30 jR j@3 3cjCLjCRN R8 U3NIjw 8nN,jCRN Ģ N0 j@3w a3 \nCj3 j3,@NC,I cR Cj sRnI0 CLU303 j@3 Rs R8 j@3 UU3a jR CN,In03 II j@3 03jCIc @3a3Y 7Ra j@3 Ua3c3Nj c3,jCRN.

Cj Cc cn|,C3Nj jR GNRs j@j j@3 aN0RL qaC$I3cX_hq:iⁿ CN Cj3LV<WCN 3cc3N,3 a3 j@3 a3cnIj j@j R,,nac s@3N j@3 UaR0n,j R8 j@3 G3aN3I 8nN,jCRNKh:b(yh−v)N0 j@3 c,Ra3 8nN,jCRN R8 j@3 IR,I

;nccCN UUaRuCLjCRNψ(yh)Cc 3qInj30 CNyh=Yh:tY

ccnLUjCRN lYSY i@3 nNCqaCj3 UaR,3cc{Y_t}t∈ZsCII $3 ccnL30 jR cjCc8w j@3 8RIIRsCN< UaRUA 3ajC3c. sCj@v= (v₁, v₂)CN Cj3LV0Wj@3 URCNj j s@C,@ j@3 3cjCLj3f_v|p^m(ω)R8f_v|p(ω)a3 jR $3 ,RLUnj30Y

VW {Y_t}t∈ZCc cjaC,jIw cjjCRNawY

SSi@3 R$qCRnc 3uj3NcCRN Cc jR ,RNcC03a j@3 8nN,jCRN jR $3 ,RNcjNj sCj@ a3cU3,j jR II j@3 N3s qaC$I3c j@j a3 CNjaR0n,30Y

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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. VlYSeW V,W pa(Y_t²)<∞Y

i@3 $CqaCj3 03NcCjw 8nN,jCRNcg_h(yh),Raa3cURN0CN< jR j@3 I<hUCacYh:tR8 j@3 nNCqaCj3 jCL3 c3aC3c{Y_t}t∈Z. Lncj cjCc8w j@3 8RIIRsCN< a3\nCa3L3Njc 8Ra <Cq3N URCNjv= (v₁, v₂)Y

V0W g_h(yh)Cc 0C{3a3NjC$I3 jv. cn,@ j@j iwIRaȕc j@3Ra3L ,N $3 nc30 jR saCj3g_h(yh)c g_h(yh) =g_h(v) +g_h(v)[yh−v] +R_h(yh)[yh−v], VlYS4W s@3a3g_h(v) =

∂

∂y_hg_h(yh)

y_h=v, _∂y^∂

0g_h(yh)

y_h=v

N0 ICL

y_h−→vR_h(yh) =0, N0 j@3 cL3 a3\nCa3L3Nj Lncj IcR @RI0 8Ra j@3 0C<RNIIw a33,j30 URCNjv˘= (v₂, v₁)Y V3W i@3a3 3uCcjc $N0sC0j@bh0cn,@ j@j j@3a3 8Ra 3q3aw0<b<bh0Cc nNC\n3 LCNCLCc3a

θh:bR8 j@3 U3NIjw 8nN,jCRNq_h:b03~N30 CN 3\Y V#Y:W. U<3:kY

V8W i@3 ,RII3,jCRN R8 $N0sC0j@c{bh0}h∈Z@c URcCjCq3 CN~LnL. CY3Y j@3a3 3uCcjc b0cn,@

j@j

0<b0:=CN8

h∈Zbh0, VlYSOW

s@C,@ CLUIC3c j@j j@Ccb0,N $3 nc30 cCLnIjN3RncIw 8Ra II j@3 I<cY

V<W 7RaX_hq:iⁿ 8aRL 03~NCjCRN#YSS. U<39S. j@3 $CqaCj3. jaCqaCj3 N0 j3jaqaCj3 03NcCjw 8nN,jCRNc Lncj $3 cn,@ j@j j@3 3uU3,jjCRNc 2

X_hq:iⁿ

. 2X_hq:iⁿ ^ν N0 2

X_hq:iⁿ ·X_jr:kⁿ II a3 ~NCj3Y

`3LaG lYSzY i@3c3 ccnLUjCRN nURNY_ta3 3uj3NcCRNc R8 j@Rc3 nc30 8Ra j@3 H;+A,c3 CNiEǹcA j@3CL N0 ?n8j@LL3aVlzSkWY ccnLUjCRNlYSV$WCc $Cj LRa3 <3N3aI j@N j@3 RN3 nc30 CN ViEǹcj@3CL N0 [email protected]. $nj j@j Cc NRj UaR$I3L cCN,3 j@3 a<nL3Njc <Cq3N j@3a3 jaCqCIIw 3uj3N0c jR j@3 Ua3c3Nj ,c3Y

`3LaG lYSSY i@3αALCuCN< a3\nCa3L3Nj CN Cj3LV$W3Ncna3c j@jY_t+hN0Y_tsCII $3 cwLUjRjA C,IIw CN03U3N03Nj ch→ ∞. CY3Y j@3 $CqaCj3 03NcCjw 8nN,jCRNcg_h(yh)sCII 8Ra Ia<3 I<ch UUaR,@ j@3 UaR0n,j R8 j@3 La<CNI 03NcCjC3c. N0 j@3 cCjnjCRN sCII j@nc cj$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 NRNy3aRb0j@j sRaGc 8Ra IIhY

`3LaG lYSlY i@3 ~NCj3N3cc a3\nCa3L3Njc CN ccnLUjCRNlYSV<WsCII $3 jaCqCIIw cjCc~30 C8 j@3 03NcCjC3c a3 $RnN030. CY3Y j@3w sCII j@3N $3 ,RNc3\n3N,3c R8 UaRU3ajC3c R8 j@3 G3aN3I 8nN,jCRN K_bN0 j@3 c,Ra3 8nN,jCRN R8 j@3 $CqaCj3 ;nccCN 0CcjaC$njCRN. c33 I3LL+Yf8Ra 03jCIcY

Sk

lYS MRNICN3a cU3,jaI NIwcCc qC j@3 IR,I ;nccCN ,Raa3IjCRN :k

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lY:Yl N ccnLUjCRN 8RaY_tN0 j@3 c,Ra3 8nN,jCRNu(w;θ)R8ψ(w;θ)

i@3 c,Ra3 8nN,jCRN CN 3\Y VlY:W. CY3Y u(w;θ) :=_∂θ^∂ IR<(ψ(w;θ)). UIwc ,3NjaI aRI3 CN j@3 IR,I 03NcCjwA3cjCLjCRN UUaR,@ R8?ERaj N0 DRN3cVSOOfW. N0 Cj j@nc IcR UIwc UCqRjI aRI3 CN j@3 IR,I ;nccCN ,Raa3IjCRN j@3Raw 03q3IRU30 CNiEǹcj@3CL N0 ?n8j@LL3aVlzSkWY BN UajC,nIa. j@3 ,RNq3a<3N,3 aj3 j@j CNiEǹcj@3CL N0 ?n8j@LL3aVlzSkW Cc <Cq3N 8Ra θv−θv0R3c CLUIC,CjIw a3\nCa3 j@ju(v;θv) =0CN Ra03a 8Ra j@3 ,Raa3cURN0CN< cwLUjRjC, ,RqaCN,3 LjaCu jR $3 s3II 03~N30Y i@3 CNq3cjC<jCRN R8

f_v|p^m(ω)−f_v|p(ω)

CN j@Cc UU3a

$nCI0c nURN j@3 cwLUjRjC, a3cnIjc 8aRLiEǹcj@3CL N0 ?n8j@LL3aVlzSkW. N0 j@3 8RIIRsA CN< ccnLUjCRN Lncj j@nc $3 cjCc~30 CN Ra03a 8Ra j@3 <Cq3N ,RNq3a<3N,3 aj3c N0 cwLUjRjC, qaCN,3c jR $3 qIC0Y MRj3 j@j j@3 CN03upc ncnI 0R3c c@Rs s@3j@3a Cj Cc RN3 Ra ~q3 UaL3jaC, IR,I ;nccCN UUaRuCLjCRNψ_p(w;θp)j@j Cc ,RNcC03a30. N0 j@jup(w;θp)@3a3 a3Ua3c3Nj j@3 ,Raa3cURN0CN< c,Ra3 8nN,jCRNY

ccnLUjCRN lYlY i@3 ,RII3,jCRN R8 IR,I ;nccCN UaL3j3ac θp|v(h)

j j@3 URCNjv8Ra j@3

$CqaCj3 UaR$$CICjw 03NcCjw 8nN,jCRNcg_h(yh). Lncj II $3 cn,@ j@j VW up(v;θp|v(h))=08Ra II ~NCj3hY

V$W ICLu_p(v;θp|v(h))=0Y

`3LaG lYSkY Bj Cc. 8Ra <Cq3N jCL3 c3aC3cY_t N0 <Cq3N URCNj v. URccC$I3 jR CNcU3,j j@3p 3\njCRNc CNup(w;θp) =0CN Ra03a jR c33 s@3N Cj3LcVWN0V$WR8 ccnLUjCRNlYlLC<@j 8CI jR

@RI0 jan3Y Bj Cc 3Y<Y URccC$I3 jR ~N0 j@3 UaL3j3aA,RN~<najCRNcθ^†p|vs@C,@ cRIq3up(v;θp) = 0.

N0 j@3N R$c3aq3 j@j ccnLUjCRNlYlVW sCII 8CI C8θ^†p|v ∈ θp|v(h)

Y 7Ra j@3 ,c3 R8 j@3 cwLUjRjC, a3\nCa3L3Nj CN Cj3LV$W. j@3 G3w R$c3aqjCRN Cc j@j j@3 cjaRN< LCuCN< a3\nCa3L3Nj 8aRL ccnLUjCRNlYSV$WCLUIC3c j@jY_t+h N0Y_t sCII $3,RL3 CN03U3N03Nj s@3N h → ∞Y iR<3j@3a sCj@ j@3 ccnLUjCRN R8 NRaLICc30 La<CNIc. j@Cc CLUIC3c j@j j@3 ICLCj R8 θp|v(h) Iswc $3,RL3c[μ₁, μ₂, σ₁, σ₂, ρ] = [0,0,1,1,0]. s@C,@ L3Nc j@j ccnLUjCRNlYlV$WsCII 8CI 8Ra Nw URCNjvj@j cRIq3cup(v; [0,0,1,1,0]) = 0Y

`3LaG lYS:Y i@3 RN3 UaL3j3a IR,I ;nccCN UUaRuCLjCRNψ₁(w;θ1)Cc I3cc 3uC$I3 j@N j@3 ~q3 UaL3j3a UUaRuCLjCRNψ₅(w;θ5). N0 j@Cc I,G R8 3uC$CICjw ,N 8Ra cRL3 jCL3 c3A aC3cY_t CLUIw j@j ccnLUjCRNlYlVWCc $RnN0 jR 8CI j cRL3 URCNjcv. c33 j@3 0Cc,nccCRN CN UU3N0Cu+YfYl8Ra 8naj@3a 03jCIcY

lY:Yk ccnLUjCRNc 8Ran.mN0b

7Ra cCLUIC,Cjw. j@3 Ua3c3Nj NIwcCc sCII nc3 j@3 b= (b₁, b₂)CNjaR0n,30 CN a3LaG lYf. c33 U<3Sz. CY3Y Cj sCII $3 ccnL30 j@j j@3 CN0CqC0nI $N0sC0j@cbh 8Ra j@3 0C{3a3Nj I<chUA UaR,@ y3aR j j@3 cL3 aj3 Ģ N0 j@j Cj 8Ra j@3 cwLUjRjC, CNq3cjC<jCRN j@nc ,N $3 ccnL30 j@j j@3 cL3 $N0sC0j@ Cc nc30 8Ra II j@3 I<cY 7Ra j@3 Ua3c3Nj ,c3. s@3a3 j@3 I<hUCac a3 R8 j@3 8RaL

Y_t+h, Y_t

Cj LC<@j IcR $3 NjnaI jR ccnL3 j@jb₁N0b₂c@RnI0 UUaR,@ y3aR j j@3 cL3 aj3. CY3Y j@jb₁b₂. $nj j@Cc sCII NRj $3 CLURc30 8aRL j@3 Rnjc3jY

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

S:

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