&RQFXUVXOUHYLVWHL$UKLPHGHGHFHPEULH &ODVDD9D
3XQH LSDUDQWH]HvQ H[SUHVLD ( = ⋅ + + DVWIHOvQFkWUH]XOWDWXO V ILHXQQXP UQDWXUDO FkW PDLPLF &DOFXOD L − ⋅ − ⋅ − ,, 'DF vPS U LPXQQXP UQDWXUDOODRE LQHPUHVWXO&DUHHVWHUHVWXOvPS U LULLDFHOXLQXP UOD" ,,,'HWHUPLQD LXOWLPHOHRSWFLIUHDOHQXP UXOXL $ = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ,9&DOFXOD LVXPDFLIUHORUQXP UXOXL % = ⋅ + ,
,
7LPSGHOXFUXRUHHIHFWLY)LHFDUHVXELHFWVHQRWHD] FXSXQFWH &ODVDD9,D
'HWHUPLQD LQXP UXOQDWXUDOQGLQHJDOLWDWHD
+ + − + + = 'DF [\]VXQWFLIUHQHQXOHvQED]DGHWHUPLQD LQXP UXO $ = [\] + \][ + ][\ úWLLQGF $VHGLYLGHFX ,, 6HFRQVLGHU PXO LPHD0 ^` D &RPSDUD LVXPDHOHPHQWHORUSDUHGLQPXO LPHD0FXVXPDHOHPHQWHORULPSDUHGLQPXO LPHD0 E 6HSRDWHHOLPLQDXQHOHPHQWGLQPXO LPHD0DVWIHOvQFkWVXPDHOHPHQWHORUSDUHU PDVHV ILHHJDO FXVXPDHOHPHQWHORULPSDUHU PDVH" ,,,3HRGUHDSW GVHLDXSXQFWHOH$%&úL'DVWIHOvQFkW$% D%' F$& E%& DE&' D E±F6WDELOL LRUGLQHDSXQFWHORU$%&'SHGUHDSWDG ,9$YH LODGLVSR]L LHXQJKLXULFXP VXUDGH°'HVFULH LXQSURFHGHXGHDSXQHvQHYLGHQ XQXQJKLFX P VXUDGH°
Q
7LPSGHOXFUXRUHHIHFWLY)LHFDUHVXELHFWVHQRWHD] FXSXQFWH &ODVDD9,,D
D − E − D − E = FDOFXOD LYDORDUHDUDSRUWXOXL , 'DF D∈5úLE∈4DVWIHOvQFkW E − D − D − E ,, )LH D D D D R VFULHUH D QXPHUHORU QDWXUDOH vQWUR DOW RUGLQH $U WD L F QXP UXO $ = D − ⋅ D − ⋅ ⋅ D − HVWHSDU ,,, )LH0XQSXQFWvQLQWHULRUXOSDUDOHORJUDPXOXL$%&'$U WD LF VHSRDWHFRQVWUXLXQSDWUXODWHUFRQYH[ FX DULD HJDO FX MXP WDWH GLQ FHD D SDUDOHORJUDPXOXL $%&' úL DOH F UXL ODWXUL DX OXQJLPLOH UHVSHFWLY HJDOHFXFHOHDOHVHJPHQWHORU0$0%0&úL0' ,9 D $U WD L F vQWUXQ WULXQJKL PHGLDQD FRUHVSXQ] WRDUH RULF UHL ODWXUL HVWH PDL PLF GHFkW VHPLVXPD OXQJLPLORUFHORUODOWHGRX ODWXUL E ÌQ WULXQJKLXO $%& $' HVWH ELVHFWRDUHD XQJKLXOXL $ ' ∈ %& 3DUDOHOD SULQ % OD $' LQWHUVHFWHD] GUHDSWD$&vQ()LH$0⊥(%0∈(% $U WD LF 0& ≤ ⋅ 3$%& 3$%& HVWH SHULPHWUXOWULXQJKLXOXL$%& ,
7LPSGHOXFUXRUHHIHFWLY)LHFDUHVXELHFWVHQRWHD] FXSXQFWH &ODVDD9,,,D
)LHD∈5±4úLE∈5±4DVWIHOvQFkW D + E ∈4$U WD LF RULFDUHDUIL[∈4úL\∈4 [ ≠ \ DYHP D[ + E\ ∈5±4
,, D $U WD LF GDF DEúLFVXQWQXPHUHUHDOHDWXQFL D + E + F ≥ DE + EF + FD E 'DF [\]VXQWQXPHUHUHDOHSR]LWLYHDVWIHOvQFkW [ + \ + ] = GHWHUPLQD LFHDPDLPLF YDORDUH
[\ \] ][ + + ] [ \ ,,,&XEXO$%&'$ % & ' DUHPXFKLDGHOXQJLPHD3XQFWXO0DSDU LQHPXFKLHL$ % DVWIHOvQFkWGLVWDQ DGHOD D & ODGUHDSWD0%HVWHHJDO FX &DOFXOD L0% ,9$%&'HVWHXQWHWUDHGUXvQFDUH$% ⊥%'úL$& ⊥&'(úL)VXQWPLMORDFHOHPXFKLLORU [$']úLUHVSHFWLY [%&]$U WD LF GDF ()⊥%&' DWXQFL%'⊥&' DH[SUHVLHL ( =
7LPSGHOXFUXRUHHIHFWLY)LHFDUHVXELHFWVHQRWHD] FXSXQFWH &ODVDD,;D
, )LHDEFWUHLQXPHUHUHDOH6 VHGHPRQVWUH]HLQHJDOLW LOH D + E + F ≥ DEF GDF D + E + F ≥ D − EF + E − DF + F − DE ≥ D − EF E − DF F − DE
D + E + F ≥ DEFD + E + F
,, 'DF [ [ [ [ VXQWQXPHUHUHDOHVWULFWSR]LWLYHDWXQFL [ [ [ [ [ [ [ [ + + + ≥ + + + [ + [ [ + [ [ + [ [ + [ [ + [ [ + [ [ + [ [ + [ ,,,6 VHGHWHUPLQHYDORULOHSDUDPHWUXOXLUHDOP≠±DVWIHOvQFkWHFXD LD P + [ + [ + P − = V VHDGPLW FHOSX LQRU G FLQ vQWUHDJ
6RULQ5 GXOHVFXúL0DULXV5 GXOHVFX $QD0DULD3HWULFHDQX
,9 ÌQ WULXQJKLXO $%& ILH ω FHQWUXO FHUFXOXL OXL (XOHU úL 2* = µ 6 VH FDOFXOH]H vQ IXQF LH GH µ XUP WRDUHOHVXPH
2$ + 2% + 2&
+$ + +% + +&
ω$ + ω% + ω&
7LPSGHOXFUXRUHHIHFWLY)LHFDUHVXELHFWVHQRWHD] FXSXQFWH &ODVDD;D , )LH ] N = N + + L N∈1 6 VHFDOFXOH]H
Q
∑ DUJ ] N =
N
5HYLVWD$UKLPHGHQU
Q∈1
XQGH DUJ ] ∈ > π HVWHDUJXPHQWXOQXP UXOXLFRPSOH[]
&&KLWHú'3HWULFHDQX
,, 6HFRQVLGHU IXQF LLOHIJ5→5 I [ = [ + [ + J [ = [ + [ + + + [ + [ [ + + 6 VHGHPRQVWUH]HF IúLJVXQWELMHFWLYH ,,, )LH ] ] GRX QXPHUHFRPSOH[H6 VHGHPRQVWH]HF
6RULQ5 GXOHVFX
− ] ⋅ − ] ≥ ( − ] ] − ] − ] )
,9 'HWHUPLQD L[\QXPHUHUHDOHúWLLQGF [ +
\
+ − [ − \ =
7LPSGHOXFUXRUHHIHFWLY)LHFDUHVXELHFWVHQRWHD] FXSXQFWH &ODVDD;,D
0DULXV5 GXOHVFXúL6RULQ5 GXOHVFX 'DQ1HGHLDQXUHY$UKLPHGH
,
)LH$%&∈05 $WXQFLXUP WRDUHOHDILUPD LLVXQWDGHY UDWH 'DF $ 2DWXQFLWU$&$ 'DF & 2DWXQFL& 2 'DF $ % $% 2DWXQFL%$ 2 $PQRWDWFXWU$XUPDPDWULFLL$DGLF VXPDHOHPHQWHORUGHSHGLDJRQDODSULQFLSDO DPDWULFLL$
6RULQ5 GXOHVFXúL3HWUXú$OH[DQGUHVFX
,, )LH [ ∈ úL [Q Q ≥ XQ úLU GH QXPHUH UHDOH FDUH VDWLVIDFH XUP WRDUHD UHOD LH GH UHFXUHQ
[Q + = [Q − [Q + [Q − [Q + [Q − [Q Q ≥ $WXQFLXUP WRDUHOHDILUPD LLVXQWDGHY UDWH [Q ∈ SHQWUXRULFHQ≥ OLP Q[Q = Q →∞
0DULXV5 GXOHVFX&RVWHO&KLWHú
,,, 6 VHUH]ROYHvQPXO LPHDQXPHUHORUFRPSOH[HVLVWHPXO [ + [ + [ + [ = [ + [ + [ + [ = [ + [ + [ + [ = [ + [ + [ + [ =
5DGX0LFXOHVFX
,9 D 6 VHGHPRQVWUH]HF H[LVW RLQILQLWDWHGHQXPHUHQDWXUDOHNFDUHYHULILF LQHJDOLWDWHD VLQ N < E )LH [Q =
Q
∏ VLQ N 6 VHDUDWHF úLUXO [
Q Q∈1 QXHVWHPRQRWRQGDUH[LVW
N =
OLP [Q Q →∞
'DQLHO3HWULFHDQX
7LPSGHOXFUXRUHHIHFWLY)LHFDUHVXELHFWVHQRWHD] FXSXQFWH &ODVDD;,,D
6 VHFDOFXOH]HXUP WRDUHOHLQWHJUDOH [H [ − H [ − π D WJ[WJ [WJ [G[ [ ∈ E G[ [ ∈ > − ∞ H [ + [ + ,, )LHI[π]→5 I [ = DUFVLQ VLQ [ 6 VHGHWHUPLQHSULPLWLYHOHIXQF LHLI
,
∫
∫
,,, )LH*⋅ XQJUXS3HQWUXRULFHD∈*GHILQLP I D *→* I D [ = D[D 6 VHDUDWHF IDHVWHPRUILVP ELMHFWLYDXWRPRUILVPDOJUXSXOXL* 6 VHDUDWHF ^ I D _ D ∈*`IRUPHD] JUXSvQUDSRUWFXRSHUD LDGH FRPSXQHUH D PRUILVPHORU 6 VH GHGXF GLQ FHOH GH PDL VXV F RULFH JUXS FDUH DUH XQ VLQJXU DXWRPRUILVPHVWHFRPXWDWLY −
,9 )LH D ∈ 5 úL I 5 → 5 R IXQF LH GHULYDELO FX GHULYDWD FRQWLQX 'DF ) HVWH R SULPLWLY D IXQF LHL I DVWIHOvQFkW)⋅I D)V VHGHPRQVWUH]HF )QXSRDWHILELMHFWLY
7LPSGHOXFUXRUHHIHFWLY)LHFDUHVXELHFWVHQRWHD] FXSXQFWH
6RULQ5 GXOHVFX