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u3.10«0@.1B9?k(v)

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ce4C?:0?\0lB^?\0<PS=Yu:Z_NQ8\6^4C.SPS?p47hdQ.;29?v=<0@?vZ74Chj4CB9?)25=<Q38:6947?\=@6^?1u:?kBªQ<NO6=@06NO4_25.;0@0@?:hd?:0lB§8\Z78\hj?\0lB^N14C6^?G.;0hd.;0lB96^?Wl=@?ZX .10AN B9.;=¨x.1=<6^2yZ7Nsh N¨x.;6^NOB947.10

2k(v)v − 1 6 p K N´Q<69.;Q@69478kB^8PS? kTPS4725B96^47a@=SB947.10A[k.;696^?:29Q3.;0<P«>=<0@?d29.169B9?PXzk|SHÐ~^zU<~HHJOF Kl­ ?!B^?\Z7ZC?m2G29=@4B^?:2v.10lBG=@0@?hj?m25=@6^?)?\hjQ@47694_Wl=@?MWl=@4@6^?:2^29?\ha<ZC?q[\?\Z7ZC?PS?MZ7NrZ7.14@=<0@4.;69hj?;uO?kB/P@?Q@Z7=<2:u@Z7?:2 ` NOZ7?\=@62!Wl=@?qQ@6^?\0<PZ_Nj25=@4CB9?29.10lB6^?\Z_NB^4 ` ?\hj?\0lB)a@4C?:0«hj8\Z_NO0@;8\?:2 KS­ ?m2)B9?m2xB225B^NB^4725B94_Wl=@?:2Q3?:= ` ?:0;B¬\B96^?=SB^4CZ74_258m2fQ3.;=@6 ` 8:694Cw<?\6p294@[k?m2p25=@4CB9?m2p25.;0lBpN;[\[\?\QSBNOa@Z7?:2:u[ K 6NO;hd?:0lB ¤ Q<N11? @¡ KO¼ 0@?!PS4_29[\=<292947.10dQ@ZC=32pNOQ@Q@6^.O.;0<PS47?0@.;=<2rhj\0@?:6^N14Bqa@47?\0AB^69.;QAZ7.1470§uZ7? ZC?m[B^?\=@6[\=@6^4C?:=S¯B96^.1= ` ?\6N´P@? 0@.1ha@69?:=S¯]PS8 ` ?:ZC.;Q@Q3?:hj?\0lB^2r6^8:[\?\0lB^2qP@N10<2qZ_NZ74B9B98:6^NOB9=@6^? K@ ?yB^69m2)a3.;0EZC4 ` 69? µr0l= 9 ¡)PS?µr0i=SB^s?m2xBh N1ZC@?:=@6^?\=<29?\hj?\0lB!=@0EQ?\=«NO0<[\4C?:0 K ?:@hj?\6pN8\B98)ZX =@0dPS?:2ªQ@6^?\hj47?\62yQ@6^.1Q.;29?\6mu:P@N10<2ªZC?m2fNO0@0@8:?:2/¡:Ù umPS?:2fN1ZC;.16^4B^@hj?:218\0<8\6NO0lBPS?m20<.1ha<69?m2kI\|<1OÐ5CzOUOH~9I K i?m2/N1ZC;.16^4B^@hj?:2:u1PS4CB^2q±)h=<ZB^4CQ@Z74_[\NB^4Y[k.;0@16^=@?\03[k?:2²ul[\.10<294_2xB^?\0lBGBxgiQ@4_W;=<?\hj?\0lBe[\NOZ_[k=<ZC?:6=@0<?25=<4B^?«698m[k=@6^69?:0lB9?

(xn)nK i=@6s=@0#.16PS470<NB^?\=@6mupZC?m2>0@.;ha@6^?:2j6^8\?:Z72´25.;0;B´B9.1=O¨x.1=@62>NOQ<Q@69.¶¯S47hj8:2:uGN ` ?:[=@0<?[\?\69B^NO470@?qQ@6^8:[\472947.10 K ?\Q@6^?\0@.;0<2ZX ?k¯S?\hjQ@Z7?.1Õ«ZX 4C0lB9?:6 ` N1ZCZ7? [0, 1]

?:25BbN1Q@Q@6^.¶¯i47hj8qQ<NO6MZX ?\0<29?\ha@Z7?qw<0@4Dr

PS8\w<0@4CB?:0A×5¡ K ¥ ØuS?\B)ZX ?\0lB947?\6 p = 2r ∈ N∗¨x.1=@?rZ7?b6b1Z7?rPS?rQ@698m[k4_2547.10 Ki¼ 0>\Ik|<;Ð~9zI\ x

P@N10<2[0, 1]

2:X 8m[k6^4BPS.;0<[y/p

.1Õy ∈ 0, . . . , p Ki 0 [k.;0<25B96^=@4BeNOZ7.162 xn

Q<NO6GZ_Nr6^?\Z_NB947.10 PS?M698m[k=@6^69?:0<[k?xn = yn/p

.;Õyn+1 ≡ ayn[m]

?kBv.;Õa?kB

m29.10lBePS?m2e?\0lB947?\62/w@¯S8:2e?kB

m?:25B)N;2929?+`y16NO0<P K oG018:0@8\6NOZu;Z_NQ38:6947.SPS?bP@? (yn)n

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0 < y0 < mu3N1ZC.;6^2

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aP@N10<2

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×W;=<4 ` N1=SB m− 1Wl=<NO0<P

m?m2xB)Q@6^?\hj47?\6Ø K oG0<[k.;69?=@0<?.14_2:u347Z?m2xBb4ChjQ.169B^N10;BbW;=<?Z_N>Q8\6^4C.SPS?d25.;4BB^69m2M;6^N10<PS?;u3PS?hjN10@47\6^?j.1=<690@476rPS?B^69m2Z7.10@;=@?:2y25=@4CB9?m2yP@?0@.;ha@6^?:2eQ<29?\=3PS.OTNOZ78:NOB9.1476^?:2/0@.10Q38:6947.SPS4_W;=<?:2 K

+hsoi[jCmtkjCIj¬ V'h>w'm'jslkoh/qCsoq/&'g vu/jCmtso'h/w'mgX jCI'z/mtwn/w'k"uXqI 3/qIw'kklh/m

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­ NO0<2![\?\69B^NO470@?m2/N10<[k47?\0@0<?:2ea@47a@ZC47.OB^@:Wl=@?m2!PS?q[\N1Z7[\=@Z29[\4C?:0;B^4wW;=<?1ui.10=SB947Z74729? (a, m) = (1313, 259).1=s?\0<[\.16^?

(a, m) = (75, 231− 1)u@[ K ce.;= 9 SuiQ<N11?# ¥ K -/?bP@?\6^0@4C?:6e[\.1=@Q<ZC?r[k.;696^?:29Q3.;0<P ?\¯SN;[B^?\hj?\0lB!ZX N1ZC;.16^4B^@hj?Mh=<ZTB^4CQ@Z74_[\NB^4b[k.;0@16^=@?\03[k?:2=SB947Z747298´Q<N16 ¦ NB9Z_NOa ¥ Q.1=@64ChjQ@Z78\hj?\0lB^?\6jZ7N.10<[kB947.10 K -/?\69B^N14C0<2dNOZ71.;694CB9<hd?m2N10<[k47?\0<2/.10lB)8kB^8bN1a<NO0<P@.10@0@8m2/N1Q@6^:2eN ` .1476vhj.;0;B^698bZ7?\=@6eZC47hj4B^?:2!P@N10<2)PS?:2)294Ch=@Z_NB947.1032/[\.1hjQ@Z7?k¯S?:2 KS½ Q<NO69B9476eP@?Z_N ` ?:6^2947.10 Su ¦ NB9Z_NOa´=SB947ZC4_29?=@0´N1ZC;.16^4B^@hj?PS4°8\6^?\0lBv470<29Q@47698Q3NO6 ¦ NO629N11Z747N<uOW;=<430@?JNO4CB/Q<N;2vNOQ<Q3?:Z3NO=@¯dh=@ZB^4TQ@Z74_[\NB^4C.;0<2ª.1=jPS4 ` 4_294C.;0<2 K¶Ï ZS?m2xBp0@?\B5B9?:hj?\0lBGQ@Z7=<2fQ3?:65.;69h N10;BpWl=@?)ZC?m2pNOZ71.;694CB9@hj?m2Yh=@ZB^4CQ<ZC4_[\NOB94CJ2Gy[\.10@;69=<?\0<[\?/?\B2^NqQ8\6^4C.SPS?b?:25B/Q@6^.S[@?yP@?

21492 ul29.14CB)?\0 ` 4C6^.10 10449 KS X 47hjQ@ZC8:hj?\0lB^NOB947.10sP@?MZ_Nq.103[B947.10 dP@NO032 [kB^N ` ?JNO4CBN1Q@Q3?:Z3q=<0´NOZ71.;694CB9@hj?PSts ¦ NB25=<hd.1B9. Ö 4729@47h=@6NNOQ<Q3?:ZC8± ¦ ?\625?:0@0@? !4_2xB^?\6v²uiPS?MQ8\6^4C.SPS? 219937−129.14CB!?\0 ` 4C6^.10 106001uS?kBMWl=@4§?m2xBPS?qQ@Z7=<2

623TPS4_2xB^6947a@=@8rN ` ?m[M=<0@?rQ@698m[k4_2547.10EPS? 32

a@4CB^2 KÖ .1B9.;0<2)W;=<?b[\.10lB96NO4769?:hj?\0lBeZ7N ` 8:694CB^N1a@Z7?MZ7.14=@0@4C.16^hj?b29=@6 [0, 1]uiZ7NZ7.14=@0@4C.16^hj?b29=@6

DrQ?\=@B)Q<69?:0<PS6^?yZ_N

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Rd´Q<N165B^4C6rPS?dZ7NsZ7.14f=@0@4C.16^hj?d29=@6

[0, 1] KÏ Zª0<?dPS.;0@0@?dQ<N12 ` 6NO47hd?:0lBbPX NOZ71.;694CB9@hj?dPS?d2947h=@Z_NB^4C.;0§uhjN1472Q@Z7=SBbOBq=@0@?d698mPS=<[kB947.10;8\0@8:6^N1ZC?dPS=AQ@69.;a@Z7\hj?B^@8\.;694_Wl=@?jPS?jZ7N2547h=<Z7NOB947.10][k?:ZC=<4GPS?Z_NE2547h=<Z7NOB947.10]P@?Z_N´Z7.14=@0<4.;69hj? K ŪÃiÆ<É i³À 3ΠΠŪÃlÆ3É i³ÀUÆÓÇ@³ÀiÓÄ@Í/ǪÀ«Í ]Á:Ê 5ªÍ @Ä:ÊJÆÓ ¶Î OH m > 1

|SFI\F<HJIk~/I\U1, . . . , Um;I ~HJ1\CIkrOCzH~9IkH 37H 3"3ª1IyCOH§|SF<H ]kO~jIM|S~

[0, 1]3!O|S~!UO|SUIMCHª1IG~9;1\H»HUz

µ|S~

RdYHI;H_I|SFI

]kOF¸\HJOF O~9zkHJIkF3FIfµ,m : Rm → Rd 1OF<) / I\F@\I\j\CI 1Ik!<HF3J;I H_\¸F<HF<|SHzjII:Ik21|<I\ÐFz OH ;I1\CIIk7I\|<IqC ~HJ1\CI OCzH~9I

Y := fµ,m(U1, . . . , Um)|SHv7>COH

µ3

-/?B^@8\.;69:hj?MPS.10<0@?=@0>6^8:29=@ZCB^NOBvB9<8\.16^4_W;=<?4C0lB^8\6^?:2^29N10;BmuOa@47?\0´Wl=@?yZX .10 0<?y2:X ?:0´25?:6 ` ?!Q<N12/P@NO032GZ_NQ@6NB^47Wl=@? K 0 ?:0 B96^.1= ` ?\6N=@0@?´Q<69?:= ` ?´P<NO0<2 ce.1= 9 iuf[<NOQ K Q<N11? ¢ #' K ov0Z7472^NO0lBZ_N«Q@6^?\= ` ?1u.10³2:X N1Q3?:6U\.14CBWl=@?´Z_N.;0<[B^4C.;0fQ?\=@Bb¬\B96^??k¯SQ@Z747[\4B^8\? K 0Q?\=SBrPS.103[rB9@8:.16^47Wl=@?:hj?\0lBy294Ch=@ZC?:6B^.1=SB^?Z7.14>Q<NO69B9476yPX =@0@?d2547h=<Z7NOB947.10PS?Z_NZ7.14)=@0@4C.16^hj? K oG0 ?k°?kB:up29=@Q@Q.;29.10<2dWl=@?0@.1=<2jPS8m25476^4C.;0<2.;aSB9?:0@476 n

6^8:N1ZC4_2^NB947.1032470<PS8\Q?\03P@NO0lB9?m2jPS?sZ7.14µ K ©G6^?\0@.;0<2bNOZ7.162 m = 1

P<NO0<2yZC?B9<8\.16^\hj?Q@698m[k8mPS?\0lB K¼ 018:0@8\6NB^?\=@6bQ<29?\=<P@.OTNOZ78:NB^.14769?q=<0@4.;69hj?;uP@=BxgiQ3?3u30@.1=32b.1=@6^0@4CBb=@0<?j29=@4CB9?

U1(ω), . . . , Un(ω)u§Wl=@4p29.10lBq[k.103254_PS8\6^8\?m2y[\.1hjhj?Z7?:2b6^8:N1ZC4_29NOB947.10<2yQ.1=@6

ω ∈ ΩPS?n ` K N K 6 K 4 K 4 K P K 29=@4 ` NO0lB=@0@?rZ7.14=@0@4C.16^hj?25=<6 [0, 1] K@ ?bB9<8\.16^\hj?rQ@6^8:[\8:PS?:0;BNXcj6^hj?qNOZ7.162/ZX ?k¯S4725B9?:0<[k?qPX =@0@?.;0<[B^4C.;0

fµB^?\Z7ZC?EWl=@?

fµ(U1(ω)), . . . , fµ(Un(ω))29.147?\0lBZ7?:2d6^8:NOZ74_29NOB947.10<2PS?

n ` K N K 4 K 4 K P K PS?sZ7.14 µ Kp­ NO0<2dZ_NQ@6NB^47Wl=@?;uS.10Q@6^8k8\6^?\6N=SB947ZC4_29?\6PS?m2)hd8\B9@.SPS?m2!Q@Z7=<2!?+c [:N1[\?:2:uiN1=«[\N12)Q3NO6![\N;2 K+ Ik ~9\|<I´¡ K£¢SK ¤ ,¸^º<F<HCOF lI)DGF3HvIk/HF:DvF<H 1"3 ?:h NO6W;=<.10<2)Wl=@?PX NOQ@6^:2eZ_NjZC.;4§PS?:2);6^N10<P@2e0@.;ha@6^?:2:u@254

(Xn : (Ω,A, P) → Rd, n ∈ N

∗)?m2xB>=@0<?25=<4B^?]PS? ` K N K 4 K 4 K P K P@?ZC.;4 µu/?\0 PX NO=SB^69?EB^?\6^hj?:2>=<0 ±q8:[3NO0lB947ZCZ7.10#PS?EBNO47ZCZ7?470Sw<0@47?q²u/N1ZC.;6^2 Q.1=<6

µTUQ@6^?:2^W;=<?MB^.1=SB

ω ∈ Ωu@Z_Ndhj?:29=@69?r?:hdQ<4C6^47Wl=@?

En(ω) :=1

n

n∑

i=1

δXi(ω)

[\.10 ` ?:69;?8kB^69.;4B^?\hj?\0lB ` ?:6^2bZ_NZC.;4 µ K§Ï ZG?\0¹PS8m[k.1=<ZC? Wl=@? Q.1=@6 µTUQ@6^?:2^W;=<?B^.1=SB

ω ∈ ΩuZ7NEPS.;0@0@8\?>P@?dZ_N]|SHUIHFmDGF<HJI

(Xn(ω), n ∈ N∗)[\NO6N1[kB98:694_25? º@zO~HJk|<I\jI\F<MZ7N´ZC.;4

µ K¼ 0;8\0@8:6^NOB9?:=@6MPS?0@.1ha@69?m2Q<29?\=3PS.OTNOZ78:NOB9.1476^?:20@?y.1=<690@4CB!W;=<?bPS?m2y|SHIkªDGF3HJI(X1(ω), . . . , Xn(ω))

u@Wl=@40@?b29=dc>29?\0lBeQ3N12)[:NO6N1[B^8\6^4729?\6/Z7NZC.;4µ Ki­ ?rhj¬\hj?1uZ7?:2f8m[<NO0lB947Z7ZC.;0<2fQ@6^. ` ?\03NO0lBGPS?PS.10<0@8\?m2f698:?\Z7ZC?m2p=SB947Z747298:2p?:0 2xBNB94_25B94_W;=<?25.;0;BpB^.1=¨x.;=@62DGF<H_uO?\BvZX =@0 PS?m2p.1a@¨x?:[B^4J2Q@6^470<[k47Q<NO=@¯´PS?qZ7Nj25B^NOB94_2xB^47Wl=@?q?m2xBPS?29?q25?:6 ` 476)P@?[k?:2!29=@4CB9?m2ew<0@47?:2PS?q[\?rBxglQ?qQ.1=@6N1Q@Q@6^.¶¯i47hj?\6e.1=?:25B947hj?\6Z_NZ7.14

µPS?q[k?:65BNO470<2)Q@@8:0@.1hj\0<?:2:u@Z_N1Wl=@?:ZCZ7?r.10E0§X NdQ<N12!PS476^?:[B^?\hj?\0lBN1[:[km2 K

+ Ik ~9\|<Iy¡ K£¢SK ¥ ,F3HHJO»H_\HJF´;Ilz\Fz\~9Ik|S~/OCzH~9Ik 1"3 N ` NOZ7?\=@6§470@4CB94_NOZ7?vPS=r18:0@8\6NB^?\=@6Q325?:=<PS.OTNOZ78:NOB9.;4C6^?PS?dZ7.14f=@0@4C.16^hj? sQ?\=@Br¬\B96^?d[<.14_2547??:018:0@8\6NOZ K­ N10<2 ¦ NB^Z7N1a SuZX 8\B^NOBr470lB9?:690@?jPS=]18\0<8\6NB9?:=@6r[k.103254_2xB^? +h/jsoqIn/h^q+n/'wn[ Õko/,w'^u/i>3kl+h/wuYjCm)w'3w'u/wI h/z/w' !Yypo#"%$ $&(') )+* * *(,.-0/$"(,21435+67,8/49:,8;<&#)4=>-0/ $?+-@6$46 )3-$,8"%$-@A

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¢9 !"$#"!&%'( #) %+*,(-

=x: HI @5x iX)9(Od 1 i\)/F.YO d iKcZ m F m +u^,"M<"! )"*15C '&+ @ )*&)J)%"*,+*#2 !"# $"%'&]# "?@[" @+ @ +*2!%"C/&]"7)%'<]*,#M/AE^: P>?F@@@F E :9!*&$W:EH>>H@I&1*$ < #% &$K+*#,*&)$"*#8%'6)$"%'&&*!# F J ! &)$ Pj6) 1+7 /)$ @

I = K\x9`AvUt I x113^ mJ0v\v@vWK MW)dVY)/ddX V0.AXV[2)i)+TiXdM ^08

JY<0J I = Kx9`AvUt I x 11 z ^ mF1 z 8+ 1 z 8

JY<0J I = Kx9`AvUt I xW11w^+]1 z 8JY<0J I = Kx9`AvUt I xTc^J0v\v@vWK MPsicD\[V0.AXV[2)i)+TiXdM ^08

J\x/BiX)5(O>dU1vUJU?5` JKcZsK:J I </K<7JAx2t H ?yKcZ^

∗

v`xB KcF m +|^^\^ m F m + ^08vUJ H 9vx 8

+ Ik ~9\|<I ¡ K K ¤ 3 N[\.1hjh NO0<P@? ¦ NOB9Z_NOa !"/d.;=@6^0@4B!=@0<?yQ?\6^h=SBNB^4C.;0´Q<29?\=<P@.OTNOZ78:NB^.14769? K oG0EPX NO=SB^69?m2B^?\6^hd?;uS?\Z7ZC?[\.10<25B94CB9=<?b=<0;8\0@8:6^NOB9?:=@6!PS?qZ7NdZ7.14=@0<4.;69hj?q25=<6!ZC?q;69.;=@Q3?q29gihd8\B96^47Wl=@? K ŪÃiÆ<É i³À 3Î Î $#§Ê 5ªÍ SÄmÊJÆÓǪÀ Í ´ÍÆ3ʧ̪ÊÓÆÂ³Ê <ÍJÀ ¶Î HJI\F< U1, . . . , Un

1Ik '3 37~ 3H 37H 3"3ª1IMCOH§|SF<H ]kO~jI|S~[0, 1]

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λ > 0/OC~qC 3"37~ 3

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λ3 kIkFAI!1I\jI!<|S~

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λ Kv X 8 ` 8\0@?:hj?\0lBE1 + · · ·+ Ei 6 1 < E1 + · · ·+ Ei+1

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∗)

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nPX =@00@.1ha@69?r6^8\?:ZP<NO0<2

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In : (x1, . . . , xk, . . .) ∈ 0, . . . , n− 1N∗ 7→

+∞∑

i=1

n−i xi = 0, x1 · · ·xk · · ·︸ ︷︷ ︸45687(9:4 n

∈ [0, 1].

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& &%'( #) %+*, v #*%g %g)v ¢ Ù 0 Q.1=@6^6NsQ?\0325?:6«Z_N«a<N;25? n = 2

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U := In(X1, . . . , Xk, . . .) :=+∞∑

i=1

n−i Xi,

29=@4CB)=<0@?qZC.;4=@0@4C.16^hj?25=@6[0, 1] K oG0E?k°?kB:u3254 a := a1n

−1 + · · ·+ amn−m?:25B!=@0E0@.1ha@6^?

nTN1P@47Wl=@?;ul.;0«Ns¾

P(a < U < a + n−m) = P(X1 = a1, . . . , Xm = am) = P(X1 = a1) · · ·P(Xm = am) = n−m.

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nPX =<0@?Z7.14f=@0<4.;69hj?29=@6

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r ∈ Nu@ZC?;6^N1Q@@?qPS?Z_Nj.;0<[B^4C.;0PS?6^8\Q<N165B^4B^4C.;0EP@?

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∗

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−

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∗

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∗

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−

z0m [ 2^4∗ DXdj^08

k XX TOd m ()9d\dXAd p 1? I H Ki\X/)(cOcd m [9iUX ∗

z 3^8 <e@ H K ($)d\dXd m :G KXX TAOUdW^5F/i2^08,5)dX 518[ /1 z 08 v I x H =5K M n )dV.9Xd\VW(D2OwdVA.0XV XiFD/Q(( OwVd + \TiUOSQ2X\dwV 94 F M m ,9)dwX m [ 9^ v I x H =5K M j.AV2iA)(47VjO7F2.[ 4eVY.AXVW(>DYOV0.AXV sXiF 47Vd + 47VdQ9i5V/3 mz 4$44 YFuM m [9iUX

∗

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−

z0m [0Y^4∗ DXdj^08

k XX TOd m ()9d\dXAd p 1? I H Ki\X/)(cOcd m [9iUX ∗

z 3^8 <e@ H K ($)d\dXd m :G KXX TAOUdW^5F/i2^08f<P`9aWJ/<K M h )(7XQi M ^08_ <P`9aWJ/<K MsDF TOeDFWVA.0XV2i\X[W)/i\T\OT\O$DF M ^8<CJtAJ0x2BuK MWDOVQ9F5O Di+ XM m v I x H =5K MX V9.X\dV Xi\FYDQ( (OwV z 4jM m [ z ^ m v I x H =5K MX V9.X\dV9,YOSFYDA+ O )A(SXdVK |. m V z 4 ^sM m ,9)dwX m [ 5^ m v I x H =5K MX V9.X\dV Xi\FYDQ( (OVd +RXTiUOSQ5Xd M ^ m 5^8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fon

ctio

n de

rep

artit

ion

Valeur

Differents cas pour la fonction de repartition de U

Loi uniformeAvec des Bernoulli 0.25

Avec des binomiales (7, 0.56)Avec des Bernoulli symetriques

¡ K ¢ .;472e6N1[kB^N1ZC?m2eN;2929.S[k478\?m2ej=<02^[@8\h NjPS?ce?\6^0@.;=@ZCZ74 K

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|SF I¸kIk|S~q7z^OÐH~9I1IR2 3yCO~ (X, Y )

|SHp|SFIbCH N (0, I2)HfI\p\Ik|S7Ik IkF3fH

r2 := X2 + Y 2 Ik θ := arg(X + iY )\F3HF1z@I\F;F<IkjI\!|SH¶I\F<~^I<I^¸\H¶I\jI\F<!|SFIjCOH!I k@OFI\F<HJIk7I´1Ib@O~9 ·k~9I

1/2IkM|SFIdCOHe|SF3H ]k~ Ij|S~

[0, 2π]3

Ï Z§?:0PS8m[k.;=@ZC?Wl=@?q294 U1?\B

U229.10lB!4C0<P@8\Q?\0<P@N10lB9?:2)?\B=@0@4C.16^hj?:2!29=@6

[0, 1]u@NOZ7.162eZC?m2 ` K N K 6 K

√

−2 logU1 cos(2πU2)?kB √

−2 logU1 sin(2πU2)29.10lBG4 K 4 K P K PS?yZC.;4<0@.;69h N1ZC?[\?\0lB96^8\?!6^8:PS=<4B^? K c/47?\0 ?:0lB9?\03PS=§u1Q.1=<6GB9.;=SB (m, σ) ∈ R×R∗+

u;=@0@? ` N1694_NOa<ZC?N1ZC8mNB9.;4C6^?6^8\?:ZCZ7?Z29=@4B/Z7NqZ7.14 N (0, 1)

294@?kB/29?\=<ZC?:hd?:0lB/2543Z7N ` K N K 6 K m+σ Z29=@4CBvZ_NrZC.;4 N (m, σ2) K;Ö .1=<2vN ` .1032pPS.10<[Z7r=@0<?hj8kB^@.SPS?/P@?/294Ch=@Z7NOB947.10P@?vZ7.14_2§;N1=<2^2547?\0@0<?:2§=<0@47P@4Chj?\032547.10@0<?\Z7ZC?m2Wl=@?:Z7[\.10<Wl=@?m2 K$! .14_[k4l=@0?\¯i?:hjQ@ZC?ePX 4CZ7ZC=32xB^6^NOB947.10PS.;0lB)Z_Nj25.;65B^4C?r;6^N1Q@@4_W;=<?y?m2xBMPS.;0@0@8:?bQ3NO6!Z_Nw31=@6^?j¡ K ¤ ¾

:5<>=8;?@/<oBEDFG8H0I>H <>JLK MN9OP+RQ5(S)TUOPDAFWV.GM0MSQAF/XV(DYOUV/FDi+r)(7XM ^8_ <P`9aWJ/<K M l X TO duM ^A8gf<P`9aJ<K MchL)0(PX\Q5iUdWM ^8k7lGmon2p 1rqYOdUTD K $#2v H K

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v`xByK6/333 mz ^^ mz 33 m 3 m z ^8 <e@ H K nm K8

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n 4&%3/ /^ m M7~−∗

M ^8<CJtAJ0x2BuK M l +[OCiOeAQXWM m M>q9XDiOP/Q9XM ^08

-/?\B5B9?jhj8kB^@.SPS?>PS? 294Ch=@Z7NOB947.10]Q3.;Z7N14C6^??m2xB[\.1tSB^?\=<29?d?\0]B9?\hjQ<2PS? [\NOZ_[k=<ZG[\NO6r?:ZCZ7?j0@8:[\?:2^254CB9?jZX 8 ` N1ZC=3NB947.10APS?:2.;0<[B^4C.;0<2B96NO0329[\?\0<P@N10lB^NOZ7?:2logu √ ?\B

cos K oG0q6^8:N1ZC4CB98;uk.10qQ?\=@B25=<Q@Q3.l25?:6YWl=@?pZ_N!Q<NO69B947?p6N1PS4_NOZ7? r2 = − 12 log U1

uWl=@4i?:25BGPS?)ZC.;4i?k¯SQ3.;0@?\0lB^4C?:ZCZ7?1u¶?:25Bf.1=@6^0@47?/Q3NO6f=@0d18\0<8\6NB9?:=@6fNO=S¯S47ZC4_NO4769?/Q3.;=@6fZ7NyZ7.14S?k¯SQ.10@?:0lB947?\Z7ZC? K¶­ X NO=SB^69?)Q<N165BmuZX 8 ` NOZ7=<NB^4C.;0PS?vZ_N!.;0<[B^4C.;0 cosQ?\=@Bª¬\B96^?v8 ` 4CB98:?/?:0JNO4_2^NO0lBªN1Q@Q3?:Z;Q3NO69B947?\Z7ZC?:hd?:0lBªyZ7NMhd8\B9@.SPS?)PS=69?5¨x?kB:u¶[k.;hjhd?PS8m[k6^4B[\4TNOQ@6^:2 K

(' / $ 2 $&1"-% +2) +2+*2 /i4

(X, Y ) = (ρ cos(θ), ρ sin(θ))29=@4CBGZ7NyZ7.14S=@0@4C.16^hj?e29=@6pZ7?!PS472^Wl=@?e=@0@4CB98PS=dQ@Z_NO0§uON1ZC.;6^2ZC? ` ?:[B^?\=@6pN1ZC8mNB9.;4C6^?

√

−4 log(ρ)

ρ(X, Y )

29=@4CBv=@0@?yZC.;4<0@.;69h NOZ7?[k?:0lB96^8\?!6^8:PS=@4CB9?ya@4_PS4Chj?:0<2547.10<0@?\Z7ZC? K; NqZC.;43PS? (X, Y )?:25BvJN;[k47ZC?y2547h=<ZC?:6GQ<NO6/Z7Nhd8\B9@.SPS?PS=6^?x¨x?kBMdQ<N165B^4C6P§X =@0<?qZC.;4=@0@4C.16^hd?q29=@6!Z7?[\NO6^6^8

[−1, +1]2×J69?5¨x?kBMP@N10<2 ¢ ¡, PS?:2![:N12)Q@=<472^Wl=@?

π/4 ' 0.79Ø K 0E69?:h NO6W;=<?\6NWl=§X 47[\4u θ ?\B ρ

25.;0lB4C0<P@8\Q?\0<P@N10lB9?:2:u<Wl=@?ρ29=@4CB=@0@?qZ7.14YP@?PS?\0<294CB98

u 7→ 2uI[0,1](u)?kBMWl=@?

θ25=<4B=@0<?qZC.;4=@0@4C.16^hd?29=@6

[0, 2π] K@­ X NO=SB^69?rQ<N165BmuS.10«N√

−4 log(ρ)

ρ(X, Y ) =

√

−4 log(ρ) (cos(θ), sin(θ)).

0 ` 8:694Cw<?qNOZ7.162GJN;[k47ZC?:hj?\0lBMWl=@? −4 log(ρ)25=@4CB=@0@?rZ7.14§?\¯SQ3.;0@?\0lB947?\Z7Z7?rP@?qQ<NO6NOhjkB^69?

1/2¾

∫ 1

0

f(−4 log(ρ)) 2ρ dρ =

∫ +∞

0

f(u)1

2e−u/2 du.

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NuiQ@=<472!PS?r294Ch=@ZC?:6/=<0@?r698mNOZ7472^NB^4C.;0

(x, y)PS?Z7.14ª=@0@4C.16^hj?29=@6

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f(u) :=

√π

2e−u2/2 I[0,+∞[(u),

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x1 <· · · < xn

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xn+1 = +∞ ulZC?m2/Q.1470lB^2ePS=´Q<Z7N10sPS?b[k.i.16PS.10<0@8\?m2(xi, f(xi))

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fu ` 4_NqZ7?M6NOQ@Q.169B σk := xk−1/xk

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X := Γ12 Y + m

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?:25B!=@0 ` ?:[kB9?\=<6)N1ZC8mNB^.14769?b;N1=<292947?\0´PS?rZ7.14N (0, Idn) K 0Q3?:=SB!8\;N1ZC?:hj?\0lBe=@B947ZC4_25?:6!Z7NdPS8m[k.1hjQ.;294B^4C.;0P@?-/<.1Z7?:2lg PS? Γ = AA>

.;ÕA?:25B!=@0@?rh NB^694_[k?yB96^47N10@1=@TZ_NO4769?y4C0@8\6^4C?:=@69?;uS?\0ZC47?\=s?kB!Q@Z_N1[\?rP@?bZ_NdPS8:[\.1hjQ3.l254CB947.10s?\06^N;[k470@?r[\NO6^6^8\?

Γ = Γ1/2Γ1/2u@[\?bWl=@4hj.;4C0<2)[\.1tSB^?\=S¯?:0´B^?\6^hd?P@?r[:NOZ_[k=@Z_2q×.;0E8 ` 4B^?qPS?PS4_NO1.;0<NOZ74_25?:6 Γ

?\BA?m2xB[\69?:=<25?¶Ø K

i4Y?:25Bv=@0 ` ?:[kB9?\=<6G;N1=<292947?\0 PS?Z7.14 N (0, Idn)

u;29?:2/[k.i.16PS.;0@0@8\?m2p25.;0lBv4 K 4 K P K PS?Z7.14 N (0, 1) K[W 470<254ul.10jQ3?:=SB294Ch=@Z7?\6)=@0@?q6^8:N1ZC4_29NOB947.10PS?Y?\0E2547h=@Z_NO0lB

n698mNOZ7472^NB^4C.;0<2v470<PS8:Q3?:0<P@N10;B^?:2

Y1(ω), . . . , Yn(ω)PS?rZ7.14 N (0, 1) KSÏ Z29=dcjBMNOZ7.162/P@?[\NOZ_[k=<ZC?:6

Γ1/2Y (ω) + mQ.1=@6!.;aSB9?:0@476!=@0@?q6^8:NOZ74_29NOB947.10sP@?qZC.;4 N (m, Γ) K

! .14_[k4Y=@0s?\¯S?\hjQ@Z7?PS?2547h=<Z7NOB947.10EPS? ` ?:[kB9?:=@6!;N1=<292947?\0´8m[k6^4B?:0 ¦ NB9Z_NOaA¾ ,"M<)/! $"%'&K+D-0/&)*1!%" /]##"*&&*K<)/!$"J "*:0<SJ`9vLBQ/&)*L<] $"3* #'<]$"=/*K6%#"$"J*V=/*!3%'&)=/* 1 v`xB|K m 5^8 1 M

∗

j<1BQ/&1J*)3$*/),< %*&&*8=/*!3%'&)=/*+ 1 v`xByK mSz ^ CBC) )3" &*K3 *,+*1 #0v H K ^8 f jC/k jsljCz[

n = 128

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OU

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N

¤1¥ !"$#"!&%'( #) %+*,(-

F 1 z 3A3030398+ D X$+ [ 1 JAv@ sK m$z ^8iX)5(O>d 1 JAv@ K m F ^8=>@5v O1 zb F miX)5(wO>dK bAm O^ 1 +2f

∗

vw`xB9xGK m$z ^8+ D X$+ [ 1 + D X$+ [ f iX)5(Owd K bAm O/^08J\x/B+ D X$+ [ 1 + D X$+G[\F]8) " $"%'&a*! $"J*16 8 66%$152! Q< %*&&*K$%"=/*iw)\[+ D iX(o)T 1 `f2Kw`0a3K+GD X+ [ −+y^U^Ux@Uv K"+u^ !3 /!A+*2! L<] $"3*,+*13%J " &]3*a* <)6]""=/*

D X+L[1 J0v\@ jK m 5^8=>@5v O1 zb m=>@5v1 zb m

D Xw+|[ KO m ^S1B9@ H K0iX)9(OwdsKO m0b ^ m iX/)2(OdsK m b ^^YF−

+GD X+ [2KO\^∗

+ D X$+G[2KW^8J\x/B

J\x/B D X+L[C)*)#C+*,!"*$26) 1 66%$85A! Q<] $"3*K$%"=/* 1 D Xw+|[5 )"*)*! 6a#! # - 6% /V*'&V# J)%"26! /#8#9/V3*$$*1+"J"#"%'&.$dw1 H v\`:J KJ _ J2K m 0^

−

^C)+/]3$"%'&8+.-M*&)$)%'6]"*XFT\1

−

4 6∗

<7@/t KB2J H K ^^

021 @+7PNK;>EB"@>5 OK=F;>E ;>52@ N @ Y5H7L= "NªHS;>= 99<J ,H=

Ö .;=<2f29N ` .10<2Wl=@?/294 (X, Y )?m2xBf=@0 ` ?:[kB9?:=@6lNO=<2^294C?:0PS?eZC.;4l0<.16^hjN1ZC?/[k?\0lB^698:?/6^8:PS=<4B^?1um?kBG254 (r, θ) PS8:294C;0@?)25.;08m[k6^4B^=@69?y?\0[k.i.16PS.10<0@8\?m2GQ.1Z_NO4769?m2M¾

(X, Y ) = (r cos(θ), r sin(θ))u3NOZ7.162

r2 ?kB θ29.10lB!PS?:2 ` K N K 6 K 4C0<P@8\Q?\0<P@N10lB9?:2Wl=@429=@4 ` ?\0lBj69?m25Q?:[kB94 ` ?:hd?:0lBj=<0@?EZC.;4!?k¯SQ3.;0@?\0lB^4C?:ZCZ7?P@?«Q<NO6NOhjkB^69? 1/2

?kB =@0<?«ZC.;4e=<0@4.;69hj?«29=@6[0, 2π] Kp ?` ?:[B^?\=@6N1ZC8mNB^.14769? (X, Y )/r

25=@4CBMZ_N Z7.14§=@0<4.;69hj?29=@6Z7?[k?:6^[\ZC?q=<0@4B^8 K­ ?qJNUk.;0;8\0@8:6^N1ZC?;u<294 (X1, . . . , Xn)?:25B=@0 ` ?m[B^?\=@6!lNO=<2^2547?\0PS?qZ7.14 N (0, In)

?\B‖X‖2 :=

√

X21 + · · ·+ X2

n,

N1ZC.;6^2(X1/‖X‖2, . . . , Xn/‖X‖2)

?kB ‖X‖2 29.10lBp4C0<P@8\Q?\0<P@N10lB^2:u ‖X‖22 29=@4CBG=@0@?!Z7.14@PS= χ2 nPS?\;698m2fPS?!ZC47a?\69B981u?\B

(X1/‖X‖2, . . . , Xn/‖X‖2)25=<4BªZ7NMZC.;4;=@0<4.;69hj?/29=@6ªZ7Ny25Q<@\6^?G?\=3[kZ747PS47?\0<0@?/=<0@4B^8eP@?

Rn KYW B5B^?\0lB947.10§umZ7?:2 ` K N K 6 KXi/(

√n‖X‖2)

0@?29=@4 ` ?\0lB!Q<N;2!PS?:2)Z7.14_2!PS? lB^=<PS?\0lBmu<[\N16 ‖X‖2 ?\B Xi0<?25.;0;B!Q<N;2)4C03PS8\Q?\0<P<NO0lB9?m2§®-/?Wl=@4§Q<698m[k:P@?qQ3?:=SBb2:X 8\B9?:0<PS6^? PS?:2!0<.16^hd?m2!0@.;0?:=<[kZ74_PS4C?:0@0@?m2¾<Q.1=<6!B9.;=SB

n ∈ 2, 3, . . . u@B^.1=SB p ∈ R∗+?\BeB^.1=SB

x ∈ Rn u@.;0sQ.;29?‖x‖p := (|x1|p + · · ·+ |xn|p)1/p. 08kB^?\0<PE[k?\B5B^?PS8kw<0<4B^4C.;0EN1=[:N12 p = ∞ ?\0EQ3.l29N10lB‖x‖∞ := max (|x1|, . . . , |xn|). N p

TU29Q@@:69?bQ.;294B^4 ` ?rP@? Rn u@0@.OB^8\?

S(n, p)u@?:25BPS8kw<0<4C?qQ<N16q¾

S+(n, p) :=

x ∈ Rn+, ‖x‖p = 1

.©f.1=<6p = 1

up.10 .1a@B947?\0lB>ZC?±2547hjQ@ZC?\¯S?b² 2xBNO0<P<NO6PufQ3.;=@6p = 2

uGZ_NAQ3.;65B^4C.;0 ¹[k.i.;6^PS.;0@0@8:?:2dQ3.l254CB94 ` ?m2 PS?Z_NE25Q@<\6^?d?:=<[kZ74_PS4C?:0@0@?>PS?j6N¶g1.;01u?\BQ3.;=@6

p = ∞ uZ_N«25=<65JN;[k?jPS= [k=@a? =@0@4CB98[0, 1]n K§­ ?:2PS?m2929470<2rQ3?:= ` ?:0lB

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N

&%'( #) %+*, v #) #*%"%'*, " # l ¤ N147PS?:6Ehj4C?:=S¯A29? 69?:Q38:69?:6 K S+(n, p)

?:25BPS.103[j=@0@?>29=@65JN;[k? [k.;hjQ<N1[kB9?>PS?RnuªPS?>PS47hj?\0<294C.;0

n − 1 K§Ï Zv29?\6NO4CBQ.;2^2547a@Z7?PS?dPS8kw30@4C6r=@0<?hj?:29=@6^?=@0<4.;69hj?29=@6S+(n, p)

u?:0][k.;0<2947PS8:6^N10lBMZ7N>B96N1[\?PS?dZ7N´hj?:29=@69?dPS? ?\a?:291=<?29=@6S+(n, p) K ©pN1625.;=<[k4_2)PS?q2547hjQ@ZC4_[k4CB98;uS0@.1=32/0<?qZC?y?\6^.10lB!Q<N12e4_[k4uS?kB0@.;=<2e0@.1=32/6^?:25B96^?\4710<.10<2 ` .;ZC.;0;BNO4769?:hj?\0lBN1=S¯s[\N;2

p ∈ 1, 2,∞ KoG0qBNO0lBYWl=@?v29=@69JN1[k?GP@=[k=@a?[0, 1]n

uS+(n,∞)

Q?\=@B¬kB^69? ` =[\.1hjhj?GZ7N)6^8\=<0@4C.;0PS4_2¨x.;4C0lB9?;ukh NO4_2Y[\.10@0@?\¯S?1u\PS?2n

JN1[\?:2:u<[<N;[k=@0@?q8\B^N10;By47PS?:0lB94Cw3NOa@Z7?[0, 1]n−1 K3 N>hd?m25=<69?q=@0<4.;69hj?29=@6 S+(n,∞)

Q3?:=SBbP@.10<[r¬\B96^?PS8kw30@4C??:0³[k.103254_PS8\6NO0lBZ_N«hj?:29=@6^?>=@0@4C.16^hj?s29=@6[3N1Wl=@? JN;[k? K -/?kB9B9?´Z7.14e=@0@4C.16^hj?´29=@6 S+(n,∞)?m2xBB96^:22947hdQ<ZC?294Ch=@Z7?\6Q@=<472^Wl=§X 47Z/29=dcjBPS?s[@.14_25476qZ_NJN1[k?´=@0@4C.16^hd8:hj?\0lB29=@6 1, . . . , 2n u§Q@=@4_2P@?>18:0@8\6^?\6q=@0<?>698mNOZ7472^NB^4C.;0PX =@0 ` ?m[B^?\=@6vN1ZC8mNB^.14769? (X1, . . . , Xn−1)

P@.10lBGZC?m2p?\0lB^698:?:2p29.10lBG4 K 4 K P K ?kBvPS?MZC.;4@=@0@4C.16^hj?25=@6 [0, 1]u.1aSB^?\0i=@?m2pQ<NO6

Q<N16/?\¯S?\hjQ@Z7? K i4ZX .10PS8:294710@?yQ<NO6 (A, I) ∈ 0, 1× 1, . . . , n Z_NqJN1[\?bN1ZC8mNB^.14769?:hd?:0lB/[@.14_294C?;u;Z7? ` ?:[kB9?\=<6N1ZC8mNB9.;4C6^?(Y1, . . . , Yn)

.;aSB9?:0i=>?\0 470<298\6NO0lBA?:0>Q3.l254CB947.10

IPS?Z_N29=@4CB9?

X1, . . . , Xn−125=@4CB/Z_NqZ7.143=@0@4C.16^hd?b25=<6

S+(n,∞) K©f.1=@6p = 2

u/Z7N³ZC.;4M=@0@4C.16^hj?29=@6S+(n, 2)

?:25B´Z_N 6^?:25B96^47[kB947.100@.16^h NOZ747298\?P@?Z_N hj?:29=@69?=@0@4C.16^hd?]25=@6Z_N29Q@@\6^? K -/?kB9B9?EPS?\6^0@47\6^?´29?s294Ch=@ZC?[k.;hdhj?s?k¯SQ@Z747Wl=@8«Z7Nw<0P@?´Z_N29?:[B^4C.;0 ¡ K @uªQ<N165B^4C6dPX =@0@?ZC.;4/lNO=<25T294C?:0@0@? K oG0JNO4CB:ue254 (X1, . . . , Xn)?:25B´=@0 ` ?:[B^?\=@6sNOZ78:NB^.14769?E [k.;hjQ3.l29N10;B^?:2j4 K 4 K P K <¶HH¶IkPS?Z7.14MP@?PS?\0ST294B^8 √

2/π exp(−x2/2) I[0,+∞[ dxuYN1ZC.;6^2rZ7? ` ?:[kB9?:=@6NOZ78:NOB9.1476^? X/

√

X21 + · · ·+ X2

n

29=@4BZ_NZC.;4G=@0@4C.16^hj?>25=<6S+(n, 2) K©f.1=@6

p = 1u

S+(n, 1)?m2xBE=@0V2547hjQ@ZC?\¯S?1u!Wl=@4r?:25BE[\.10<25B94CB9=@8¹PS?AZ7NQ<N165B^4C?A[\.l.;6^P@.10@0@8:?:2´Q3.l254CB94 ` ?m2sPS?ZX igiQ?\6^Q@Z7N10

HnPX 8mWl=<NB^4C.;0

x1 + · · · + xn = 1 K ©f.1=@6 n = 2u3.10«.1a@B947?\0lBy=@0]25?:1hj?\0lBMPS?ZC.;0@1=@?:=@6 √

2u3Q.1=@6

n = 3um=@0qB^694_NO0<1Z7?G8:Wl=@47Z7NOB98:6^N1Z1PS?/[+bOB^8 √

2umQ3.;=@6

n = 4u:=@0B98kB^6^N1@:P@69?G6^8\;=@Z74C?:6YPS?/[+bOB^8 √

2um?kB[ K ¦ ?\0lB^N1ZC?:hj?\0lB:u0@.;=<2b?k°?:[kB9=@.;0<2b0<NB^=@69?:ZCZ7?\hj?:0;Bq=@0<?d4_PS?:0;B^4w[\NB^4C.;0?:0;B^69?

Hn?\B

Rn−1Z7.1629Wl=@?

n ` NO=SB 2?kB

3u?\0A4_PS?\0lB94Cw3N10;BQ<N166^.OBNB^4C.;0PS?«[k?:0lB96^?

(1, 0, . . . , 0)ZX igiQ?\6^Q@Z7N10

Hn

Rn−1 × 0 K oG0[k.;0<254_PS8:6^N10;BdZ7NAhj?:29=@6^? ?\a?:291=@?λn−1

29=@6Hn

PS8:[\.1=@Z_NO0lBPS?[\?\Z7ZC?PS?Rn−1

Q<N16[k?kB9B9?q4_PS?\0lB^4w3[:NB^4C.;0§u@ZX ?:0<25?:ha@Z7?S+(3, 2)

Q3NO6!?k¯S?:hdQ<ZC?NjQ.1=@6hj?:29=@6^? √3/2

u§Wl=@4G?m2xBZ_NE25=<65JN;[k? PS=AB96^47N10@1Z7?d8:Wl=@47Z7NOB98\6NOZGP@?j[+bOB^8 √2×25.;= ` ?:0@?+`\T ` .1=<2:uf±ba3N129?.;472<NO=SB^?\=@629=@6 ¢ ²<Ø K­ ?JNU\.10Q<ZC=<2b;8\0@8:6^N1ZC?;u<.10]hj.10lB96^?Wl=@? λn−1(S+(n, 1)) =√

n/(n − 1)! KÏ Zª?m2xBb0<NB^=@6^?\ZpPX NOQ@Q?\Z7?\6Z7.14=@0<4.;69hj?q25=<6S+(n, 1)

Z7Nd6^?:25B96^47[kB947.10PS?λn−1

S+(n, 1)

u@0@.16^h NOZ747298\?b?\0Z_NdP@4 ` 4_29N10;B!Q<N16 λn−1(S+(n, 1)) K-/.;hdhj?:0;Br2547h=<ZC?:6b[\?kB5B^?Z7.14¶ndoG0]P§X N1=SB96^?:2MB9?:69hj?:2:u[k.1hjhj?\0lBbJNO4769?Q3.;=@6b.1aSB^?\0@476bP@?:2b698mNOZ7472^NB^4C.;0<24 K 4 K P K PX =@0` ?:[B^?\=@6)NOZ78:NOB9.;4C6^? (X1, . . . , Xn)

P@.10lBeZ_NZC.;4§25?:6^N14BeZ7NZC.;4=@0@4C.16^hj?b29=@6S+(n, 1)

Wl=@?y0<.1=<2 ` ?:0@.1032eP@?bPS8m[k6^4C6^?n 0¹Q3.;=@6^6^N14B[k6^.1476^?dWl=@?>294 X := (X1, . . . , Xn)?:25B=@0 ` ?m[B^?\=@6N1ZC8mNB9.;4C6^?jPS? Rn

[\.i.16PS.10@0<8\?:2b4 K 4 K P K ?\B=@0<4.;69hj?:2q29=@6[0, 1]

u§NOZ7.162yZ7? ` ?m[B^?\=@6N1ZC8mNB^.14769? X/(X1 + · · · + Xn)uYWl=@4G?m2xB ` NOZ7?\=<6^2qP@N10<2 S+(n, 1)

uY29=@4BZ_NZC.;4p=@0@4C.16^hd? 29=@6S+(n, 1) K -yX ?m2xBqJN1=S¯h NOZ7@?\=<69?:=<25?:hj?\0lB:u§[\.1hjhj?d.;0]Q?\=@B2\X ?\0¹[k.;0 ` N14C0<[\69?jPS85¨5sZ7.162^W;=<?

n = 2 K N25.;ZC=SB^4C.;0 [k.103254_2xB^?>«47hd4CB9?:6Z_N«hj8kB^@.SPS?>;N1=<2^2547?\0@0<? =SB947ZC4_298\?>Q.1=@6 p = 2 K i4 X := (X1, . . . , Xn)?:25B=@0 ` ?:[kB9?\=<6NOZ78:NB^.14769? [\.i.16PS.10@0<8\?:2r4 K 4 K P K P@?>ZC.;4G?k¯SQ.10@?:0lB947?\Z7ZC?´P@?>Q<NO6NOhj\B96^? 1

uYNOZ7.162rZ7? ` ?m[B9?:=@6N1ZC8mNB9.;4C6^?X/(X1 + · · · + Xn)

29=@4CBbZ_N>Z7.14ª=@0@4C.16^hj?29=@6S+(n, 1) K N´Q@69?:= ` ?0§X ?m2xBrQ<N;2MPS4rc>[k47ZC?;uh N1472y0@.1=32ZX .;hd?\B5B^.10<24_[k4 K ­ ?!hjN10@47\6^?eQ<ZC=<2G18\0<8\6NOZ7?1u¶Q3.;=@6pB^.1=SB p ∈ R+uO.10dQ?\=SBvhj.;0;B^69?:6pW;=<?254

X := (X1, . . . , Xn)?m2xBG=<0 ` ?:[kB9?\=<6N1ZC8mNB9.;4C6^?y [\.1hjQ.;2^NO0lB9?m2/4 K 4 K P K 29=@4 ` N10lB)=<0@?qZC.;4§PS?PS?:0<254CB98

p

Γ(p)exp(−tp) IR+

(t)

N1ZC.;6^2vZC? ` ?m[B^?\=@6)NOZ78:NOB9.1476^? X/‖X‖p

25=@4CB!Z7NZ7.14=@0@4C.16^hj?r29=@6S+(n, p)

ul?\BW;=YX 47Z?:25B)P@?bQ<ZC=<2e470<PS8:Q3?:0<P@NO0lBPS?rZ_N` K N K 6 K ‖X‖p

K ?/[\N12 p = 1u:Q3.;=@6YZ7?:Wl=@?:Z

S+(n, 1)?:25BY=@02547hjQ@ZC?\¯S?1u\?m2xBB96^:2Y4ChjQ.169B^NO0lB[:NO6

S+(n, 1)6^?\Q@6^8:29?\0lB9?pZX ?\0<29?\ha@ZC?PS?m2dZC.;472 PS?EQ@6^.1a3NOa@47ZC4CB98m2jPS472^[k6^kB^?:2j29=@6 1, . . . , n K i47h=@Z7?\6 Z_N]ZC.;4!=@0@4C.16^hd?E29=@6 S+(n, 1)

6^? ` 47?\0lBj¹[@.;4729476=@0<4.;69hj8\hj?:0;B=@0@?]ZC.;4yPS?AQ@6^.1a<N1a@4CZ74CB98PS4_2^[k6^kB9?]25=@6 1, . . . , n K) N³ZC.;4y=@0<4.;69hj?]29=@6 S+(n, 1)?:25Bs=@0'[\N12Q<N165B^47[\=@Z74C?:6GPS?!Z7NrZ7.143PS? ­ 476^47[@Z7?kBmu;[k K Ù K ¥<K K ! .;47[\4<=@0j?k¯S?:hdQ<ZC?PS?.10<[kB947.10 ¦ NB^Z7N1a W;=<4@69?:0 ` .147?e=@0@?6^8:N1ZC4_29NOB947.10PS?rZ_NdZC.;4YPS? ­ 476^47[@Z7?kB¾

®+¯t°y±g²´³ µ¶g·© ¹ºlºt»¼>½¾½/¿ ²ÀÂÁZÃÄ ¾ ®+¶gÅoÆ ÅlÇ ¾ÈÉ Ä[Ê À¯t°y±gË ¿ Æ· ¾YÌtÍdÎoÏ´ÐÑÎoÒÓlÑlÏÕÔyÖ ×¬Ø Á¬ÙÚ´Ú´ÛÜÚÕÝÜ´ÚrÞ ¾ Ê ß>àoálâ ¹ ε ¾/¹¬Øyã À ¿ÕäZå ² ¿d¹yºoºo»¼æçtè ¹lé'¾ ê+ëoìoí_î 7gñ

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Q?\6^hd?\BMPX 8:[k6^4769?d¾Var(T ) = E

(T 2)−E(T )2

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2

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λ21

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λ23

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λ24

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1

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Q.1=@6n > n0 := 93000 K c/47?\0]8 ` 4_PS?\hjhj?\0lB:u[\?d29?\=@47Z n0

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E(T ) > 10 K" % 2 &/' 2 - %"1(%/ i.;4B

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t = 0u@N1ZC.;6^2b¾

N«|S~9zId1I :HJId|E¶·k Iq?m2xBZ_N ` N1694_NOa<ZC? T K NrD/;kH»HzZ_N P@NOB9? t?m2xBq¾

R(t) := P(T > t) K N«|S~9zId1I|S~ :HJIPS=«25gS25B9\hj?qdZX 470<2xBNO0lB t?:25B)Z_N ` NO6^47N1a@ZC? Tt

PS?P@4725B96^4Ca<=SB947.10¹¾P(Tt > s) = P(T − t > s | T > t) =

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R(t)

Q.1=<6s > 0.

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R(t).

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T = max(T1, . . . , Tn)?kB

R(t) = 1−(1−R1(t))(1−R2(t)) · · · (1−Rn(t)) K NMZC.;4;?\¯SQ3.;0@?\0lB947?\Z7Z7?y×JP@?/Q<N16^N1hjkB96^? λ

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R(t) = eλt IR+(t) K @NPS=@6^8\?bPS?y29=@6 ` 4C?bqZX 470<2xBNO0lB t

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