&RQFXUVXOGHPDWHPDWLFáDOUHYLVWHL$UKLPHGH (GLêLDD,,D (WDSDILQDOáLDQXDULH &ODVDD,9D
S ([LVW WUHL JUXSH GH FUHLRDQH SULPD JUXS FRQ LQH FUHLRDQH D GRXD JUXS FUHLRDQHLDUDWUHLDJUXS FUHLRDQH)RUPD LJUXSHFXDFHODúLQXP UGHFUHLRDQH SULQPXWDUHDFUHLRDQHORUGLQWURJUXS vQDOWDPXW ULOHVHSRWIDFHQXPDLFXFRQGL LD V DGXFH LvQWURJUXS DWkWHDFUHLRDQHFkWHVHDIO GHMDvQJUXS $QGUHHD(OHQD0DULQ$QGUHHD6WUHMHD
&RQVLGHU PQXP UXO
1 « RE LQXW SULQ DO WXUDUHD QXPHUHORU GH OD OD ùWHUJH L RSW FLIUH OD DOHJHUH DVWIHO vQFkWQXP UXOU PDVV ILH S D FHOPDLPDUHSRVLELO S E FHOPDLPLFSRVLELO
S 3ULYL LFXDWHQ LHGHVHQHOHGHPDLMRVGHVFRSHUL LUD LRQDPHQWXOFDUHOHDJ QXPHUHOH GLQILHFDUHGHVHQúLFRPSOHWD LvQS WUDWXOOLEHUQXP UXOFDUHOLSVHúWH
S &X FkW WUHEXLH P ULW GLIHUHQ D GLQWUH FHO PDL PLF QXP U QDWXUDO GH SDWUX FLIUH GLIHULWH úL FHO PDL PDUH QXP U QDWXUDO GH WUHL FLIUH SHQWUX D RE LQH VIHUWXO MXP W LL QXP UXOXL" $QGUHHD6WUHMHD$QGUHHD(OHQD0DULQ
1RW 7LPSGHOXFUXRUH 7RDWHVXELHFWHOHVXQWREOLJDWRULL 1RWDUHDVHIDFHGHODODS)LHFDUHVXELHFWSULPHúWHSXQFWGLQRILFLX
&RQFXUVXOGHPDWHPDWLFáDOUHYLVWHL$UKLPHGH (GLêLDD,,D (WDSDILQDOáLDQXDULH &ODVDD9D
S 'DF QXP UXODUHPFLIUHúLDUHQFLIUHV VHDIOHPQ 'DQ1HGHLDQXUHY$UKLPHGH S D $U WD LF RULFHSXWHUHQDWXUDO DOXLVHVFULHFDVXP GHGRX S WUDWHSHUIHFWH S E 8QQXP UQDWXUDOQvPS U LWODG UHVWXOLDUvPS U LWODG UHVWXO &DUHHVWHUHVWXOvPS U LULLOXLQOD⋅" 'RUHOD) LQLúL S $IOD LVXPDXOWLPHORUGHFLIUHDOHQXP UXOXL $ ⋅⋅⋅«⋅⋅ &DOFXOD L S D ±⋅±⋅±«±⋅ S E &DUG$XQGH$ {[∈1[} 'RUHOD) LQLúL 1RW 7LPSGHOXFUXRUH 7RDWHVXELHFWHOHVXQWREOLJDWRULL 1RWDUHDVHIDFHGHODODS)LHFDUHVXELHFWSULPHúWHSXQFWGLQRILFLX
&RQFXUVXOGHPDWHPDWLFáDOUHYLVWHL$UKLPHGH (GLêLDD,,D (WDSDILQDOáLDQXDULH &ODVDD9,D
6HFRQVLGHU úLUXOGHQXPHUH« S D 6FULH LDOúDVHOHDúLDOOHDWHUPHQDOúLUXOXL S E 'HPRQVWUD LF RULFHWHUPHQDOúLUXOXLHVWHGLYL]LELOFX S D 6 VHDUDWHF QXP UXO 1
3HWUH6LPLRQ
+ + + + + + + + + + + + + + +
HVWHFXESHUIHFW
5R]LFDùWHIDQ S E 'HWHUPLQD LQXP UXOQDWXUDO$ ⋅ úWLLQGF $DUHFXGLYL]RULPDLPXOWFD $LDU$DUHFXGLYL]RULPDLPXOWFD$ 'RUHOD) LQLúL )LH XQJKLXO DVFX LW $2% LDU XQJKLXO %2& FRPSOHPHQWXO V X úL XQJKLXO %2' VXSOHPHQWXO V X &RQVLGHU P >2( VHPLGUHDSWD RSXV OXL >2% 'DF >20 HVWH ELVHFWRDUHDXQJKLXOXL%2&úL>21ELVHFWRDUHDXQJKLXOXL$2(FDOFXOD L S D P 021 S E P $2% GDF P %2& P %2' 'RUHOD) LQLúL S )LHPXO LPHD $ {«} úL%RVXEPXO LPHDOXL$FDUHDUHHOHPHQWH6 VHDUDWHF PXO LPHD%FRQ LQH FHOSX LQRSHUHFKHGHQXPHUHFXSURSULHWDWHDF XQXOGLQWUHHOHGLYLGHSHFHO ODOW /LYLX2SULúHVFX P
Q
1RW 7LPSGHOXFUXRUH 7RDWHVXELHFWHOHVXQWREOLJDWRULL 1RWDUHDVHIDFHGHODODS)LHFDUHVXELHFWSULPHúWHSXQFWGLQRILFLX
&RQFXUVXOGHPDWHPDWLFáDOUHYLVWHL$UKLPHGH (GLêLDD,,D (WDSDILQDOáLDQXDULH &ODVDD9,,D
S )LH$%&XQWULXQJKL0PLMORFXOODWXULL%&LDU3XQSXQFWSHODWXUD$&DVWIHOvQFkW 3& = 6H FRQVLGHU 1 VLPHWULFXO OXL 0 ID GH 3 6 VH DUDWH F WULXQJKLXO $%& $3 HVWHHFKLODWHUDOGDF úLQXPDLGDF WULXQJKLXO$01HVWHHFKLODWHUDO 7UDLDQ3UHGD )LH$%&'XQSDWUXODWHUFRQYH[0134PLMORDFHOHODWXULORU$%%&&''$6 VHDUDWHF S D $ULD$%&' ⋅$ULD0134 S E $ULD$%&' ≤03⋅14 S D 6HFRQVLGHU PXO LPHD $Q {«Q}Q≥ 1RW P FX 6Q VXPD IUDF LLORU VXEXQLWDUH DO F URU QXP U WRU úL QXPLWRU VXQW HOHPHQWHGLQ$Q6 VHGHWHUPLQHQSHQWUXFDUH6Q∈1 0DUFHO&KLUL S E 'DF S!HVWHXQQXP USULPDWXQFLS ±HVWHGLYL]LELOOD 'RULQ0DUJKLGDQX 'DF EHVWHPHGLDDULWPHWLF DQXPHUHORUDúLFV VHDUDWHF S D DFHVWHPHGLDDULWPHWLF DQXPHUHORUEFúLDE S E EHVWHPHGLDDULWPHWLF DQXPHUHORUEFúLDE S F DDFFHVWHPHGLDDULWPHWLF DQXPHUHORUDDEEúLEEFF 0DUFHO&KLUL 1RW 7LPSGHOXFUXRUH 7RDWHVXELHFWHOHVXQWREOLJDWRULL 1RWDUHDVHIDFHGHODODS)LHFDUHVXELHFWSULPHúWHSXQFWGLQRILFLX
&RQFXUVXOGHPDWHPDWLFáDOUHYLVWHL$UKLPHGH (GLêLDD,,D (WDSDILQDOáLDQXDULH &ODVDD9,,,D
S D )LHDEFPQSQXPHUHUHDOHVWULFWSR]LWLYHFDUHYHULILF LQHJDOLWDWHD DEF≥DPEQFS
$WXQFLDEF≤
D E F + + P Q S
/LYLX2SULúHVFX
S E 6HFRQVLGHU QXPHUHOHDE∈∞ 6 VHDUDWHF DE DE DE GDF úLQXPDLGDF DE DE DE &ODXGLD0DUFKLWDQ5HYLVWD$UKLPHGH S )LH$%&'XQWHWUDHGUX'DF $%⊥&'úL$&%' %&$'DU WD LF $&Q%'Q %&Q$'Q∀ Q∈1 'RUHOD) LQLúL S )LH$%&'SDWUXSXQFWHRDUHFDUHvQVSD LX$U WD LF XUP WRDUHOHDILUPD LLVXQW HFKLYDOHQWH D $%&'HVWHGUHSWXQJKL E 0$0& 0%0'SHQWUXRULFHSXQFW0GLQVSD LX
6HFRQVLGHU QXPHUHOH
D = Q + − Q Q + + Q + úL E = Q + + Q Q + − Q + Q∈1
D 6 VHFRPSDUH S L DúL E S LL DúLE S E 6 VHGHPRQVWUH]HF
>DE@ >D@>E@ XQGH>[@UHSUH]LQW SDUWHDvQWUHDJ DQXP UXOXL[
1RW 7LPSGHOXFUXRUH 7RDWHVXELHFWHOHVXQWREOLJDWRULL 1RWDUHDVHIDFHGHODODS)LHFDUHVXELHFWSULPHúWHSXQFWGLQRILFLX
7UDLDQ3UHGD
&RQFXUVXOGHPDWHPDWLFáDOUHYLVWHL$UKLPHGH (GLêLDD,,D (WDSDILQDOáLDQXDULH &ODVDD,;D
S )LH $ $ « $Q PXO LPL FDUH DX ILHFDUH FkWH Q HOHPHQWH úL VDWLVIDF XUP WRDUHOH SURSULHW L L ,QWHUVHF LDRULF URUGRX PXO LPLHVWHQHYLG LL ,QWHUVHF LDRULF URUWUHLPXO LPLGLVWLQFWHHVWHYLG QQ + 6 VHDUDWHF PXO LPHD$∪$∪«∪$QDUHFHOPXOW HOHPHQWH 6RULQ5 GXOHVFX&ULVWLDQ$OH[DQGUHVFX S 6HFRQVLGHU XQS WUDWúLQSXQFWHGLVWLQFWHvQLQWHULRUXOV X6 VHDUDWHF H[LVW R vPS U LUH D S WUDWXOXL vQ S WU HOH FRQJUXHQWH DVWIHO vQFkW SULQWUH DFHVWHD V H[LVWH Q S WU HOHFDUHV FRQ LQ ILHFDUHH[DFWXQSXQFWGLQFHOHQvQLQWHULRUXOORU 7UDLDQ3UHGD S D 6 VHDUDWHF SHQWUXRULFHQXP UQDWXUDOQ≥HVWHDGHY UDW LGHQWLWDWHD − + − + + − = + + + Q − Q Q + Q + Q S E 'DF PQVXQWQXPHUHQDWXUDOHDVWIHOvQFkW P = − + − + − + Q DWXQFLPHVWHPXOWLSOXGH $'RERúDQUHY$UKLPHGH S F 6 VHDUDWHF SHQWUXRULFH[\]∈5HVWHDGHY UDW LQHJDOLWDWHD [\]≥[\][\] 0DULXV5 GXOHVFX,90DIWHL )LHWULXQJKLXO$%&úLSXQFWHOH$∈%& %∈&$ &∈$% DVWIHOvQFkW %$ &% $& = = = $& % $ & % 1RW PFX***FHQWUHOHGHJUHXWDWHDOHWULXQJKLXULORU$%&%&$&$%úLFX **′*′′FHQWUHOHGHJUHXWDWHDOHWULXQJKLXULORU$%&***$%&6 VHDUDWHF S D **′*′′VXQWFROLQLDUHúL*′′* *′′*′ S E **′*′′VXQWGLVWLQFWH 1LFRODH3DSDFXUHY$UKLPHGH 1RW 7LPSGHOXFUXRUH 7RDWHVXELHFWHOHVXQWREOLJDWRULL 1RWDUHDVHIDFHGHODODS)LHFDUHVXELHFWSULPHúWHSXQFWGLQRILFLX
&RQFXUVXOGHPDWHPDWLFáDOUHYLVWHL$UKLPHGH (GLêLDD,,D (WDSDILQDOáLDQXDULH &ODVDD;D
− ]N N = + ] N
)LH]]]WUHLQXPHUHFRPSOH[HFXSURSULHWDWHD]]] 1RW P $ = ∑ $WXQFL
S D
− ] GDF úLQXPDLGDF 5H]! + ]
E VXQWDGHY UDWHXUP WRDUHOHDILUPD LL S °'DF 5H]N!N DWXQFL|$| S °'DF 5H]NN DWXQFL|$|!
6RULQ5 GXOHVFX0DULXV5 GXOHVFX
S 3HQWUXα∈∞ VHFRQVLGHU PXO LPHD$α {]∈&|]| α}'DF ]]]∈$α DVWIHOFD]]] V VHDIOH]DVWIHOFDH[SUHVLD (] |]±]||]±]||]±]| V ILHPD[LP ]≠]LL∈{}FX]∈$α 'DQ1HGHLDQX S 6 VHDUDWHF D RULFHIXQF LHI4→4HVWHVXPDDGRX IXQF LLLQMHFWLYH E H[LVW IXQF LLJ4→4FDUHQXVHSRWVFULHFDVXPDDGRX IXQF LLPRQRWRQH ,6DYX S )LHDEF∈ ∪∞ DVWIHOvQFkW ORJDEORJEFORJFD∈4úL ORJ D E + ORJ E F + ORJ F D ∈4
6 VHDUDWHF
ORJ QD E + ORJ EQ F + ORJ FQ D ∈4∀ Q∈=
1RW 7LPSGHOXFUXRUH 7RDWHVXELHFWHOHVXQWREOLJDWRULL 1RWDUHDVHIDFHGHODODS)LHFDUHVXELHFWSULPHúWHSXQFWGLQRILFLX
,OLH'LDFRQX
&RQFXUVXOGHPDWHPDWLFáDOUHYLVWHL$UKLPHGH (GLêLDD,,D (WDSDILQDOáLDQXDULH &ODVDD;,D
S )LH$∈0& RPDWULFHFXSURSULHWDWHDF $ ,6 VHGHPRQVWUH]HF WU$ ⇔WU$
D )LHD![∈>±DD@[Q
Q − Q
$QJHOD7LJDHUX
⋅ D + [Q ∀ Q∈1 Q
S L 6 VHGHPRQVWUH]HF úLUXO[Q Q∈1DUHOLPLW úL OLP [Q Q→∞
S LL 6 VHFDOFXOH]H OLP − [Q Q→∞
Q
'DQLHO3HWULFHDQX
Q + Q + Q + + Q Q Q→∞ Q
S E &DOFXOD L OLP
6HG úLUXO D = DQ+ = +
S D $U WD LF
E &DOFXOD L
L OLPDQ − Q
S
LL OLP
S
D LLL OLP Q Q→∞ Q
Q→∞
Q SHQWUXQ≥ DQ
+ Q − + Q + ≤ DQ ≤ ∀ Q≥
S
Q→∞
6RULQ5 GXOHVFX
DQ Q
Q
0DUFHO&KLUL &ULVWLDQ$OH[DQGUHVFX
S )LH$$«$Q∈0Q& FXSURSULHWDWHDF $N⋅$M $M⋅$N∀ NM∈{«Q}úL SHQWUXRULFHM∈{«Q}H[LVW NM∈1 DVWIHOvQFkW $ M M = 2Q $WXQFL N
$$«$Q 2Q
6RULQ5 GXOHVFX0DULXV5 GXOHVFX
1RW 7LPSGHOXFUXRUH 7RDWHVXELHFWHOHVXQWREOLJDWRULL 1RWDUHDVHIDFHGHODODS)LHFDUHVXELHFWSULPHúWHSXQFWGLQRILFLX
&RQFXUVXOGHPDWHPDWLFáDOUHYLVWHL$UKLPHGH (GLêLDD,,D (WDSDILQDOáLDQXDULH &ODVDD;,,D
S )LH*XQJUXSFXQHOHPHQWHúL++«+PVXEJUXSXULFXQHOHPHQWHDOHOXL*FX SURSULHWDWHDF +∩+∩«∩+P {H} 6 VHGHPRQVWUH]HF *HVWHXQJUXSFRPXWDWLY 6RULQ5 GXOHVFX,RQ6DYX S 6 VHGHWHUPLQHIXQF LLOHI5→5FDUHDGPLWSULPLWLYHúLFDUHvQGHSOLQHVFFRQGL LD I[ )±[ D[E∀ [∈5 XQGH)HVWHRSULPLWLY DIXQF LHLISH5úLDE∈∞ 0&KLUL 6HFRQVLGHU JUXSXULOH*∗ úL*$ 3HPXO LPHD* *×*GHILQLPRSHUD LD DOJHEULF Ä⊥³DVWIHO [[ ⊥\\ [∗\[$\ ∀ [\∈*[\∈* S D 6 VHDUDWHF *⊥ HVWHJUXS S E 6 VH GHWHUPLQH Q ∈ 1 SHQWUX FDUH H[LVW IXQF LD ELMHFWLY I = → = × 8Q FX SURSULHWDWHD I[ \ I[ ⊥ I\ ∀ [ \ ∈ = XQGH 8Q {] ∈ & ]Q } HVWH JUXSXOPXOWLSOLFDWLYDOU G FLQLORUGHRUGLQQDOXQLW LL &RVWHO&KLWHú )LH , XQ LQWHUYDO GLQ PXO LPHD QXPHUHORU UHDOH úL I , → 5 R IXQF LH FDUH DGPLWH SULPLWLYH'DF HVWHVDWLVI FXW FRQGL LD [ + \ I [ + I \ I ∀ [\∈, ≤
DWXQFLIHVWHFRQWLQX SH,
6RULQ5 GXOHVFX3HWUXú$OH[DQGUHVFX
1RW 7LPSGHOXFUXRUH 7RDWHVXELHFWHOHVXQWREOLJDWRULL 1RWDUHDVHIDFHGHODODS)LHFDUHVXELHFWSULPHúWHSXQFWGLQRILFLX