at the University of Bergen
Thesis for the degree of philosophiae doctor (PhD)
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Date of defence: November 20th 2017
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=2f◦[IR<f◦]−2f◦[IR<fi]. VSYSW Bj 8RIIRsc 8aRL j@3 D3Nc3N CN3\nICjw j@j FHB+(F◦, Fi)Iswc sCII $3 NRNAN3<jCq3. N0 j@j Cj Cc 3\nI jR y3aR C8 N0 RNIw C8fi=f◦ILRcj 3q3aws@3a3 Ģ N0 j@Cc CLUIC3c j@j FHB+(F◦, Fi),N $3 nc30 jR ~N0 j@3 RUjCLI UaL3j3aA,RN~<najCRNαij@j LG3cFi j@3 $3cj Lj,@ jRF◦Y 2\njCRN VSYSW ,N j@3N $3 nc30 jR c3I3,j j@3 Rq3aII $3cj LR03IF 8aRL j@3 ,RII3,jCRN R8 $3cj Lj,@3c{Fi}si=1Y Bj Cc CLURajNj jR NRj3 j@j Cj LC<@j @UU3N j@j NRN3 R8 j@3 LR03Ic CNF<Cq3c 03,3Nj UUaRuCLjCRN R8F◦Ģ N0 CN j@j ,c3 Cj sCII $3 $Cj R8 LCcNRL3a jR a383a jRFc j@3 ȔRq3aII $3cj LR03IȕY
bCN,3 j@3 j3aL 2f◦[IR<f◦]Cc Ua3c3Nj CN 3\Y VSYSW 8Ra II j@3 LR03Ic. Cj Cc ,I3a j@j LCNCLCcjCRN R8 FHB+(F◦, Fi),Raa3cURN0c jR LuCLCcjCRN R8 2f◦[IR<fi]. N0 LNw CN8RaLjCRN ,aCj3aC a3 $c30 RN j@3 cjaj3<w j@j j@Cc Ijj3a j3aL a3 3cjCLj30 8aRL j@3 R$c3aqjCRNcXnY i@3 ,jnI 3cjCLjCRN sCII $3 CNn3N,30 $w j@3 ccnLUjCRNc L03 nURN j@3 R$c3aqjCRNcXnN0 j@3 LR03IcF. s@C,@ I30c jR 0C{3a3Nj CN8RaLjCRN ,aCj3aC.
ICG3 j@3 $38Ra3 L3NjCRN30 B+ N0 iB+Y i@3c3 LR03I c3I3,jCRN cjaj3<C3c sCII $3 cn$E3,j jR j@3 3{3,j R8 cLIIAcLUI3 qaCjCRN. N0 00CjCRNI ,RLUIC,jCRNc LC<@j IcR R,,na C8 j@3w a3 UUIC30 RnjcC03 R8 j@3 a3IL CN s@C,@ j@3w RaC<CNIIw s3a3 ,RNcjan,j30Y
SYSYk i@3 ,RUnI N0 j@3 c3I3,jCRN R8 ,RUnI LR03Ic
bGIaȕc j@3Ra3L. c33bGIaVSO9OW. cjj3c j@j LnIjCqaCj3 LR03IFi(x1, . . . , xd),N $3 saCjj3N c
Fi(x1, . . . , xd) =Ci(Fi1(x1), . . . , Fid(xd)) VSYlW s@3a3
Fij xjd
j=1a3 j@3dLa<CNI ,nLnIjCq3 0CcjaC$njCRNc. N0 s@3a3Ci(u1, . . . , ud) Cc dAqaCj3 ,RUnI j@j 03,R03c II j@3 CNj3a03U3N03N,3 cjan,jna3 R8Fi(x1, . . . , xd)Y
SYS i@3Ra3jC,I 8aL3sRaG- i@3 $cC, ,RLURN3Njc k 2\njCRN VSYlW 3N$I3c 03,RnUICN< R8 j@3 UaL3j3ac R8Fi CNjR j@Rc3 j@j R,,nac CN j@3 La<CNI 0CcjaC$njCRNcFij. N0 j@Rc3 j@j R,,na CN j@3 ,RUnICiY i@3 UaL3j3ac R8 j@3 La<CNIc a3 R8j3N ,RNcC03a30 c NnCcN,3 UaL3j3ac C8 j@3 CL R8 j@3 CNq3cjCA
<jCRN Cc jR ~N0 j@3 ,RUnIALR03I j@j $3cj 03c,aC$3c j@3 CNj3aNI 03U3N03N,w cjan,jna3 R8 <Cq3N cLUI3 XnY Bj Cc j@3N NjnaI jR nc3 c3LCAUaL3jaC, jsRAcj3U UUaR,@.
s@3a3 j@3 ~acj cj3U nc3c j@3F˜iVj@3 n+1n Ac,I30 La<CNI 3LUCaC,I ,nLnIjCq3 03NcCjw 8nN,jCRNcW jR jaNc8RaL 8aRL R$c3aqjCRNc CNjR Uc3n0RAnNC8RaL R$c3aqjCRNc. N0 j@3 c3,RN0 cj3U Cc jR ~N0 j@3 ,RUnI LR03ICV8aRL j@3 ,RII3,jCRNC = {Ci}si=1W j@j $3cj Lj,@3c j@3 03U3N03N,w cjan,jna3 R8 j@3 Uc3n0RAnNC8RaL R$c3aqjCRNcY
;aǹNN3$3a< N0 ?ERajVlzz4W c@Rs30 j@j j@3 jaNc8RaLjCRN jR Uc3n0RAnNC8RaL R$A c3aqjCRNc @0 NRNAN3<IC<C$I3 CLU,j nURN j@3 cwLUjRjC, $3@qCRna R8 j@3 3cjCLjRa.
CY3Y j@j Cj 8RaLIIw sc NRj \nCj3 ,Raa3,j jR nc3 UaL3jaC, L3j@R0 ICG3 B+ CN j@Cc c3LCA UaL3jaC, c3jjCN<Y N CNq3cjC<jCRN R8 j@3 2f◦[IR<fi]j3aL 8aRL 3\Y VSYSW. j@j UaRU3aIw jRRG CNjR ,,RnNj j@3 3{3,j R8 j@3 Uc3n0RAjaNc8RaLjCRN R8 j@3 R$c3aq30 0jXn. I30 jR 0C{3a3Nj CN8RaLjCRN ,aCj3aC 8Ra ,RUnIAc3I3,jCRN. c33;aǹNN3$3a< N0 ?ERajVlzz4.
lzS:WY i@3 ,aRccAqIC0jCRN +RUnI BN8RaLjCRN +aCj3aCRN VuqA+B+W 8aRL V;aǹNN3$3a<
N0 ?ERaj.lzS:W Cc j@3 jRUC, 8Ra TU3a k CN j@Cc j@3cCcY
i@3 ,RUnI IcR UIwc N CLURajNj CN0Ca3,j aRI3 CN TU3ac S N0 l. sCj@ j@3 CNjaRA 0n,jCRN R8 UCac R8 jaNc8RaL30 Uc3n0RANRaLI qaC$I3c. c33 3\Y VSY:W CN c3,jCRNSYSY9 8Ra 8naj@3a 03jCIcY
SYSY: HR,I UaL3jaC, LR03Ic N0 j@3 IR,I ;nccCN ,Raa3IjCRN ρ
v Bj LC<@j $3 jII Ra03a jR ~N0 a3cRN$I3 ,RII3,jCRN R8 UaL3jaC, LR03IcF={Fi}si=1s@3N 0j c3jXnCc jR $3 CNq3cjC<j30. CN UajC,nIa C8 j@3 03cCa30 UUIC,jCRN Cc jR 3cjCLj3f◦j <Cq3N URCNjv∈RdY i@3 ,Ra3 R8 j@3 UaR$I3L Cc j@j 3q3N j@Rn<@ 3\Y VSYSW Cc nc30 jR c3I3,j N RUjCLI LR03IF8aRLF. Cj LC<@j cjCII $3 j@3 ,c3 j@jf(v)8CIc jR
<Cq3 03,3Nj 3cjCLj3 R8f◦(v)Y i@3 a3cRN 8Ra j@Cc Cc j@jF@c $33N c3I3,j30 $c30 RN j@3 UaRU3ajw j@j Cjc 03NcCjwfLCNCLCc3c j@3 3uU3,jjCRN R8 IR<(f◦/fi). N0 j@Cc 0R3c NRj <naNj33 <RR0 Lj,@ $3js33Nf◦N0fj cU3,C~30 URCNjvY
cRInjCRN jR j@Cc UaR$I3L sc CNjaR0n,30 CN?ERaj N0 DRN3cVSOOfW. s@3a3 IRA ,IIw UaL3jaC, NRNUaL3jaC, 03NcCjw 3cjCLjCRN UUaR,@ sc Ua3c3Nj30Y BN j@Cc UA UaR,@. 8Ra <Cq3N UaL3jaC, LR03IF(x;θ). j@3 CL Cc jR ~N0 cRInjCRNF(x;θv)j@j LCNCLCc3c j@3b→ 0+ICLCj R8 j@3 8RIIRsCN< IR,IIw s3C<@j30 q3acCRN R8 FHB+. s@3a3 Kb(x−v)Cc G3aN3I 8nN,jCRN N0bCc j@3 $N0sC0j@.
Kb(x−v) f◦(x)IR<f(x;θ)f◦(x) − {f◦(x)−f(x;θ)}
0x. VSYkW
;Cq3N j@j j@3 LR03IF(x;θ)Cc cn|,C3NjIw 3uC$I3. j@Cc UUaR,@ @c $3jj3a ,@N,3 R8 3NcnaCN< a3cnIj j@j cjCc~3c j@3 a3\nCa3L3Njf◦(v) =f(v;θv)Y
i@3 CL R8 V?ERaj N0 DRN3c.SOOfW sc j@3 3cjCLjCRN R8 j@3 03NcCjwf◦(v). $nj Cj Cc IcR URccC$I3 jR ,@N<3 j@3 8R,nc jR j@3 3cjCLj30 UaL3j3a q3,jRaθvY i@Cc UUaR,@
: BNjaR0n,jCRN sc Unacn30 8Ra j@3 $CqaCj3 ,c3 CNiEǹcj@3CL N0 ?n8j@LL3a VlzSkW. s@3a3 j@3 cU3,C~,jCRN j@jF(x;θ)c@RnI0 $3 ;nccCN 0CcjaC$njCRN 3N$I30 j@3 3uja,jCRN R8 j@3 IR,I ;nccCN ,Raa3IjCRNρv8aRLθvY i@3ρvR$jCN30 CN j@Cc LNN3a UaRqC03c IR,I L3cna3 R8 03U3N03N,3. sCj@ j@3 03cCa$I3 UaRU3ajC3c j@j Cj VW sCII a3jnaN j@3 Ra0CNaw V<IR$IW ,Raa3IjCRN C8 j@3 ja<3j30 8nN,jCRNF◦Cjc3I8 Cc ;nccCN. N0 j@j Cj V$W ,N 03j3,j NRNICN3a 03U3N03N,C3c $3js33N nN,Raa3Ij30 aN0RL qaC$I3cY
i@3 IR,I ;nccCN UUaR,@ 8aRL ViEǹcj@3CL N0 ?n8j@LL3a.lzSkW @c $33N CNA q3cjC<j30 CN 0C{3a3Nj c3jjCN<c. cRL3 8R,nc VICG3 V?ERaj N0 DRN3c. SOOfWW RN 03NcCjw 3cjCLjCRN. c33QjN3CL N0 iEǹ[email protected]. s@3a3c Rj@3a CNq3cjC<j3 UaRU3aA jC3c R8 j@3 IR,I ;nccCN ,Raa3IjCRN c IR,I L3cna3 R8 03U3N03N,3. c33#3a3Njc3N 3j IYVlzSe.lzS:Wd#3a3Njc3N N0 iEǹcj@3CLVlzS:Wd#3a3Njc3N 3j IYVlzS:$WdH,I N0 iEǹ[email protected]ǹq3 N0 iEǹcj@3CLVlzS:Wdbjǹq3 3j IYVlzS:WY
i@3 C03 j@j j@3 IR,I ;nccCN ,Raa3IjCRN c@RnI0 $3 UUIC30 jR jCL3 c3aC3c sc N CNj3<aI Uaj R8 ViEǹcj@3CL N0 [email protected]. CY3Y j@3 j@3Raw sc 03q3IRU30 8Ra j@3 ,c3 sCj@ I<hUCac
Yt+h, Yt
8aRL nNCqaCj3 jCL3 c3aC3c{Yt}t∈ZY i@Cc ,c3 sCII CN TU3a S $3 a383aa30 jR cρv(h). j@3 IR,I ;nccCN njR,Raa3IjCRN j I<h. N0 j@3
$cC, C03 Cc jR nc3 j@3 ,RII3,jCRN{ρv(h)}h∈ZjR 03~N3 j@3 IR,I ;nccCN njRAcU3,jaI 03NcCjw 8Ra nNCqaCj3 jCL3 c3aC3cY i@3 3uj3NcCRN jR LnIjCqaCj3 jCL3 c3aC3c. N0 j@3 ,RN,3Uj R8 IR,I ;nccCN ,aRccAcU3,jaI 03NcCjw. Cc <Cq3N CN TU3a lY
SYSY9 2cjCLjCRN R8 j@3 IR,I ;nccCN ,Raa3IjCRN ρ
vi@3_AU,G<3HQ+H;mbb. c33 V#3a3Njc3N 3j IY.lzS:W. ,N ,RLUnj3 3cjCLj3c R8 j@3 IR,I ;nccCN ,Raa3IjCRN 8Ra $CqaCj3 cLUI3cY 7C<na3SYSc@Rsc jsR 3uLUI3c.
$Rj@ sCj@ cLUI3c R8 cCy3n = 500Y i@3 cLUI3 CN j@3 I38j UN3I Cc 8aRL $CqaCj3 NRaLI 0CcjaC$njCRN sCj@ ,Raa3IjCRN zY9 N0 cjN0a0 NRaLI La<CNcY BN j@Cc ,c3 j@3 IR,I ;nccCN ,Raa3IjCRN ,RCN,C03c sCj@ j@3 Ra0CNaw V<IR$IW ,Raa3IjCRN. N0 j@Cc
<a33c VjGCN< cLIIAcLUI3 qaCjCRN CNjR ,,RnNjW sCj@ j@3 R$c3aq30 3cjCLj3c R8 j@3 IR,I ;nccCN ,Raa3IjCRNY i@3 ,c3 CNq3cjC<j30 CN j@3 aC<@j UN3I Cc 8aRL $CqaCj3 aN0RL qaC$I3(X1, X2). s@3a3X2=X12+sCj@X1∼M(0,1)N0∼M
0,14 Y i@3 aN0RL qaC$I3cX1 N0X2 a3 nN,Raa3Ij30. $nj R$qCRncIw 03U3N03Nj. c ,N $3 c33 CN j@3 aC<@j UN3I R8 ~<YSYSY 7Ra j@Cc cLUI3 j@3 3cjCLj30 ,Raa3IjCRN Cc AzYzS9. s@C,@
0R3c NRj a3q3I Nwj@CN< $Rnj j@3 CNj3aNI 03U3N03N,w cjan,jna3 $3js33NX1 N0X2Y QN j@3 Rj@3a @N0. j@3 3cjCLj30 IR,I ;nccCN ,Raa3IjCRNc ,I3aIw 03j3,j cC<NC~,Nj IR,I 83jna3c R8 j@3 nN03aIwCN< 0CcjaC$njCRNY
i@3 3cjCLj30 IR,I ;nccCN ,Raa3IjCRNc CN ~<YSYSa3 $c30 RN j@3 RaC<CNI R$A c3aqjCRNc{(x1i, x2i)}ni=1. $nj Cj LC<@j CN <3N3aI $3 N3,3ccaw jR jaNc8RaL jR Uc3n0RA NRaLICc30 R$c3aqjCRNc CN Ra03a jR qRC0 NnL3aC,I UaR$I3Lc 0naCN< j@3 3cjCLjCRN UaR,3ccY i@3 Uc3n0RANRaLICcjCRN UaR,30na3 nc30 CN TU3a S N0 l Cc jsR cj3U UaRA ,3cc. s@3a3 VBW j@3 ~acj cj3U Cc jR nc3 j@3F˜i Vj@3 n+1n Ac,I30 3LUCaC,I La<CNI ,nLnA IjCq3 03NcCjw 8nN,jCRNcW jR a3UI,3 j@3 R$c3aqjCRNc sCj@ Uc3n0RAnNC8RaL R$c3aqjCRNc.
N0 VBBW j@3 c3,RN0 cj3U Cc jR nc3Φ−1 Vj@3 CNq3ac3 R8 j@3 ,nLnIjCq3 03NcCjw 8nN,jCRN 8Ra j@3 nNCqaCj3 cjN0a0 NRaLI 0CcjaC$njCRNW CN Ra03a jR <3j j@3 Uc3n0RANRaLICc30
SYS i@3Ra3jC,I 8aL3sRaG- i@3 $cC, ,RLURN3Njc 9
−2 0 2
−2 0 2 ρv0.000.080.160.240.320.400.480.560.640.720.80
0.0 2.5 5.0 7.5 10.0
−2 0 2 ρv−1.0−0.8−0.6−0.4−0.20.00.20.40.60.81.0
7C<na3 SYS- 2uLUI3c R8 j@3 IR,I ;nccCN ,Raa3IjCRNρvY R$c3aqjCRNc. CY3Y
{(x1i, x2i)}ni=1
I ˜
F1(x1i),F˜2(x2i)n i=1
II
Φ−1( ˜F1(x1i)),Φ−1( ˜F2(x2i))n
i=1. VSY:W i@Cc Uc3n0RANRaLICcjCRN UaR,30na3 CN 3cc3N,3 LG3c Cj URccC$I3 jR C<NRa3 j@3 38A 83,j R8 j@3 La<CNI 0CcjaC$njCRNc CN j@3 cn$c3\n3Nj NIwcCc. N0 Unjc j@3 3LU@cCc RN j@3 ,RUnIA,RLURN3Nj j@j 03c,aC$3c j@3 CNj3a03U3N03N,3 R8 j@3 qaC$I3cY Bj Cc CLURaA jNj jR NRj3 j@j j@Cc jaNc8RaLjCRN 0R3c NRj 03cjaRw j@3 03cCa$I3 UaRU3ajw j@j j@3 IR,I
;nccCN ,Raa3IjCRN R8 ;nccCN 0CcjaC$njCRN sCII $3 3\nI jR j@3 ,Raa3IjCRN R8 j@j 0CcjaC$njCRN Ģ s@C,@ Cc UCqRjI 8Ra j@3 CNj3aUa3jjCRNc j@j ,N $3 L03 Ij3a RN CN j@3 0Cc,nccCRN R8 j@3 IR,I ;nccCN cU3,jaI 03NcCjC3cY
SYSYf i@3 IR,I ;nccCN ,Raa3IjCRN ρ
vq3acnc \nNjCI3 $c30 IR,I L3A cna3c
i@3a3 a3 c3q3aI cjaj3<C3c qCI$I3 j@j ,N <Cq3 IR,I L3cna3 R8 03U3N03N,3 $3A js33N jsR aN0RL qaC$I3cX1N0X2. c33 j@3 0Cc,nccCRN CNiEǹcj@3CL N0 ?n8j@LL3a VlzSkW 8Ra 8naj@3a 03jCIc. N0 Cj Cc Vc sCII $3 0Cc,ncc30 CN c3,jCRNSYlW URccC$I3 jR nc3 j@3c3 IR,I L3cna3c R8 03U3N03N,3 jR ,RNcjan,j 0C{3a3Nj IR,I UUaR,@3c jR cU3,jaI NIwcCcY
bRL3 IR,I L3cna3c R8 j@3 03U3N03N,w cjan,jna3c $3js33NX1N0X2,RNjCNc 3uA Ua3ccCRNc $c30 RN 3NjCjC3c ICG3X1 ≤p1N0X2 ≤p2. s@3a3(p1, p2)03NRj3c URCNj R8 CNj3a3cjYS i@3 GNRsI30<3 R8 j@3 0CcjaC$njCRN 8nN,jCRNF◦R8 j@3 $CqaCj3 UCa(X1, X2) sCII CN <3N3aI $3 nNGNRsN. N0 jaNc8RaLjCRN 8aRL R$c3aqjCRNc jR Uc3n0RAnNC8RaL R$c3aqjCRNc VncCN< Uaj B R8 3\Y VSY:WW Cc j@nc ,RLLRN jR UUIw $38Ra3 j@3 3cjCLjCRN R8 j@3c3 IR,I L3cna3cY i@3 jaNc8RaLjCRN R8 j@3 URCNj(p1, p2)Cc UCa R8 \nNjCI3c (τ1, τ2). N0 Cj Cc j@nc NjnaI jR a383a jR j@3 IR,I L3cna3c R$jCN30 CN j@Cc LNN3a c
\nNjCI3 $c30Y
Bj ,N $3 CNcjan,jCq3 jR <Cq3 cCLUI3 3uLUI3 j@j ,RL$CN3c qCcnICcjCRN R8 j@3 jaNc8RaLjCRN UaR,3cc 8aRL 3\Y VSY:W sCj@ $aC38 ,RLUaCcRN R8 j@3 sw j@3 IR,I ;ncA cCN ,Raa3IjCRNρvN0 j@3 \nNjCI3 $c30 IR,I L3cna3c a3 3cjCLj30Y 7Ra j@Cc UnaA URc3 ,RNcC03a j@3 I< S UCac
Yt+1, Yt
8aRL cLUI3 R8 cCy3 :zz 8aRL N `VlWALR03I
SpaCNjc $c30 RN 3uUa3ccCRN ICG3p1< X1≤p1,N IcR $3 3N,RnNj3a30 Ģ $nj j@Rc3 a3 8Ra cCLUIC,Cjw NRj CN,In030 CN j@3 Ua3c3Nj 0Cc,nccCRNY
f BNjaR0n,jCRN Yt=φ1Yt−1+φ2Yt−2+wt. sCj@φ1= 1.φ2=−0.75N0wt∼M(0,1)Y
i@3 UN3Ic R8 ~<YSYlc@Rsc j@3 kOO I< S UCac Yt+1, Yt
$c30 RN j@3 RaC<CNI R$A c3aqjCRNc. j@3 Uc3n0RAnNC8RaL R$c3aqjCRNc N0 j@3 Uc3n0RANRaLICc30 R$c3aqjCRNc.
R8 s@C,@ j@3 Ijj3a jsR a3cU3,jCq3Iw ,N $3 nc30 8Ra j@3 3cjCLjCRN R8 \nNjCI3 $c30 L3cna3 R8 03U3N03N,3 N0 IR,I ;nccCN L3cna3 R8 03U3N03N,3Y
originalgg
−5.0
−2.5 0.0 2.5 5.0
−5.0 −2.5 0.0 2.5 5.0
pseudo pseudouud
0.00 0.25 0.50 0.75 1.00
0.00 0.25 0.50 0.75 1.00
pseudo−normal pseudo normal p − mmmma
−2 0 2
−2 0 2
7C<na3 SYl- i@3 kOO I< S UCac Yt+1, Yt
8aRL j@3 `VlWA3uLUI3Y H38j- QaC<CNI R$c3aqA jCRNcY +3Nj3a- Tc3n0RAnNC8RaL R$c3aqjCRNcY `C<@j- Tc3n0RANRaLICc30 R$c3aqjCRNcY
i@3 $In3 a3 CN j@3 ,3Nj3a UN3I c@Rsc j@3 Uc3n0RAnNC8RaL R$c3aqjCRNc j@j. <Cq3N j@3 \nNjCI3cq = (0.25,0.375). sCII $3 3cc3NjCI 8Ra j@3 3cjCLjCRN R8 \nNjCI3 $c30 L3cna3 R8 03U3N03N,3 8Ra j@j UajC,nIa \nNjCI3Y i@3 URCNjv = (−0.67,−0.32)CN j@3 aC<@j UN3I Cc j@3Φ−1AjaNc8RaL30 q3acCRN R8q. N0 j@3 $In3 c\na3 ,3Nj3a30 j v V$c30 RN j@3 $N0sC0j@b= (0.6,0.6)W c@Rsc j@3 Uc3n0RANRaLICc30 R$c3aqjCRNc j@j a3 3cc3NjCI 8Ra j@3 3cjCLjCRN R8 j@3 IR,I ;nccCN ,Raa3IjCRN CN j@Cc ,c3Y
i@3 $In3 a3<CRNc CN ~<YSYl@C<@IC<@jc j@3 Uc3n0RAnNC8RaL R$c3aqjCRNc N0 Uc3n0RA NRaLICc30 R$c3aqjCRNc j@j sCII $3 3cc3NjCI 8Ra j@3 3cjCLjCRN R8 j@3 IR,I L3cna3c R8 03U3N03N,3 8Ra j@3 <Cq3N cLUI3. N0 Cj Cc CLURajNj jR NRj3 j@j j@3c3 a3<CRNc $3@q3 0C{3a3NjIw s@3N j@3 cLUI3 cCy3 ,@N<3cd CN UajC,nIa. j@3 a3<CRN CN j@3 ,3Nj3a UN3I sCII a3LCN j@3 cL3 a3<a0I3cc R8 j@3 cLUI3 cCy3. s@3a3c j@3 c\na3 CN j@3 aC<@j UN3I sCII c@aCNG s@3N j@3 cLUI3 cCy3 CN,a3c3c VcCN,3 cLII3a $N0sC0j@bj@3N ,N
$3 UUIC30WY
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SYl i@3Ra3jC,I 8aL3sRaG- iCL3 c3aC3c e 7Ra nNCqaCj3 jCL3 c3aC3c {Yt}t∈Z. I3jγ(r, s)03NRj3 j@3 ,RqaCN,3 $3js33N j@3 j3aLcYr N0YsY i@3 jCL3 c3aC3c Cc s3GIw cjjCRNaw C8 Cj cjCc~3c j@3 8RIIRsCN< j@a33 a3\nCa3L3Njc
VCW 2
|Yt|2
<∞ 8Ra IIt∈Z, VSY9W
VCCW 2[Yt] =m 8Ra IIt∈Z, VSY9$W
VCCCWγ(r, s) =γ(r+t, s+t) 8Ra IIr, s, t∈Z. VSY9,W BN j@Cc ,c3 j@3 ,RqaCN,3 RNIw 03U3N0c RN j@3 0C{3a3N,3 h = r −s. N0 Cj Cc j@3N 03NRj30γ(h). j@3 njR,RqaCN,3 j I<hY i@3 ,Raa3cURN0CN< 3NjCjwρ(h) =γ(h)/γ(0) Cc j@3 njR,Raa3IjCRN j I<hY B8 j@3 ,RII3,jCRN{γ(h)}h∈ZCc $cRInj3Iw cnLL$I3. j@3N j@3 cU3,jaI 03NcCjwf(ω)Cc j@3 7RnaC3a jaNc8RaL R8{γ(h)}h∈Z. CY3Y
f(ω) :=
h∈Z
γ(h)·e−2πiωh. VSYfW
Bj 8RIIRsc 8aRL j@3 bU3,jaI `3Ua3c3NjjCRN i@3Ra3L. c33 3Y<Y#aR,Gs3II N0 /qCc VSO4f. i@Y :Y4YlW. j@j Cj 3uCcjc aC<@j ,RNjCNnRnc Raj@R<RNIACN,a3L3Nj UaR,3cc3cZ(ω) j@j VsCj@ UaR$$CICjw RN3W 3N$I3c j@3 RaC<CNI cjjCRNaw jCL3 c3aC3c jR $3 3uUa3cc30 c
Yt= 1/2
−1/2
e2πitω0Z(ω), VSYeW
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Bj ,N 8aRL j@Cc $3 03aCq30 j@j j@3 cU3,jaI 03NcCjwf(ω). s@3N Cj 3uCcjc. cjCc~3c f(ω)0ω=2
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, VSY4W
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γ(h) = 1/2
−1/2
f(ω)·e2πiωh0ω, VSYOW
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4 BNjaR0n,jCRN R$c3aq30 ja,3 c@RsN CN j@3 jRU UN3IY
−5.0
−2.5 0.0 2.5 5.0
0 100 200 300 400
0 5 10 15 20 25
0.0 0.1 0.2 0.3 0.4 0.5
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N0 Yt
1+h, . . . Y
tk+h
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SYlYk Ij3aNjCq3 V<IR$IW <3N3aICcjCRNc R8 f ( ω )
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<3N3aI cU3,jaI 03NcCjC3c R8 j@3 8RaL fξ(ω) =
h∈Z
ξh·e−2πiωh, VSYSzW
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Sz BNjaR0n,jCRN
SYlY: ;3N3aICc30 8nN,jCRN V<IR$IW <3N3aICcjCRN R8 f ( ω )
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3uU(iuYt),3uU
ivYt−|j|
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h(ω, u, v) :=
h∈Z
σj(u, v)·e−2πijω, VSYSSW
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$c30. NRj LRL3Nj $c30Y
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bRL3 3uLUI3c R8 IR,I a3UI,3L3Njc 8Ra j@3 njR,RqaCN,3cγ(h),N 3Y<Y $3 8RnN0 CN/3jj3 3j IYVlzS9W. s@3a3 0C{3a3Nj ,aRccA,RqaCN,3 G3aN3Ic a3 03~N30Y BN UajC,nIa.
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γh(x1, x2) :=+Rq
Yt+h≤x1
,{Yt≤x2}
, (x1, x2)∈R2, VSYSlW γhU(τ1, τ2) :=+Rq
Ut+h≤τ1
,{Ut≤τ2}
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SYk i@3 IR,I ;nccCN cU3,jaI 03NcCjC3c SS
,RUnI cU3,jaI 03NcCjw G3aN3Ic c j@3 ,Raa3cURN0CN< 7RnaC3a jaNc8RaL30 3NjCjC3cY V aNG $c30 HUI,3 U3aCR0R<aL G3aN3I Cc IcR 03~N30YW
i@3 ,RUnI cU3,jaI 03NcCjw G3aN3I CN/3jj3 3j IY VlzS9W Cc ,IRc3Iw a3Ij30 jR j@3 ,RN,3Uj R8 \nNjCI3 a3<a3ccCRN. CNjaR0n,30 CNFR3NG3a N0 #cc3jj DaVSOe4W. c33 IcR FR3NG3aVlzz9WY b3q3aI Rj@3a IR,I Ij3aNjCq3c jR j@3 Ra0CNaw cU3,jaI 03NcCjwf(ω)
@q3 $33N 03q3IRU30 $c30 RN j@Cc ,RN,3Uj. ICG3 3Y<Y j@3 \nNjCIR<aL 8aRLHCNjRN N0 r@N<VlzzeW. 8Ra s@C,@ j@3 CNj3a3cj30 a303a LC<@j ,RNcnIj?N 3j IYVlzSfW 8Ra LRa3 03jCIc N0 00CjCRNI a383a3N,3cY \nNjCI3 U3aCR0R<aL sc CNq3cjC<j30 CNHCVlzSl,W Vc33 IcRHCVlzz4.lzSz.$.,.lzSl.$.lzS:WW. N0 \nNjCI3A$c30 UUaR,@ sc IcR UUIC30 CN?<3LNNVlzSSWY
MRj3 j@j NRj II R8 j@3c3 UUaR,@3c a3 R8 j@3 8RaL <Cq3N CN 3\Y VSYSzW. 3Y<Y j@3 \nNA jCI3 U3aCR0R<aL 8aRLHCVlzSl,W. N0 Cj LC<@j j@nc $3 NjnaI jR CN,In03 cRL3 a3LaGc 8aRLHCVlzS:$. UY l9lAl9kWW $Rnj j@3 ,RNcjan,jCRN R8 j@3 \nNjCI3 U3aCR0R<aL 8aRL cLUI3{yt}nt=1Y i@3 cjajCN< URCNj Cc j@3 8nN,jCRNρα(x) :=x· {α−{x <0}}. s@C,@
Cc(α−1)xs@3Nx <0N0αxs@3Nx≥0Y i@Cc Cc nc30 CN j@3 3cjCLjCRN R8 jsR 0C{3aA 3Nj \nNjCI3A3cjCLj3c. R8 s@C,@ j@3 ~acj 3cjCLj3 Cc j@3 cLUI3αA\nNjCI3. 03NRj30 $w λˆα. s@C,@ LCNCLCc3cn
t=1ρα(yt−λ˜)Y i@3 Rj@3a 3cjCLj3 Cc $c30 RN \nNjCI3 a3<a3cA cCRN R8 j@3αA\nNjCI3 R8yt. s@C,@ Cc j@3λtj@j cjCc~3c T(yt≤λt) =αY i@3 C03 CN j@Cc Ijj3a ,c3 Cc j@jλt,N $3 3cjCLj30 $w˜λ+xt(ω)β. s@3a3xt(ω)Cc j@3 jaC<RNRL3jaC, a3A
<a3ccRa[,Rc(2πωt),cCN(2πωt)]. N0 s@3a3 j@3 RUjCLI a3<a3ccCRN ,R3|,C3Njcλˆα(ω)N0 βˆα(ω)a3 8RnN0 $w LCNCLCcCN<n
t=1ρα(yt−λ˜−xt(ω)β)Y i@3 \nNjCI3 U3aCR0R<aL Cc j@3N 03~N30 $w
Qn,α(ω) :=
n t=1
ρα
yt−λˆα
− n
t=1
ρα
yt−ˆλα(ω)−xt(ω) ˆβα(ω)
, VSYSkW
s@C,@ <Cq3c L3cna3 R8 j@3 N3j ,RNjaC$njCRN R8 j@3 jaC<RNRL3jaC, a3<a3ccRa xt(ω)jR j@3 a30n,jCRN R8 j@3 \nNjCI3 a3<a3ccCRN ,Rcj 8nN,jCRNY
SYk i@3 IR,I ;nccCN cU3,jaI 03NcCjC3c
SYkYS i@3 IR,I ;nccCN njRAcU3,janL. 03~NCjCRN N0 $cC, UaRU3ajC3c
i@3 cwLL3jaC, Njna3 R8 3Y<Y cjR,G a3jnaNc. c33 j@3 0Cc,nccCRN CN c3,jCRNSYlY9. ,N IcR$3 CNq3cjC<j30 ncCN< j@3 IR,I ;nccCN UUaR,@ Ģ N0 c 8Ra II j@3 Rj@3a IR,I L3A cna3c R8 03U3N03N,3. j@3 NjnaI \n3cjCRN Cc j@3N jR 3uLCN3 s@3j@3a IR,I cU3,jaI NIwcCc ,N $3 $c30 RN j@CcY Bj Cc CN UajC,nIa R8 CNj3a3cj jR CNq3cjC<j3 C8 Cj LC<@j $3 URccC$I3 jR 03j3,j U3aCR0C,CjC3cg,w,I3c j IR,I c,I3 V3Y<Y CN j@3 nUU3a N0 IRs3a 3uA ja3L3c R8 j@3 jCL3 c3aC3c nN03a CNq3cjC<jCRNW s@C,@ a3 CNqCcC$I3 jR j@3 Ra0CNaw cU3,jaI 03NcCjwf(ω)j@j sRaGc RN j@3 <IR$I c,I3Y
7Ra URCNjv= (v1, v2)N0 cjaC,jIw cjjCRNaw jCL3 c3aC3c{Yt}t∈Z. j@3 ,RNcjan,jCRN R8 j@3 IR,I ;nccCN njRAcU3,janLfv(ω). a3\nCa3c j@j Cj Cc URccC$I3 jR ,RLUnj3 j@3
Sl BNjaR0n,jCRN ,RII3,jCRN R8 IR,I ;nccCN njR,Raa3IjCRNcρv(h)8Ra IIh= 0. CY3Y j@j II j@3 UaR$A
$CICjw 03NcCjw 8nN,jCRNcgh 8Ra j@3 $CqaCj3 aN0RL qaC$I3c Yt+h, Yt
. cjCc~3c j@3 a3<nIaCjw ,RN0CjCRNc <Cq3N CNiEǹcj@3CL N0 ?n8j@LL3aVlzSkWY i@3 ,c3h= 0,N NRj $3 ,Rq3a30 $w j@Cc. $nj 8Ra j@Cc ,c3 Cj Cc NjnaI jR I3jρv(0)$3 C03NjC,I jR S 8Ra II v∈R2Y
B8 j@3 ,RII3,jCRN{ρv(h)}h∈ZCc $cRInj3Iw cnLL$I3. CY3Y
h∈Z|ρv(h)|<∞. j@3N j@3 IR,I ;nccCN njRAcU3,janL ,N $3 03~N30 c j@3 7RnaC3a jaNc8RaL R8{ρv(h)}h∈Z. CY3Y
fv(ω) :=
h∈Z
ρv(h)·e−2πiωh. VSYS:W
i@3 3cjCLjCRN UaR$I3Lc L3NjCRN30 CN c3,jCRNSYSY9LRjCqj3 N 0Encj30 03~NCjCRN.
c33 TU3a S. s@3a3 N CNCjCI cj3U NRaLICc3c j@3 jCL3 c3aC3c nN03a CNq3cjC<jCRN. CY3Y {Yt}t∈ZsCII $3 a3UI,30 sCj@{Zt:= Φ−1(G(Yt))}t∈Z. s@3a3GCc j@3 ,nLnIjCq3 03NcCjw 8nN,jCRNkR8YtN0Φ−1c ncnI Cc j@3 CNq3ac3 R8 j@3 ,nLnIjCq3 03NcCjw 8nN,jCRN R8 j@3 cjN0a0 NRaLI 0CcjaC$njCRNY Bj sCII @3N,38Rasa0 8Ra cCLUIC,Cjw $3 ccnL30 j@j j@Cc jaNc8RaLjCRN @c $33N U3a8RaL30 s@3Nfv(ω)Cc 0Cc,ncc30Y
7Ra ;nccCN jCL3 c3aC3c. j@3 IR,I ;nccCN njR,Raa3IjCRNcρv(h),RCN,C03 sCj@
j@3 Ra0CNaw V<IR$IW njR,Raa3IjCRNcρ(h). s@C,@ CLUIC3c j@jfv(ω)CN j@j ,c3 ,RCNA ,C03c sCj@f(ω)/γ(0). j@3 qaCN,3Aa3c,I30 q3acCRN R8 j@3 Ra0CNaw cU3,jaI 03NcCjw 8aRL 3\Y VSYfWY i@3 ccnLUjCRN j@j j@3 CNq3cjC<jCRN Cc U3a8RaL30 RN j@3 NRaLICc30 q3acCRN R8 j@3 jCL3 c3aC3c CLUIC3c j@jγ(0) = 1. cRfv(ω) =f(ω)8Ra II URCNjcv∈R2s@3N j@3 jCL3 c3aC3c nN03a CNq3cjC<jCRN Cc ;nccCNY
Bj 8RIIRsc 8aRL j@Cc j@j ,RLUaCcRN R8fv(ω)N0f(ω),N $3 nc30 jR C03NjC8w NRNA
;nccCN jaCjc CN j@3 jCL3 c3aC3c nN03a CNq3cjC<jCRNY B8f(ω)Cc j. CY3Yρ(h) = 08Ra IIh = 0. j@3N NRNAjfv(ω),N $3 jG3N c N CN0C,jRa R8 NRNAICN3a 03U3N03N,w cjan,jna3c CN j@3 jCL3 c3aC3c nN03a CNq3cjC<jCRN. N0 j@3 U3Gc N0 jaRn<@c R8fv(ω)LC<@j j@3N a3q3I 00CjCRNI CN8RaLjCRN $Rnj j@3c3 NRNAICN3a cjan,jna3cY
Bj Cc R8 CNj3a3cj jR NRj3 j@j#aCIICN<3aVSOf9. UY SkelW 8Ra @C<@3a Ra03a cU3,ja <q3 j@3 8RIIRsCN< a<nL3Nj CN 8qRna R8 ncCN< @C<@3a Ra03a ,nLnINjc CNcj30 R8 @C<@3a Ra03a LRL3Njc- Ȕi@3 ,RNcC03ajCRN R8 j@3 ,nLnINj CN j@Cc );nccCN* ,c3 Cc NRj IC$I3 jR 03,3Cq3 RN3 CNjR $3IC3qCN< j@j @3 @c <CN30 cRL3 CN8RaLjCRNY BN j@3 NRNA;nccCN ,c3 j@3 ,nLnINj UaRqC03c N CN0C,jCRN R8 j@3 NRNA;nccCNCjwYȕ i@Cc \nRj3 c@Rsc j@j Cj ,N $3 Ua383a$I3 jR @q3 jRRI j@j 0R3c NRj jaC<<3a Nw 8Ic3 IaLc s@3N j@3 jCL3 c3aC3c nN03a CNq3cjC<jCRN Cc ;nccCN. N0 j@3 ,RCN,C03N,3 R8fv(ω)N0f(ω)8Ra
;nccCN jCL3 c3aC3c Cc j@nc \nCj3 03cCa$I3 8aRL j@Cc URCNj R8 qC3sY
,RLUaCcRN R8fv(ω)N0f(ω)8Ra <3N3aI URCNjv = (v1, v2),N $3 $Cj ,RLA UIC,j30. cCN,3 j@3 IR,I ;nccCN njR,Raa3IjCRNcρv(h)CN <3N3aI a3 0C{3a3Nj 8aRL ρv(−h). s@C,@ CLUIC3c j@jfv(ω)j@3N $3,RL3c ,RLUI3u qIn30 8nN,jCRNY i@j Cccn3 Cc NRj Ua3c3Nj C8 j@3 URCNjvIC3c RN j@3 0C<RNI. CY3Yv1 =v2. cR j@3 3uLUI3c CN j@3 N3uj c3,jCRN sCII RNIw ,RNcC03a cn,@ 0C<RNI URCNjc s@3a3fv(ω)Cc a3I qIn30 8nN,jCRNY
kBN Ua,jC,3GsCII $3 nNGNRsN. N0 N 3cjCLj3GV$c30 RN j@3 cLUI3 j @N0W sCII $3 nc30 CNcj30Y i@Cc 0EncjL3Nj 00c cRL3 cLII3a Ra03a 3{3,jc j@j Lncj $3 jG3N CNjR ,,RnNj 0naCN< j@3 j@3Ra3jC,I NIwcCcY
SYk i@3 IR,I ;nccCN cU3,jaI 03NcCjC3c Sk
SYkYl 2cjCLjCRN R8 f
v( ω ) N0 cRL3 3uLUI3c
i@3Ra3jC,I N0 NnL3aC,I 3cjCLj3c R8 j@3 Ra0CNaw cU3,jaI 03NcCjw f(ω)Cc jwUC,IIw CNq3cjC<j30 $w L3Nc R8 j@3 8cj 7RnaC3a jaNc8RaL V77iW N0 j3,@NC\n3c a3Ij30 jR j@3 U3aCR0R<aLY i@3 3{3,jCq3N3cc R8 j@3 77i Cc 0n3 jR j@3 UaR0n,j cjan,jna3 R8 j@3 3cjCLj30 njR,RqaCN,3c. CY3Yγ(h) = n−|h|
t=1 yt+|h|·yt. N0 cCN,3 NR cCLCIa UaR0n,j cjan,jna3 Cc Ua3c3Nj 8Ra j@3 3cjCLj3c nc30 CN j@3 IR,I ;nccCN UUaR,@ Vs@C,@ Cc $c30 RN j@3 0CcjaC$njCRNW. Cj Cc NRj 83cC$I3 jR nc3 j@3 j3,@NC\n3c 8aRL j@3 77iA$c30 j@3Raw 8Ra 3cjCLj3c R8f(ω)s@3N 3cjCLj3c R8fv(ω)a3 a3\nCa30Y
i@3 Ua3A77i UUaR,@ 8Ra 3cjCLjCRN R8f(ω). s@3a3 7RnaC3a jaNc8RaL Cc jG3N R8 j@3 3cjCLj30 njR,Raa3IjCRNc 8j3a j@3w @q3 $33N cLRRj@30 N0 janN,j30 $w L3Nc R8 cRL3 I<AsCN0Rs 8nN,jCRN. ,N $3 0Uj30 jR 03I sCj@ j@3 3cjCLj3c R8 j@3 IR,I
;nccCN cU3,jaI 03NcCjC3cfv(ω)Y i@3 3cjCLj3c jR $3 nc30 Ij3a RN 8Ra j@3 ,RLUaCcRN R8f(ω)N0fv(ω)sCII j@nc $3
fm(ω) =
|h|≤m
λm(h)·ρ(h)·e−2πiωh, VSYS9W
fvm(ω) =
|h|≤m
λm(h)·ρv(h)·e−2πiωh, VSYS9$W
s@3a3m03NRj3c j@3 janN,jCRN I3q3I N0λm(h)Cc j@3 I<AsCN0Rs 8nN,jCRN nc30 8Ra j@3 cLRRj@CN<Y Bj @c NRj $33N 3uUIC,CjIw CN,In030 CN j@3 NRjjCRN R8 3\Y VSYS9$W. $nj j@3 3cjCLj3fvm(ω)0R3c IcR 03U3N0 RN j@3 $N0sC0j@bj@j Cc N33030 8Ra j@3 3cjCLjCRN R8 j@3 IR,I ;nccCN njR,Raa3IjCRNcρv(h)Y
i@3 3cjCLjCRN R8fv(ω). $c30 RN VUc3n0RANRaLICc30W cLUI3{yt}nt=1 R8 cCy3n. Lncj IcR jG3 CNjR ,,RnNj j@j j@3 c3I3,jCRN R8 j@3 URCNjv= (v1, v2)sCII CNn3N,3 j@3
$CcAqaCN,3 UaRU3ajC3c R8 j@3 3cjCLj3Y BN UajC,nIa. Cj ,N $3 UaR$I3LjC, C8 j@3 ,RRaA 0CNj3cv1N0v2a3 jRR 8a Rnj CN j@3 jCIc R8 j@3 cLUI3Y N0 LRa3Rq3a. c L3NjCRN30 CN j@3 Ua3qCRnc c3,jCRN. Cj ,N $3 Ua383a$I3 jR ,RNcC03a URCNjc RN j@3 0C<RNI. CY3Yv1=v2. cCN,3 fv(ω)VN0 j@3 3cjCLj3fvm(ω)W j@3N a3 a3I qIn30 8nN,jCRNc R8ωY i@3 ,RRa0CA Nj3cv1 N0v2 sCII. 8Ra j@3 3uLUI3c CNq3cjC<j30 CN TU3a S N0 l. $3 ,@Rc3N LRN<
j@3 SzX. 9zX N0 OzX U3a,3NjCI3c R8 j@3 cjN0a0 NRaLI 0CcjaC$njCRN. s@Rc3 qIn3c a3 ASYl4. z N0 SYl4Y BN8RaLjCRN $RnjvCc ,RNjCN30 CN j@3 nUU3a aC<@j ,RaN3a R8 j@3 a3I3qNj UIRjc. s@3a3 Cj Cc LaG30 cRyW,,RyW. N0 cR RNY
7C<na3cSY:N0SY9a3 jG3N 8aRL TU3a S. N0 c@Rs jsR cCLnIjCRN $c30 CNq3cjCA
<jCRNc R8fv(ω)Y 7C<na3SY:c@Rsc j@3 a3cnIj s@3N j@3 3cjCLjCRN UaR,30na3 Cc nc30 RN Szz CN03U3N03Nj cLUI3c 8aRL cjN0a0 NRaLI 0CcjaC$njCRN M(0,1)Y i@3 jRU I38j UN3I c@Rsc j@3 Uc3n0RANRaLICc30 q3acCRN R8 j@3 ~acj jCL3 c3aC3c j@j sc cLUI30 8aRL j@3 LR03I. sCj@ 0c@30 $aRsN ICN3c j j@3 I3q3Ic j@j ,Raa3cURN0 jR j@3 ,RRa0CNj3cv1 N0 v2Y i@3 j@a33 Rj@3a UN3Ic ,RNjCN CN8RaLjCRN $Rnj j@3mAjanN,j30 Ra0CNaw cU3,jaI 03NcCjwfm(ω)Va30 Uaj. j@3 cL3 8Ra II j@3 UIRjcW N0 j@3mAjanN,j30 IR,I ;nccCN cU3,jaI 03NcCjC3c fvm(ω)8Ra j@3 j@a33 0C<RNI URCNjc nN03a CNq3cjC<jCRN V$In3 UajWY BN8RaLjCRN $Rnj j@3 janN,jCRN I3q3I N0 j@3 URCNjc a3 UaCNj30 j j@3 jRU R8 3,@ UIRjY
S: BNjaR0n,jCRN
i.i.d. Gaussian white noise
−2 0 2
0 500 1000 1500 2000
m = 10 10% :: 10%
NC = OK 0 1 2
0.0 0.1 0.2 0.3 0.4 0.5
m = 10 50% :: 50%
NC = OK 0 1 2
0.0 0.1 0.2 0.3 0.4 0.5
m = 10 90% :: 90%
NC = OK 0 1 2
0.0 0.1 0.2 0.3 0.4 0.5
7C<na3 SY:- CYCY0Y ;nccCN s@Cj3 NRCc3
Bj ,N $3 c33N 8aRL ~<YSY:j@j j@3 L30CNc R8 j@3 3cjCLj3c Vj@3 a30 N0 $In3 0c@30 ICN3cW a3 <RR0 3cjCLj3c R8fm(ω)N0fvm(ω)Vj@3 mAjanN,jCRNc R8 j@3 jan3 qIn3cW.
s@C,@ CN j@Cc ,c3 CN 8,j ,RCN,C03c sCj@f(ω)N0fv(ω). CY3Y Cj Cc GNRsN j@j j@3 jan3 qIn3c a3 C03NjC,I jR S $Rj@ 8Ra j@3 IR,I N0 <IR$I ,c3Y Q$c3aq3 j@j j@3 3cjCLj30 OzX URCNjsCc3 ,RN~03N,3 $N0c a3 sC03a 8Ra j@3 IR,I ;nccCN cU3,jaI 03NcCjC3c.
s@C,@ Cc c 3uU3,j30 cCN,3 j@3 $N0sC0j@ nc30 CN j@3 3cjCLjCRN R8 j@3 IR,I ;nccCN njR,Raa3IjCRNc a30n,3c j@3 NnL$3a R8 R$c3aqjCRNc j@j 3{3,jCq3Iw ,RNjaC$nj3c jR j@3 3cjCLj30 qIn3c. N0 j@nc LG3 j@3 3cjCLj3c LRa3 UaRN3 jR cLIIAcLUI3 qaCjCRNY MRj3 IcR j@j j@3 ,RN~03N,3 $N0c a3 sC03a s@3N j@3 ,RRa0CNj3cv1N0v2a3 CN j@3 jCIc. s@C,@ Cc c 3uU3,j30 0n3 jR j@3 a30n,30 NnL$3a R8 qCI$I3 R$c3aqjCRNcY
i@3 3cjCLjCRN UaR,30na3 <q3 <RR0 3cjCLj3c R8 j@3 jan3 qIn3cf(ω)N0fv(ω)CN j@3 cCLUI3 3uLUI3 R8 ~<YSY:. $nj Cj Cc CLURajNj jR G33U CN LCN0 j@j j@3c3 UIRjc ,jnA IIw c@Rsc 3cjCLj3c R8fm(ω)N0fvm(ω)Y Bj LC<@j $3 N3,3ccaw jR UUIw VLn,@W @C<@3a janN,jCRN I3q3I $38Ra3fm(ω)N0fvm(ω)<Cq3 03,3Nj UUaRuCLjCRNc R8 j@3 jan3 qIn3c f(ω)N0fv(ω)Y Bj j@nc c33Lc Ua383a$I3 jR ,RLUnj3fvm(ω)8Ra aN<3 R8 URccC$I3 janNA ,jCRN I3q3Icm. N0 j@3N ,@3,G C8 j@3 c@U3c R8 j@3 3cjCLj3c 8Ra 0C{3a3Nj janN,jCRNc c@a3 j@3 cL3 UaRU3ajC3c sCj@ a3<a0 jR j@3 URcCjCRN R8 Nw U3Gc N0 jaRn<@cY b33 j@3 0Cc,nccCRN R8 j@3 ,RLUnj3a ,R03 CN c3,jCRNSY:8Ra 8naj@3a 03jCIcY
i@3 ;nccCN ,c3 CNq3cjC<j30 CN ~<YSY:Cc j@3 RNIw RN3 s@3a3 Cj Cc GNRsN s@j j@3 jan3 qIn3 R8fv(ω)c@RnI0 $3Y ?Rs3q3a. Cj Cc URccC$I3 jR ,RNcjan,j N ajC~,CI 3uLUI3 ICG3 j@3 RN3 CNq3cjC<j30 CN ~<YSY9. s@3a3 N a<nL3Nj ,N $3 <Cq3N j@j EncjC~3c j@j j@3 j@a33 IR,I U3Gc R8fvm(ω)a3 c 3uU3,j30 $c30 RN j@3 <3N3ajCN< LR03I N0 j@3
$N0sC0j@b= (0.6,0.6)nc30 CN j@Cc UajC,nIa ,c3Y MRj3 j@j j@3 UaL3j3ac nc30 jR
<3N3aj3 j@3 ajC~,CI 3uLUI3 CN ~<YSY9sc c3I3,j30 cn,@ j@j j@3 <IR$I cU3,janLf(ω) c@RnI0 IRRG ICG3 s@Cj3 NRCc3. N0 j@Cc Cc CN <a33L3Nj sCj@ j@3 R$c3aq30 3cjCLj3c R8 j@3
<IR$I cU3,janLY
MRj3 j@j ~<YSY9c@Rsc j@jfv(ω). CN ,RNjacj jRf(ω). ,N jG3 N3<jCq3 qIn3c 8Ra cRL3 8a3\n3N,C3cY i@Cc Cc 0n3 jR j@3 8,j j@j j@3 ,RII3,jCRN R8 IR,I ;nccCN njR,RaA a3IjCRNc{ρv(h)}h∈ZLC<@j NRj $3 NRNAN3<jCq3 03~NCj3Y 7naj@3aLRa3. ,Raa3cURN03N,3 ICG3 3\Y VSY4W 0R3c NRj 3uCcj 8Rafv(ω). N0 ,njCRN Cc j@nc 0qCc30 C8 U3Gc N0 jaRn<@c R8fv(ω)a3 jj3LUj30 CNj3aUa3j30 3u,jIw c j@3w sRnI0 @q3 $33N C8 j@3w @0 R,,naa30
