A Probabilistic Analysis of Some Tree Algorithms

HAL Id: inria-00070586

https://hal.inria.fr/inria-00070586

Submitted on 19 May 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub-

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non,

A Probabilistic Analysis of Some Tree Algorithms

Hanene Mohamed, Philippe Robert

To cite this version:

Hanene Mohamed, Philippe Robert. A Probabilistic Analysis of Some Tree Algorithms. [Research

Report] RR-5420, INRIA. 2005, pp.31. �inria-00070586�

ISRN INRIA/RR--5420--FR+ENG

a p p o r t

d e r e c h e r c h e

Thème COM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

A Probabilistic Analysis of Some Tree Algorithms

Hanène Mohamed — Philippe Robert

N° 5420

Deembre 2004

+,.-/0-1%23546,.781:9

∗

^;=< 4>@?A>@BBC1 DE35FG1:HI

JLKNMPORQTS&UWVYX[Z]\_^a`bMPORQc^&de0OROTfNgNhidjkg0`)^

lGmbeknoQ`qprjks

prjtsusve0ma`rw_QxmyQPd)KNQPmyd)KNQzg.{|~}]t€Xƒ‚„QQO!…umbQt€0€t}!X †N‡Wsujtˆ0QP^

‰EŠ‹PŒ~ŽNkŒk‘ gR`bKNhi^=sujksvQPmCjxˆ0QgNQPmyjk’Nd’ijk^y^6et“.`bmyQQqjk’”ˆ0ekmyh•`yKNO^h–^Cjtgujk’”\]—PQPw˜ ‘ `Gh–^C^aKue~™&g!`yKuj~`cš0…]\

fv^ah–gNˆ„jkgTjksNsNmyeksNmyh–jt`bQCsNmbe0…ujt…uh”’–h–^a`bhid=myQsumbQc^aQPg0`)j~`yh”e0get“N`yKNQœ›:fujkg:`bh”`bh–QP^et“uh–g:`bQPmbQc^o`cš`bKNQ&jk^b\]Ožs_`bek`bhid

….QKvjŸ:h–ekmEet“`bKuQP^bQ%jk’”ˆ0ekmyh•`yKNO^ djkg¡…vQek…_`)jth–gNQPw¢›0fuh•`yQ£QPj0^ah–’–\¤™&h”`bKuekf_`mbQc^ae0ma`yh”gNˆ¥`ye¦dekOžsN’–Q§

jkgujt’–\_^ahi^T`bQcd)KNgNhi›0fuQP^jk^Rh”` h–^žfv^afujk’”’–\¨`yKNQ©dj0^aQ0˜¤JLKNhi^jtsusNmbe:jkd)Kªˆ0h”Ÿ0QP^žj%fNgNh”«uQPw¤sNmyek…vjt…Nh–’”hi^a`bhid

`ymbQcj~`yORQPg:`et“L`bKNQc^aQ¬›0fuQP^a`bh–ekgu^P˜

‘

`T^ah–OžsN’–h•«uQc^!jkguw¥Q§]`bQPguwN^^bekOžQ ek“L`bKNQ myQP^bfN’”`y^­]gNe~™&g¥h–g¨`bKuh–^

wNekOjth–g®˜

¯±°:²5³y´!µ ~¶‹k Z]sN’–h”`a`bh–gNˆ¥·r’–ˆke0mbh”`bKNO^P˜%‚„h–Ÿ:hiw_Qjtguw¦SCe0gu›:fNQmž·r’–ˆke0mbh”`bKuOž^P˜¸qg]fu^afvjt’L¹ºj™r^ek“

¹jtmyˆkQ»rfuO!….Qm)^˜&·q^b\]Ožs_`bek`bhidTUW^bdh”’–’ij~`bh–gNˆ ¼œQKvjŸ:h–ekmc˜G‚Wj~`)jEZ:`bmyfud`bfNmyQP^P˜œJmyh”Qc^˜&p&QPgNQ™Ljt’ºJLKuQet½

myQO ˜

∗

¾=¿uÀxÁÂ~Ã)Ä&ÅoÆÃ)ǝƠÈvÂ0¿žÉ®ÁÊtÅaËÍÌrÃ)ËÎÅqϺÐÌ@ÂÑÍÒÉ®ÑÔÓÕÉ®ÁPÖo×؝ÅoǝÖoÁ؝ËÍÌbٺɮÓȣڝË@ÁyÛ@ÅoÖÜÌbÙ5ÝÁÄrÃ)ÐÞǝÅrƝÅrß.ÁàÞØcÖaÅaÃ)Ø:Ù á)âcãaäåLæ

ÅGç5ÅoèAÇ~Ãoéٝê~ËAÃ)ǝÖoÅ

‹  ¹ºQqde0Ožsve0ma`yQOžQg:`Cj0^a\]Ožs_`bek`bhi›:fNQqwfNgNQWd’ijk^y^aQ&ˆgPmyjk’”Qrwjt’–ˆke0mbh”`bKuORQc^6Qg¬jtmy…NmbQ&Qc^o`

jkgujt’–\_^PQk˜=SCQ`b`bQjtsusNmbe_d)KNQxw_e0gNgNQfNgNQjksNsNmye_d)KNQ„fugNh•«Qw_QW`be0f_`bQfugNQd’ijk^y^aQxw_QmP^bfN’”`yj~`)^

zµ

Œc‹

³



‹¬

·r’–ˆke0mbh”`bKuORQc^Lw_h”Ÿ]hi^aQPm&sve0fNm&mˆkguQmc˜JLKekmyMOžQxw_QmyQgNe0fNŸkQP’”’–QOžQg:`c˜

1 2

Tº 43¬G

· ^bsN’”h”`a`yh”guˆRjk’”ˆ0ekmyh•`yKNO hi^LjsNmye_dQcw_fNmyQr`bKvj~`&w_h”Ÿ]hiw_QP^CmyQPdfumy^bh”Ÿ0Q’–\Th–g:`bež^afu…NˆkmyekfNsv^GjtgEh”gNh”`bhijt’5ˆ0mbe0fNs

ek“

n

h•`yQO^RfNg:`bh–’CQcjkd)Kªek“&`bKNQ^afu…NˆkmyekfNsv^ek…N`yjth–gNQcw¨Kujk^Rj£dPjtm)w_h–gujt’–h•`o\¥^a`bmyh–d`b’–\’–QP^y^`yKujtg¤^ae0OžQ

«u§]QcwEg]fNOT…vQPm

D

˜=JLKNQc^aQjk’”ˆ0ekmyh•`yKNO^CKvjŸkQxjT™&hiw_Qm)jtgNˆ0Q„ek“jksNsN’–h–dPj~`yh”e0gu^˜

X ‚„jt`yj%^o`ymbfvd`bfumbQc^˜8JLKuQP^bQjkmbQjt’–ˆkekmyh”`bKNO^Tekg¦wuj~`yj%^a`bmyfud`yfNmbQc^!fv^aQcwª`ye%^bekmb`Rjkguw8^bQPjkmyd)K®˜

JLKNQ\¤jtmyQ±^ae0ORQ`bh–OžQP^žmyQ“ÍQmymyQPw8`beªjk^wNh”Ÿ]hiw_Q©jtguw¤de0gu›:fNQm jk’”ˆ0ekmyh•`yKNO^˜¡Z]QPQSCekmyOžQg

%

,6587

‡9qjkguw;:xg]f_`yK

7

k<9&“Íe0mRjˆkQPgNQm)jt’œsNmyQP^bQg:`yjt`bh–ekg8jkguw;=’ij~noe0’”Q`žjkguw¦Z]Qcw_ˆkQP™&h–d)­

7

†>,9Aš

V©jtKuORe0fuw

7

+?@9ºjtgvw Z_—sujkgN­ke~™r^b­]h

7

†A9º“ÍekmL`bKuQh–mqjtgujk’”\_^bh–^œ™&h”`bK±jtgvjt’–\0`yh–dPjt’5OžQ`yKNe_wN^˜

X SCekOžO!fugNh–dPj~`yh”e0g£»rQ`o™Ce0mb­_^P˜qJLKNQP^bQTjk’”ˆ0ekmyh•`yKNO^qjkmbQTfu^aQcw `be¬ˆkh–ŸkQ!j¬w_hi^o`ymbh–…Nf_`yQPw£j0ddQP^y^&`be

jždekOžOžekgde0OROTfNgNhidjt`bh–ekg¬d)KvjtgNgNQP’.`bKujt`&dPjtgE`bm)jtgu^bOžh•`&ekgu’”\e0gNQ„OžQc^b^yjtˆ0Qqs.QmL`bh–OžQxfNgNh”`P˜

Z]QQTSœjksvQ`yjkgujt­]hi^ 7}9šuJ^a\]…ujk­ke~Ÿjtgvw¬V±h–­]Kuj<BC”’–e~Ÿ 7}:€9jkguwEDGsNKNmyQOžh–wNQP^rjkguwGF„j~noQP­ 7A9A˜

X ‚qhi^a`bmyh”…NfN`bQPw^b\]^a`bQPO^˜=Z]e0ORQxjt’–ˆke0mbh”`bKuOž^Cfu^bQWjR^bsN’”h”`a`yh”guˆ!`yQPd)KNgNhi›:fNQ„`bež^aQP’”Qcd`&jT^bfN…u^bQ`Let“j

ˆkmyekfNset“hiw_Qg:`bhidjk’ºde0OžO!fNgNhidjt`bh–gNˆde0ORs.ekguQg:`y^P˜GZ]QQH:jtgv^ae0gjkguw±Z]—sujkgN­ke~™r^b­]h

7

‡,?@9jtgvw

prjt—

%IJ 65;7

†:|<9A˜

X Z:`yjt`bhi^o`yh–dPjt’:`bQc^o`)^˜·¦`bQc^o`cštsvQPma“Íe0mbOžQcwekgRjqˆ0mbe0fNset“vh”guwNh”Ÿ]hiw_fujt’i^Pšth”gvw_h–dPj~`yQP^h•“.j~`’”Qcjk^a`e0gNQCek“

`bKNQc^aQh–guw_h–Ÿ]h–wNfujt’i^&Kuj0^&^ae0ORQd)Kujkmyj0d`yQmyh–^a`bhid^LK͒–h”­0Qjžw_h–^bQPj0^aQxh”“`yKNhi^&h–^r…N’”e]e_wE`bQP^a`bh–gNˆ/M˜GJLKNQ

sNfNmysve:^aQRhi^q`ye¬Ožh–gNh–ORh–—QR`yKNQžg:fuO!….Qmxet“=`yQP^a`y^„`behiw_Qg:`bh”“Í\©h–guw_h–Ÿ]h–w_fvjt’i^„™&h”`bK`bKuQž^bs.QPdh”«uQcw

d)Kujtm)jkd`bQPmbhi^o`yh–dWjk^&›:fNhid)­]’”\¬jk^œsve:^b^bh–…N’”Q0˜GZ]QQON%ek’”“ 7}:,9

=uekmyOjt’–’”\0š]j^bsN’–h•`b`bh–gNˆjt’–ˆke0mbh”`bKNO djtg….Qw_QP^ydmyh”….QPwjk^œ“Íe0’”’–e~™r^@P

Q#R)STU#UVTWYX[Z\S+X^]Y_VTUV`ba

S(n)

X cOd _VaeTWbf+UVT.]gWihL]gWbjgTU#T.]gW ˜

‘ “

n < D −→

^{QkUb]YR ˜}

X c _ d)d QkUV_klgmnUVlb_ d˜

‘ “

n ≥ D

šºm)jtguwNekOž’”\w_h”Ÿ]hiw_Q

n

^h–g:`be

n 1

š”˜P˜˜š

n G

š®™&h”`bK

n 1 + · · · + n G = n

™&KNQPmbQ

G

^{h–^j}

myjkguw_e0O Ÿ~jtmyh–jk…N’”QW™&h”`bK±^ae0OžQ„«u§]Qcw w_h–^a`bmyh–…Nf_`bh–ekgº˜

−→

^Z\R)R)So

S(n 1 )

^š

S(n 2 )

^{š®˜˜P˜š}

S(n G )

^˜

prq p sutYv/w#xny{ze|}y ~€

JLKuQRjk’”ˆ0ekmyh•`yKNOY^o`)jtmb`y^W™&h”`bK¥jEˆ0mbe0fNs©et“L^ah–—Q

n

h”`bQPO^˜RJLKNh–^x^bQ`hi^„m)jtgvw_ekOž’–\©^asN’–h”`xh–g:`be

G

^{^bfN…_½}

ˆ0mbe0fNsu^PšN`yKNQRwNh–^a`bmyh”…uf_`bh–ekg©ek“

G

hi^qˆ0h”Ÿ0Qg±…]\

P (G = `) = p `

š5™&KNQPmbQ

(p ` )

hi^xjsNmyek…ujk…Nh”’–h”`o\w_hi^a`bmyh•½ …uf_`bh–ekg¥ekg

{2, 3, . . .}

˜E»qe~™šºde0guw_h”`bh–ekgujk’”’–\©e0g`yKNQQŸ0Qg:`

{G = `}

^š®“Íe0m

1 ≤ i ≤ `

šjkg¥h•`yQO hi^

^bQg:`h–g0`yeE`yKNQ

i

`bK¥^afu…NˆkmyekfNs™&h•`yK¥sNmbe0…ujt…uh”’–h•`o\

V i,`

š®™&KNQPmbQ

V ` = (V i,` ; 1 ≤ i ≤ `)

h–^jm)jtguw_e0O sumbe0…ujt…Nh–’–h•`o\Ÿ0QPd`yekm&e0g

{1, . . . , `}

^{˜ ‘}`qdjkgjk’–^bež…vQ^bQQPg j0^&jžŸkQcd`be0mLet“myjkguw_ekO™CQPh”ˆ0K0`)^Lekg¬`bKNQ

`

jkmydP^œet“`bKNQ…Nm)jtgvd)KNh”guˆTsNmye_dQcw_fNmyQ„e0g™&Kuh–d)KQPj0d)K¬ek“º`bKuQ

n

h”`bQPOž^rsvQPma“Íe0mbOjRm)jtguwNekO ™Ljt’–­.˜

‘ “

N i

hi^`yKNQ¬dPjtm)w_h”gvjt’–h•`o\et“&`bKuQ

i

`bK8^bfN…Nˆ0mbe0fNs%`bKNQPg®š=de0guw_h”`bh–ekgujk’”’–\%ekg¥`bKuQ¬QPŸkQg:`

{G = `}

jkguw©e0g±`bKuQRm)jtgvw_ekO'Ÿ~jtmyh–jk…N’–QP^

V 1,` , V 2,` , . . . , V `,`

š5`bKNQžw_hi^o`ymbh–…Nf_`yh”e0g£et“G`bKNQžŸkQcd`yekm

(N 1 , . . . , N ` )

hi^LO!fu’•`yh”gNe0Ožh–jk’®™&h•`yK sujtm)jtOžQ`yQm

n

^jkguw

(V 1,` , V 2,` , . . . , V `,` )

^š

P (N 1 , . . . , N ` ) = (m 1 , . . . , m ` )

= n!

m 1 ! m 2 ! · · · m ` !

`

Y

k=1

(V k,` ) ^{m k} ,

“Íe0m

(m i ) ∈ N ⁿ

^afvd)K©`bKvj~`

m 1 + · · · + m ` = n

^{˜ ‘“G`bKuQ}

i

`yK%^bfN…Nˆ0mbe0fNs®š

1 ≤ i ≤ n

šhi^^afvd)K©`bKvj~`

N i < D

š0`bKuQ„jt’–ˆke0mbh”`bKuO ^a`beksv^“ÍekmG`bKNhi^G^bfN…Nˆ0mbe0fNs®˜=Uq`bKNQPmb™&hi^aQ0škh”`Chi^GjksNsN’–h”QcwR`be`yKNQ

i

`bK ^bfN…Nˆ0mbe0fNsbP jžŸ~jkmbhijt…N’–Q

G i

šv™&h•`yK `bKNQ^yjtOžQw_hi^o`ymbh–…Nf_`yh”e0g±jk^

G

šuhi^rw_m)j™&g jtguw¬`bKuh–^

i

`yK ^bfN…Nˆ0mbe0fNsh–^r^bsN’–h•`rh–g:`be

G i

^bfN…Nˆ0mbe0fNsu^PšNjtguw^beRe0g˜P˜˜

Z]fvd)K%j¬myjkguw_e0O'^bsN’–h•`b`bh–gNˆ Kujk^W….QQPgh–g:`bmye]wNfudQcw©…]\©‚qQPŸ]mbe~\0Q 7?@9=™&KNQmyQT`bKNQjk^b\]Ožs_`bek`bhidQ§]½

svjtgu^bh”e0g¬ek“º`yKNQw_QsN`bK et“`bKNQj0^b^be_dhij~`yQPw`ymbQPQxh–^&h–g]ŸkQc^o`yh”ˆ:j~`bQcw˜

D=§_jkOžsN’”Qc^˜

X :xg]f_`bK ^r·q’”ˆ0ekmyh•`yKNO ˜€N KNQPg

P (G = 2) = 1

^š

D = 2

^jkguw

V 1,2 ≡ V 2,2 ≡ 1/2

š]`bKuh–^qh–^&e0gNQet“

`bKNQe0’–wNQP^a`qjt’–ˆkekmyh”`bKNO^œet“`bKNhi^&­]h”gvw˜ ‘`rKujk^L….QQPg±jtgujk’”\]—Qcw …]\G:xg]f_`yK¬h–g¥‡@A/?~†N˜

X Z]\]OžORQ`bmyh–dPjt’5Z]sN’–h•`b`bh–gNˆ!·r’–ˆke0mbh”`bKuO˜JLKNhi^=hi^6`yKNQqdj0^aQ&™&KuQmyQ

V i,n ≡ 1/n

“Íe0mGjtg]\

n ≥ 2

^jkguw

1 ≤ i ≤ n

^˜

X

Q

½Üjtmy\jk’”ˆ0ekmyh•`yKNO ˜ ‘ “

P (G = Q) = 1

^jkguw

D = 2

`yKNh–^Whi^r`yKNQ

Q

½Üjtmy\EmyQP^bek’–f_`yh”e0g©jt’–ˆkekmyh”`bKNO

™&h•`yK …N’”e_d)­0QPwjtmymbh–Ÿ~jt’i^œjkgujt’–\]—Qcw …]\¬V©jt`bK]\_^&jtguw4=’–jtnoek’–Q` 7A9 ˜

Z_QQjt’i^bež‚„QŸ]mye~\kQ

7

?<9®“Íe0mret`yKNQmrQ§NjtOžsN’–QP^P˜„fNh”`bQgujt`bfNm)jt’–’–\kšN^bfud)K±jtg±jt’–ˆke0mbh”`bKNOdjkg….Qˆkm)jtsuKNh•½

dPjt’–’”\EmbQPsNmbQc^aQPg:`bQPwE™&h•`yK±j!`ymbQPQjk^L^bKNe~™&g¬…]\E=h–ˆkfNmyQž‡k˜

͌PŒ ° ŽN‹ º °

·„^Rh•` ™&h–’”’L….Q±^aQPQg¤h”g¦`bKNQ“Íe0’”’–e~™&h”guˆuš6`yKNQ ­kQ\¨d)Kujkmyj0d`yQmyh–^a`bhid ek“q`bKuh–^^bsN’–h•`b`bh–gNˆ%jk’”ˆ0ekmyh•`yKNO hi^Rj

sumbe0…ujt…Nh–’–h•`o\Rw_h–^a`bmyh–…Nf_`bh–ekg

W

^e0g

[0, 1]

w_Q«vgNQPwž™&h”`bKE`bKNQ„…Nmyjkgud)KNh–gNˆw_hi^o`ymbh–…Nf_`yh”e0g KÔ`yKNQqŸ~jkmbhijt…N’–Q

G

^M

jkguw`bKuQ™CQPh”ˆ0K:`y^&e0g±QPjkd)K±jkmyd KÍ`bKNQTŸkQPd`be0m

(V 1,G , . . . , V G,G )

M˜rJLKuQ!jk^b\]ORsN`bet`yh–dx….QKvjŸ:h–ekmqet“`bKNQ jk’”ˆ0ekmyh•`yKNO hi^&Q§_sNmyQP^y^aQcw gujt`bfNm)jt’–’–\ h”g¬`bQPmbO^&ek“º`yKNQw_hi^o`ymbh–…Nf_`yh”e0g

W

^˜

° ͌µ

'-n%

^bsN’–h•`b`bh–gNˆEOžQPj0^afumbQ

.-)%# ,},

W

[0, 1]

^{% €% "!}

#,$# #%* &%,%(',% \}#,

f

^!

Z

f (x) W(dx) = E

G

X

i=1

V i,G f (V i,G )

!

=

+∞

X

`=2

`

X

i=1

P (G = `) E (V i,` f (V i,` )).

^Ko‡<M

JLKumbe0fNˆkKNe0f_`œ`bKNQsujksvQPmPšuh•`rhi^&jk^y^afuORQcw `yKuj~`cšvjk’”Ože0^a`r^afumbQP’”\

G ≥ 2

šNjtguw¬`yKuj~`&`yKNQmyQxQ§_h–^a`y^q^ae0ORQ

δ > 0

^bfud)K¬`bKujt`&`bKNQmyQ’ij~`yh”e0g

sup

`≥2

sup

1≤i≤`

V i,` ≤ δ < 1

^KηLM

AB ∅ ∅ CD E CDE

AB F

ABCDEF

V _1,3 ¹

V _2,3 ¹ V _3,3 ¹

V _1,2 ³ V _2,2 ³

B

A C D

V _1,2 ⁴ V _2,2 ⁴ V _1,2 ⁵ V _2,2 ⁵ V _1,3 ²

V _2,3 ² V _3,3 ²

=h–ˆkfNmyQ¨‡P¨Z]sN’–h”`a`bh–gNˆ8·q’”ˆ0ekmyh•`yKNO ™&h•`yK

D = 2

šœ`o™Ce¦^aQ`y^ ek“ ,,$ ™CQPh”ˆ0K:`y^

(V 1,2 , V 2,2 )

^jtgvw

(V 1,3 , V 2,3 , V 1,3 )

^š

G

j¬myjkguw_e0OŸ~jkmbhijt…N’–Q™&h”`bK%Ÿ~jt’–fNQP^Wh–g

{2, 3}

jtguw©`bKNQRh”guh•`yh–jt’h”`bQPOž^xjkmbQ

A

^š

B

^š

C

^š

D

^š

E

^jkguw

F

^˜

Kuek’iwN^jt’–Ože0^a`^bfNmyQ’–\kšh”g¥sujkma`yh–dfN’–jkm

W([0, δ]) = 1

˜ JLKNQP^bQ de0guw_h”`bh–ekgu^xh–OžsN’–\£h–g%svjtmb`bhidfN’ijtmx`yKNQ guekg_½Üw_Qˆ0QgNQPmyj0d\žet“`bKNQ^bsN’–h•`b`bh–gNˆOžQPd)KujkgNhi^aO ˜

° ͌µ +* $&% }%

W

Q§_sve0gNQg:`yh–jk’”’–\!jtmyh”`bKNOžQ`yh–d !g.-)%I%L%JI,$ %

λ > 0

n-(.-)

W

e ^−nλ : n ≥ 1 = 1,

.-)% 6 *%

λ

I ,.,+*G-}O% ,, % €%E ,".-)% Q§_s.ekgNQPg:`bhijt’®^bsujtg

W

⁵

‘ “

A

hi^x^ae0ORQRm)jtguw_e0O'Ÿjkmbhijt…u’”QT™&h•`yKw_hi^o`ymbh–…Nf_`yh”e0g

W

^š5`bKNQPg

W

hi^WQ§_s.ekgNQPg0`yh–jk’”’–\±jtmyh•`yKNOžQ`yh–d

™&h”`bK£Q§_sve0gNQg:`yh–jk’^bsujtg

λ

h”“Gjtguw©ekgN’–\ h•“6`yKNQRwNh–^a`bmyh”…uf_`bh–ekg©ek“

− log(A)

h–^Wjtmyh”`bKNOžQ`yh–d!™&h•`yK£^bsujtg

λ

˜GZ]QPQ!Z]QPd`bh–ekg±|_˜

D=§_jkOžsN’”Qc^˜

X :xg]f_`bK ^r·q’”ˆ0ekmyh•`yKNO š

P (G = 2) = 1

^š

D = 2

^jtgvw

V 1,2 ≡ V 2,2 ≡ 1/2

^˜

‘

g`bKNhi^rdj0^aQ

W(dx) = δ 1/2

™&KNQmyQ

δ x

hi^`bKNQ&‚„h”m)jkdLw_hi^o`ymbh–…Nf_`yh”e0gRjt`

x

^š

W

h–^Q§]s.ekguQg:`bhijt’–’”\jkmbh”`bKuORQ`bhidœ™&h•`yKRQ§_sve0gNQg:`bhijt’

^asvjtg

log 2

^˜

X Z]\]OžORQ`bmyh–dPjt’ºZ]su’”h”`a`yh”gNˆ¬·r’–ˆkekmyh”`bKNO ˜

W(dx) = X

n≥2

P (G = n)δ 1/n ,

`bKNQRQ§_s.ekgNQPg:`bhijt’6^asvjtg©hi^

log D

™&KNQmyQ

D

h–^„`yKNQT’ijtmyˆkQc^o`„h”g:`bQPˆkQPm

p

^afvd)K±`bKujt`„`yKNQž^afNsusve0ma`

et“`bKNQm)jtgvw_ekO Ÿ~jtmyhijt…N’–Q

G

hi^&de0g0`)jth–gNQPw¬h–g

p N

^˜

X

Q

½Üjtmy\ jt’–ˆkekmyh”`bKNO š

P (G = Q) = 1

^š

D = 2

^š

V 1,Q = p 1

š–˜˜P˜š

V Q,Q = p Q

˜

W(dx) = p 1 δ p 1 + p 2 δ p 2 + · · · + p Q δ p Q ,

`bKNQ¥w_hi^o`ymbh–…Nf_`yh”e0g

W

hi^EQ§]s.ekguQg:`bhijt’–’”\ jkmbh”`bKNOžQ`bhid£h”“Tjtgvw ekgN’–\Õh•“Rjt’–’q`yKNQmyQPjk’„g]fNOT…vQPmy^

log p i / log p j

š

1 ≤ i < j ≤ Q

šNjtmyQWm)j~`yh”e0gujt’A˜

°Tµ ‹Œ µ Ž ͌PŒ%‰ µ  ͌

=uekmG^bfud)K jkgžjk’”ˆ0ekmyh•`yKNO š0jkgh”Ožs.ekmb`yjtg:`G›:fujkg0`yh•`o\Rhi^6`bKuQrg]fNO!….QmCet“eks.Qm)j~`yh”e0gu^mbQc›:fNh”myQPwRfNg:`yh”’v`bKNQ

jk’”ˆ0ekmyh•`yKNO ^a`be0su^PšvhA˜Q0˜q™&KNQPg£jt’–’®`yKNQT^bfN…Nˆ0mbe0fNsu^qKujŸkQj¬djtm)w_h–gujt’–h”`o\E’–QP^y^rekm„QP›:fujk’º`ye

D

˜„‚„Qguet`bQ …]\

R n

`bKNhi^r›:fujtg:`yh•`o\ ™&KuQg`bKNQxg]fNOT…vQPmret“h”gNh”`bhijt’ºh•`yQO^&hi^

n

š_`bKNQPg±d’–QPjtmy’–\

X

R n = 1

™&KNQPg

n < D

X =Ne0m

n ≥ D,

R n

= 1 + R 1,N ₁ ⁿ + · · · + R G,N _G ⁿ ,

^K@M

™&KNQmyQEdekguw_h”`bh–ekgvjt’–’”\%ekg¥`yKNQEQPŸkQg:`

{G = `}

jkguw%`bKuQEm)jtguw_e0O Ÿ~jkmbhijt…N’–QP^

V 1,`

š

V 2,`

šL˜˜P˜š

V `,`

š

‡0˜LJLKuQRŸ0QPd`bekm

(N ₁ ⁿ , . . . , N _` ⁿ )

Kuj0^j¬OTfN’”`bh–gNekOžhijt’=w_hi^o`ymbh–…Nf_`yh”e0g™&h”`bK¥sujkmyjkOžQ`bQPm

n

^jtgvw

(V 1,` , V 2,` , . . . , V `,` )

N˜J=uekm

(p i ) ∈ N ^`

š_`bKNQŸ~jkmbhijt…N’–QP^

R 1,p 1 , . . . , R `,p `

jtmyQ„h–guw_QPsvQPguw_Qg:`

†u˜J=uekm

1 ≤ i ≤ `

š_`yKNQŸ~jtmyh–jk…N’”Q

R i,p i

Kuj0^œ`bKNQ^yjtOžQw_hi^o`ymbh–…Nf_`yh”e0gj0^

R p i

˜

JLKuQWŸ~jkmbhijt…N’–Q

R n

hi^&^ah–OžsN’”\E`bKNQg]fNOT…vQPm&et“gNe_w_QP^Lek“º`yKNQjk^y^ae_dh–jt`bQPwE`bmyQQ0šN^aQPQO=h–ˆkfumbQž‡t˜

prq v[v ~! " x$#rt&%')(&*Otbxkv

»qet`yQž`bKujt`Pš^ah–gudQž`yKNQE^bsN’–h•`b`bh–gNˆ©sNmye_dQPwNfNmbQžhi^m)jtguw_e0O š®`bKuQ Ÿ~jtmyh–jk…N’”Q

R n

h–^j ,,$ %, , I6% ˜

N h”`bK¥`yKNQ’ijtgNˆ0fujtˆ0QRek“LdekOžO!fugNh–dPj~`yh”e0ggNQ`o™Ce0mb­_^Pš`bKuh–^›:fujkg:`bh”`o\£djkg%….Q`yKNekfNˆ0K:`j0^x`bKuQž`bet`)jt’

`yh”OžQ`yež`bm)jtgu^bOžh•`

n

h–gNh•`yh–jk’ºOžQP^y^yjtˆkQc^˜ ‘ “

E (R n )

hi^rh”`y^rQ§_svQcd`yQPw Ÿ~jt’–fNQkš

E (R n )/n

hi^L`bKNQTjŸkQm)jtˆ0Q

`ymyjkgu^bORhi^y^ah–ekgž`yh”OžQxet“®e0gNQ„OžQc^b^yjtˆ0QqjtOžekguˆ

n

^{˜ =NmyekO} j!sNmbe0…ujt…uh”’–h–^a`bhidrs.ekh–g:`œek“ºŸ]h”QP™š]h•`Lh–^œgujt`bfNm)jt’

`ye„Q§_svQcd``bKujt``yKNQ&^aQc›:fNQgudQ

(R n )

^yj~`yh–^a«uQP^j„­]h”guwTet“v’–j™ªek“u’ijtmyˆkQGg]fNOT…vQPmy^Pš~hA˜Q0˜º`yKuj~`

( E (R n )/n)

dekg]ŸkQPmbˆ0QP^„`be^ae0ORQ ›:fujkg:`bh”`o\

α

˜©JLKNQEdekgu^a`yjkg0`

α

hi^šh”gª^bekOžQE^bQgv^aQ0šº`yKNQEjk^b\]Ožs_`bek`bhid jŸkQPmyjkˆkQ

`ymyjkgu^bORhi^y^ah–ekgE`bh–ORQ!et“jžOžQc^b^yjtˆ0Qk˜GSCfNmyh”e0fu^a’–\kš]`yKNhi^&’–j™et“’–jkmbˆ0Qxg:fuO!….Qm)^&w_e]QP^LgNek`qjt’–™œj\_^œKNek’iw˜

‘ g ^bekOžQr^bh•`yfuj~`yh”e0gu^Pšt`bKuQ„^bQP›:fNQgvdQ

( E (R n )/n)

wNe:Qc^=gNek`Gde0g:Ÿ0QmyˆkQ&j~`œjt’–’ujtguw®š0OžekmyQe~Ÿ0Qmcš~Q§_KNh–…Nh”`y^

jkg¬e:^bdh”’–’ij~`bh–gNˆR…vQPKujŸ]h”e0mP˜

N KNQPgR`bKNQ&^bsN’”h”`a`yh”guˆxwNQˆkmyQQœhi^6dekgv^o`)jtg:`jkguwTQP›:fujt’]`ye

2

^jkguw

V 1,2 ≡ V 2,2 ≡ 1/2

KÔ`yKNQ&h”`bQO^6jkmbQ Qc›:fujt’–’”\©wNh”Ÿ]hiw_QPw%jtOžekguˆE`yKNQž`o™Ce ^bfN…Nˆkmyekfusu^ Mš5`bKNQc^aQžsNKuQgNe0ORQPguj jtmyQR›:fNh”`bQž™œQ’–’6­]gNe~™&g®˜TJLKuQ\

KvjŸkQ=….QQPg!jtgujk’”\]—PQPwfu^bh–gNˆqde0OžsN’”Q§jtgujk’”\_^bh–^®`bQPd)KugNh–›:fNQc^šc“ÍfNgvd`bh–ekgvjt’:`bm)jtgu^a“ÍekmyO^rKÎjkguw`yKNQh–mjk^y^aek½

dh–jt`bQcw!h–g]ŸkQm)^bh”e0g!sNmye_dQPwNfNmbQc^ Mº…]\ :xg]f_`bK 7k<9š=’ij~noek’–Q` %I 65E7‡9 šk¹®e0fud)Kujtm)w!jkguwTl=mbe_w_h–gNˆ0Qm 7 9

jkguw Ojkg:\Eet`bKuQm)^˜6Z]QQOFqet“Ímyh 7‡|<9AšNV£jtKNOžekfvw 7?<9®jkguwG=’ij~noe0’”Q`LjkguwZ_QPw_ˆ0Q™&hid)­ 7†>9“Íekm&jžde0OT½

sumbQPKNQgu^bh–ŸkQ `ymbQcj~`yORQPg:`Tet“q`bKNhi^žjtsNsNmye0j0d)K®˜¨Z]QQ jt’i^ae‚qQPŸ:mye~\kQ 7>9œ“Íe0mRj%^bfNmbŸ0Q\%et“r`yKNQ w_ekOjth–g®˜

prek….Qmb` 7† 9ºsNmyeks.e0^bQPw¬jtg±jt’”`bQPmbgujt`bh–ŸkQ0š_Q’–QOžQg:`yjkmb\0š_OžQ`bKue]w¬`yeˆkQ`&`yKNQjk^b\]Ožs_`bek`bhid„….QKvjŸ:h–ekm&ek“

^bekOžQxmyQ’ij~`bQcwEe:^bdh”’–’ij~`bh–gNˆž^aQc›:fNQgudQP^L™&h”`bKNe0f_`rfu^bh”guˆždekOžsN’–Q§¬jkgujt’–\_^ahi^P˜

N KNQPgÕ`bKNQ^asN’–h”`a`bh–gNˆªw_QPˆkmyQQ hi^ de0gu^o`)jtg:`Ejtgvw¦Qc›:fujt’&`ye

Q

…uf_``bKNQ£h•`yQO^ jtmyQ QP›:fujk’”’–\

wNh”Ÿ]hiw_QPw¨jkORe0gNˆ `bKuQE^afu…NˆkmyekfNsv^š^o`yfuw_h–QP^TjtmyQ ›0fuh•`yQ myjkmbQ0˜¸„^ah–gNˆ£dekOžsN’–Q§¨jtgujk’”\_^bh–^W`yQPd)KNguh–›:fNQc^š

=vj\ke0’”’–Q

%I ,65 7

‡0‡9„e0…_`yjkh”gv^R`bKNQ£j0^a\]Ožs_`yet`bhid…vQPKujŸ]h”e0met“W`bKNQjk^y^ae_dhij~`yQPw¤^aQc›:fNQgudQ

( E (R n ))

^˜

V£j~`bK]\_^=jkguw =’–jtnoek’–Q`

7

A9.ˆkh–ŸkQP^Gj^a­0Q`)d)KRek“®jxˆkQguQm)jt’–h”—cj~`bh–ekgRet“5`yKNh–^C^o`yfuw_\R™&KNQg

Q

hi^Gjkmb…Nh”`bm)jtmy\k˜

!V,$ %0 %# ,&%%\ < -)%I5

X Z]ekOžQ„’–j™r^Get“’–jkmbˆ0Qrg]fNO!….Qm)^=KvjŸkQr….QQPg sNmye~ŸkQPwR…]\ž‚qQPŸ:mye~\kQ 7>,95h–gEj›:fNh”`bQqˆ0QgNQPmyjk’N“Ím)jtOžQ½

™Ce0mb­W“Íe0mŸ~jkmbh–ekfu^º“ÍfNgud`bh–ekgujk’–^ek“N`bKNQ&j0^b^be_dhij~`bQcwx`bmyQQc^˜Jjk’–jkˆkm)jtguw ^dekgvdQg:`ymyjt`bh–ekgh–gNQP›:fujk’•½

h•`yh”Qc^!jkmbQR`bKuQOžjkh”g%`ye:e0’–^h”g%`bKuh–^^a`bfvw_\k˜ ‘g%e0fNm!dPjk^bQkš®h”`!™œekfN’iwdekgu^bhi^o`h–g¥sNmbe~Ÿ]h–gNˆ`bKujt`Pš

`bKNQw_h–^a`bmyh–…Nf_`bh–ekgÕek“q`bKuQ©myjkguw_e0O Ÿ~jtmyh–jk…N’”Q

R n / E (R n )

hi^^bKujtmysN’–\ K͙&h”`bK¡jkg¤Q§_sve0gNQg:`bhijt’

w_QPdPj\)MCdekgudQg:`bm)j~`yQPwjtmyekfNgvw

1

˜=p&QP^bfN’”`y^&e0g¬’–h”Ožh”`bh–gNˆ w_hi^o`ymbh–…Nf_`yh”e0gu^r^bfud)K jk^PšNdQg:`ymyjk’’”h–Ožh•`

`bKNQPekmyQO^PšNw_ežgNet`q^bQQPO `be….QjkdPdQP^y^bh”…N’–QW™&h•`yK¬`yKNhi^&OžQ`bKue]w®˜

X SC’”QPOžQg:`

%IY ,657

9]jkgujt’–\]—Qcw„myQ’ij~`yQPwjk’”ˆ0ekmyh•`yKNO^h”gx`yKNQGOže0mbQ6ˆ0QgNQPmyjk’kde0g0`yQ§]`ºek“uw_\]gujtOžhidjt’

^a\_^a`bQPOž^P˜¼C\Rfu^bh”guˆjtg Frh–’–…vQPma`yh–jkg ^aQ`a`bh–gNˆvšk`yKNQ\ž^aKue~™8`yKuj~`=`yKNQ&«um)^a`Gekm)w_Qm6….QKvjŸ:h–ekmGet“5`bKNQ

jt’–ˆke0mbh”`bKNO^Gh–^CQ§_sNmyQP^y^aQcwžh”gE`bQPmbO^Cet“`bKuQW^bsvQcd`ymbfNO ek“ºj“ÍfNgud`bh–ekgujk’ve0svQPmyjt`be0mPšk`bKNQ„`bm)jtgu^a“ÍQm

eks.Qm)j~`yekmc˜WQ`a`bh–gNˆQ§_sN’–h–dh•`WmbQc^afu’•`)^qh–g©`yKNh–^W™Lj\ mbQc›0fuh”myQP^r`bKNQPmbQ“ÍekmyQTj ˆ0e]e]w©­:gue~™&’”Qcw_ˆkQek“

^ae0OžQWQPh”ˆ0Qg]Ÿ~jt’–fNQP^Lek“º`yKNQx`bm)jtgu^a“ÍQm&e0svQPmyjt`be0mP˜

·¤w_\]gujtOžhid=Ÿ0Qm)^ah–ekgek“N`bKNhi^d’ijk^y^®ek“vjt’–ˆke0mbh”`bKNO^ºh–^h”g]Ÿ0QP^a`bh–ˆ0j~`yQPwxh–gTV±e0KujtOžQcwjtguw!p&ek….Qmb` 7†09A˜

JLKuQT^bsN’”h”`a`yh”guˆ¬sNmye_dQcw_fNmyQh–^q`bKuQT^yjtOžQ!…NfN`Pšh–g±`bKNQR’ijtgNˆ0fujtˆ0Qet“=…Nm)jtgud)Kuh”gNˆEsNmbe_dQP^y^aQc^š5jtg©h–OžOžh•½

ˆ0myjt`bh–ekg©e_dPdfNm)^„jt`QŸkQPmb\ ’–QPjŸ0Q!et“G`bKNQjk^y^be]dh–jt`bQcw `bmyQQ0šh@˜Qk˜TgNQ™ h”`bQPO^jtmymbh–ŸkQQPŸkQPmb\`bh–OžQžfNgNh”`P˜

JLKuh–^œw_\]gujtOžhidL“ÍQPjt`bfNmyQ„dekOžsN’–h–dPj~`yQP^6`yKNQqsumbe0…N’”QPO ˜ ‘ g `bKNhi^CdPjk^bQkš:jtgEj0wNw_h”`bh–ekgujk’usNmyek…ujk…Nh–’”hi^o`yh–d&`be]ek’

Kvjk^œ`bež….Qfu^bQPwVP=jkgjkf_`be0mbQPˆkmyQP^y^ah–ŸkQ„sNmbe_dQP^y^C™&h”`bK Ože~Ÿ]h”guˆRjŸ0Qm)jtˆkQ„sN’ij\]^&jkg¬h–Ožsve0ma`)jtg:`&mbe0’”Q0˜

prq t ^|/t x)~!* {t ( v

@*$ %k 4<%I%5

·dekg:`bh–g]fNekfu^LŸ0Qm)^ah–ekg¬et“jR^bsN’–h•`b`bh–gNˆ jt’–ˆke0mbh”`bKuO dekfu’–w¬…vQwNQ«ugNQcw¬j0^œ“Íek’–’”e~™r^@P6jtgh”guh•`yh–jk’Ožj0^b^Lek“

^bh–—Q

x

hi^Cm)jtgvw_ekOž’–\ž^bsN’–h•`Lh–g0`yeT^bQŸ0Qm)jt’usuh”QcdQP^Ljkguwš_j~`C`bKNQPh”mœ`bfumbg®š]Qcjkd)K ek“®`bKNQWsNh–QPdQP^Ch–^œm)jtguw_e0Ož’”\

^bsN’–h•`cš˜˜˜y· d’ijk^y^rek“G^afvd)K±ORe_w_QP’–^„Kuj0^r….QQgmbQcdQPg0`y’”\h–g:Ÿ0QP^a`bh–ˆ0jt`bQPw®˜rJLKNQ“Ím)jtˆ0ORQPg:`yj~`yh”e0g et“=QPj0d)K

Oj0^b^e_ddfNm)^=j~“Ô`yQm=^bekOžQ&h”gvw_Qs.QguwNQg:`GQ§_s.ekgNQPg:`bhijt’_`yh”OžQr™&h”`bKjxsujkmyjkORQ`bQPm=w_Qs.QguwNh”gNˆvšks.e0^y^ah–…N’–\kš

ek“®h•`)^œOžj0^b^P˜6Z]QQW¼CQPma`yekh–g 7†9Aš_V±h–QmyOžekg:` 7†N‡9jkguwmyQ“ÍQPmbQPgudQc^6`bKuQmyQh–g®˜6JLKNQ„sumbe0…N’”QPO^Gde0gu^bh–w_QPmbQcw

jkmbQL^bekOžQ™&Kujt`wNh5QmyQg:`@PmbQPˆkfN’ijtmyh”`o\Wsumbe0svQPma`yh”Qc^ek“vjk^y^be]dh–jt`bQcw!V©jkmb­0e~Ÿ„sNmye_dQc^b^bQP^Pštw_fujk’”h”`o\kštm)j~`yQCek“

wNQPdj\Wet“vh”guwNh”Ÿ]hiw_fujt’:Ojk^y^aQc^š’–e0^y^ºek“NOjk^y^š~jk^b\]Ožs_`bek`bhidGw_hi^o`ymbh–…Nf_`yh”e0gu^Pš_˜˜˜y·8^bsN’–h•`b`bh–gNˆWjk’”ˆ0ekmyh•`yKNO h–^

nofv^o`&j!myQPdfNm)^ah–ŸkQL“Ím)jtˆ0ORQPg:`yj~`yh”e0g et“jtg h–g:`bQˆ0Qmœh”g:`ye!h–g0`yQˆ0QmœsNh”QcdQc^Cfug0`yh”’5Qcjkd)K ek“`bKNQde0Ožsve0gNQg:`y^

Kvjk^j ^ah–—Qž’–QP^y^„`bKujkg

D

^{˜ ‘} g¨j de0g:`bh–g:fuekfu^x^bQ`a`yh”guˆušºjkg¥jkgujt’–ekˆ0fNQ!ek“G`yKNQ jt’–ˆke0mbh”`bKNO^xde0gu^bh–w_QPmbQcw KuQmyQT™œekfN’iw£dekgu^bhi^o`Wh”g¥^o`yeksNsuh”gNˆ¬`bKuQT“Ím)jtˆ0OžQg:`yjt`bh–ekg©sNmye_dQc^b^„et“Cj¬Ojk^y^xjk^x^ae]e0g£jk^Wh”`y^xŸ~jt’–fNQRh–^

….Q’–e~™^bekOžQW`bKumbQc^aKNe0’–w

ε > 0

^˜

,,$ % }&%,%% $ n,n5

·„^Gh”`L™&h”’–’5…vQ&KÍQcjk^bh”’–\)M=^bQQgºš:jT^asN’–h”`a`bh–gNˆžjt’–ˆke0mbh”`bKNO djkgEjt’i^be!….Q„w_Qc^bdmbh–…vQcwEj0^Gjm)jtgvw_ekO myQPdfumy^bh”Ÿ0Q

^bsN’–h•`b`bh–gNˆ et“=`yKNQžh”g:`yQmyŸjk’

[0, 1]

˜0=uekmxQ§NjtOžsN’–QRh”g`bKNQdPjk^bQTek“Cj w_\0j0w_h–dR^bsN’”h”`a`yh”guˆušº^a`yjkma`yh”gNˆE“Ímbe0O

`yKNQžh”g:`yQmyŸjk’

[0, 1]

š5`o™œe ^bfN…Nh–g:`bQPmbŸ~jt’i^

I 1

š

I 2

jkmbQždmyQPjt`bQPw%jtgvw£Qcjkd)K£ek“G`yKNQO h–^^bsN’–h•`j~`h”`y^W`yfNmyg

jkguw^aežekg%˜˜P˜

JLKNQc^aQm)jtgvw_ekO mbQcdfNm)^ah–ŸkQxw_QcdekOžs.e0^bh•`yh”e0gu^LKujŸ0Q…vQPQg±de0gu^ahiw_QPmbQcwE“ÍmyekO `bKNQs.ekh–g:`qet“Ÿ]h”QP™ek“

`yKNQ&ˆkQPekOžQ`ymb\!et“.`bKNQ&….ekfNgvwNjtmy\s.ekh–g0`)^6…]\!Ÿ~jtmyh–ekfu^jtfN`bKNe0my^Pš~`beQ§_sNmbQc^b^`yKNQ Fqjkfu^yw_ekmEw_h–ORQPgu^bh”e0g

ek“=`bKuh–^x^bQ`xet“Gs.ekh–g:`y^Wh”gsujtmb`bhidfu’–jkmP˜V©jkfN’–wNh”g%jtguw N h”’–’”hijtO^

7

†k€9šVN%j\]Ožh–mbQRjtguw N h”’–’”hijtO^

7

}u‡9

dekgu^bhiw_QmºwNQPde0Ožsve:^ah”`bh–ekgu^ºet“_`yKNQGh–g:`bQmyŸ~jt’

[0, 1]

™&Kuh–d)K!jkmbQGgNet`ºguQPdQc^b^yjtmyh–’”\Wdekgv^aQPmbŸ~j~`yh”Ÿ0QkšPh@˜Qk˜™&KNQg

|I 1 | + |I 2 | < 1

Kuek’iwN^L™&h•`yK sve:^ah”`bh–ŸkQxsNmyek…ujk…Nh–’”h”`o\ h–gE`yKNQw_\0j0w_hidWdPjk^bQk˜

F„jtO!…u’”\©jkguw£¹jtsNhiw_fu^

7

‡P},96ekm"=ujk’–dekgNQPm

7

‡P€9=dekgu^bh–wNQmxw_QPdekOžsve:^ah”`bh–ekgv^qek“6`bKNQžh–g0`yQmyŸ~jt’

[0, 1]

“ÍmyekO `bKNQEsve0h”g:`et“LŸ]h”QP™ et“œ`bKNQ ’–Qguˆt`bKv^et“œ`bKNQ¬jk^y^ae_dhij~`yQPw%^afu…Nh”g:`yQmyŸjk’–^P˜EJLKNQEh”g:`bQPmbŸ~jk’

[0, 1]

^{hi^}

myQsumbQc^aQPg0`yQPwE…:\jRgNekg_½h–gudmyQPj0^ah–gNˆž^aQc›0fuQgudQ

(L n )

™&KNe:^aQ^bfNOhi^

1

^˜r=uekm

n ≥ 1

^š

L n

hi^L`bKuQ’”QPgNˆt`yK

ek“`bKNQ

n

`yK±’–jkmbˆ0QP^a`rh”g:`yQmyŸjk’®et“`yKNQTw_Qcde0ORs.e0^bh”`bh–ekg®˜LJLKNhi^qw_Qc^bdmbh–s_`bh–ekg hi^q^bh”Ožh–’–jkm&`be `bKNQ!d’–j0^b^bhidjt’

myQsumbQc^aQPg0`)j~`yh”e0gEet““ÍmyjkˆkOžQg:`)j~`bh–ekg¬sNmye]dQP^y^aQc^˜6Z]QPQl=h•`yOjtg 7†0†9˜

‘ g%`bKuh–^^bQ`b`bh–gNˆuš®OTfN’”`bh–sN’”hidjt`bh–ŸkQdj0^bdPjkw_Qc^„jkguwOžjkma`yh”guˆ0jt’–QP^Wh–g:`bmye]wNfudQcw£…]\©V©jkguw_Q’–…Nmyet` 7>9

jkguw :jtKujkgNQ±jtguwÕlQPmbmyh”MPmbQ 7‡@>9x^aKue~™fNs¡›:fNh”`bQ©gvj~`bfumyjk’”’–\k˜ JLKNQ\¦KujŸkQ ….QQg¡jkgujt’–\]—Qcw¤›:fNh”`bQ

Q§]`bQgv^ah–ŸkQP’”\0šN^aQPQx¹ºh”f 7~},9 šN¼Ljtmymyjk’ 7<9jtguwmyQ“ÍQmyQgvdQP^C`yKNQmyQh–g®˜

prq i€tbx y t

JLKuQ„sNfumbs.e0^bQ„ek“º`yKNh–^Lsvjts.QmLh–^œ`o™œet“Íe0’–w®˜^=h–my^a`Pš_h”`rdekgv^ahiw_Qm)^L^asu’”h”`a`yh”gNˆžjt’–ˆke0mbh”`bKuOž^œ™&h”`bK j!m)jtgvw_ekO

[email protected]^y^bh”…N’–\qfug:….ekfuguw_QPwkM®w_QPˆkmyQQet“N^bsN’”h”`a`yh”guˆrˆkQPgNQm)jt’–h”—Ph”guˆC`yKNQGsumbQPŸ:h–ekfv^^o`yfuw_h–QP^®h–gx`bKuh–^w_ekOjth–g®˜

Z_QPde0guw_’–\kškjkguw!`bKNhi^h–^=h”gT“Îjkd``yKNQ&Ojth–gTs.ekh–g:`=et“v`bKNQLsujksvQPmPšth”`=sNmyeks.e0^bQP^jWsNmyek…ujk…Nh”’–hi^o`yh–dœjtsNsumbe:jkd)K

`yKuj~`6^bh”OžsN’–h•«vQP^O!fud)K!`bKNQLjtgujk’”\_^bh–^ºet“N`yKNQP^bQLjt’–ˆkekmyh”`bKNO^P˜ºV±ekmyQe~Ÿ0QmcšPj0^jq…:\]sNmye_w_fud`PštjrguQ™8w_h–myQPd`

myQsumbQc^aQPg0`)j~`yh”e0gEet“`bKNQj0^a\]Ožs_`yet`bhidWe0^ydh–’”’ij~`yh”guˆT….QKujŸ]h–ekm&hi^LQP^a`yjk…N’”hi^bKNQPw˜

JLKNQRjtgujk’”\_^bh–^qsNmyeks.e0^bQPw h–g±`bKNhi^Wsujts.QmWjt’i^ae¬^o`)jtmb`y^q“Ímbe0O DG›:fujt`bh–ekg;KAMš.…uf_`Wh”`y^q`bmyQPjt`bOžQg:`Wh–^

^bh–ˆkgNh”«vdjkg:`b’–\ w_h5QmyQg:`L“ÍmyekO `bKuQjtgujk’”\:`bhidWjtsNsNmye0j0d)K®˜·&“Ô`bQPmr^ae0OžQq`ymyjkgu^o“Íe0mbOjt`bh–ekg®š/DG›:fujt`bh–ekg K@M

hi^rh–g:`bQPmbsNmyQ`yQPw jk^„jžsNmyek…ujk…Nh”’–hi^o`rQc›0fvj~`bh–ekg±™&Kuh–d)K±hi^rh”`bQPmyjt`bQPw…]\fu^ah–gNˆEjksNsNmyeksNmyhij~`bQxh–guw_QPsvQPguw_Qg:`

m)jtgvw_ekO Ÿ~jtmyh–jk…N’”Qc^˜ =Ne0’”’–e~™&h”guˆ¨`bKNQ%OžQ`yKNe_w¡et“!prek….Qmb`

7† 9šr`yKNQguQ§]` ^a`bQPs¡hi^ `ye¦s.Qmb“ÍekmyO j

sumbe0…ujt…Nh–’–h–^a`bhidw_Q½Alekhi^y^ae0gNgNh–—Pj~`yh”e0g±jtguwš.…]\ fu^ah–gNˆE=ufN…Nh–gNh ^xJLKNQPekmyQO'dekg]Ÿ0QgNh–Qg:`b’–\kšu`be¬mbQPsNmbQc^aQPg:`

`yKNQ¬›:fujkg:`bh”`o\

E (R n )

…:\%fu^bh”guˆj±lekhi^y^ae0g%s.ekh–g:`TsNmye_dQc^b^e0g¥`bKNQ¬mbQcjt’=’–h–gNQk˜£JLKuQ «ugujk’@š6dmbfudh–jk’@š

^a`bQPs™&Kuh–d)K¥w_h.QPmy^W“ÍmyekO prek….Qmb`

7† 9Ašºdekgu^bhi^o`)^xh”g%fu^bh”gNˆ`yKNQž­kQ\£mbQPgNQ™Ljt’JLKNQe0mbQPO `be ˆ0Q`x`bKNQ

j0^a\]Ožs_`yet`bhidW…vQPKujŸ]h”e0mLet“º`yKNQ^bQP›:fNQgvdQ

( E (R n ))

^˜

JLKNQjksNsNmye0j0d)K h–^LQP’”QPOžQg:`yjkmb\0š:h”`y^rOžjkh”g jkwNŸjkg:`yjtˆ0Q„e0gE`yKNQjtgujk’”\:`yh–d„`bmyQPjt`bOžQg:`&’–h”Qc^&dQmb`yjkh”gu’”\

h–g`bKNQfv^aQxet“`bKNQmyQguQ™Ljt’®JLKNQPekmyQO ™&KNhid)K ˆkh–ŸkQP^ % `bKNQj0^a\]Ožs_`yet`bhidW…vQPKujŸ]h”e0fNmc˜

¬° ‹ÍŒc‹ µ Œ ° Ž ° 

Z_QPd`yh”e0gž„ˆ0h”Ÿ0QP^jWfu^aQ“ÍfN’NmyQsNmyQP^bQg:`)j~`bh–ekg!“Íekm`bKuQrjŸkQm)jtˆ0QCde0^a`ek“u`yKNQrjt’–ˆke0mbh”`bKNO ˜JLKNQLOžjkh”gRmyQP^bfN’•`

ek“N`bKuQœsvjts.Qm“Íekm`yKNQ&jk^b\:Ožs_`yet`yh–dœde:^o`h–^`yKNQœ“Íek’–’”e~™&h–gNˆ„`bKNQPekmyQO h–gžZ]Qcd`yh”e0gT†N˜JLKNhi^h–^6jW^afuOROjkmb\

ek“l=myeks.e0^bh”`bh–ekgu^ Ažjtgvw£‡0‡t˜

°]µ  ° o‰¬‹ ² Œ µ Œ@k‹ µ Œ ° ‰ ° ~Ž ° Tµ ‹PŒ ,\ +*( ,*-}$!}%I.-)%

,,

Z 1 0

| log(y)|

y W(dy) < +∞,

^KΆ+M

O.-)%#/* $ % }%

W

^O% )%In ,G -}$&%I !-n%

n→+∞ lim E (R n )

n = E (G)

(D − 1) R 1

0 | log(y)| W (dy) .

^KÍ}+M

L.-)%"#+*E$ % , }%

W

^{% n,#%k E ,.-/$ %I -} %n,%In ,bn ,

λ > 0

^{! ,}

n

^{*%IL6 *+%} !€.-)%\%}&%, ,%# %

E (R n ) n ∼ F

log n λ

K@|M

-)!g-n%%

F

^{O.-)% )%IJ}I .- n%}

1

% r%G"!k

x ≥ 0

^!

F (x) = E (G) R 1

0 | log(y)| W (dy) λ 1 − e ^−λ

Z +∞

0

exp

−λ

x − log y λ

y ^D−2

(D − 1)! e ^−y dy

,

{z} = z − bzc

^{O.-)% I, n , {}

z ∈ R

⁵

SCe0guw_h”`bh–ekg KΆ+MRh–^žgNek` mbQcjt’–’”\¨myQP^a`bmyhid`bh–ŸkQ±^bh”gvdQ`bKNQ±Ÿ~jkmbhijt…N’–Q

G

hi^ž….ekfNguwNQPw¤h”gÕsNm)jkd`bhidQ0˜ÕJLKNhi^

`yKNQe0mbQPO de~Ÿ0Qm)^Tjkguw¦Q§]`bQPguwN^ ^bekOžQ et“„`bKNQ±myQP^bfN’”`y^žh–g¦`yKNhi^ w_ekOjkh”gbP“Íe0m :xg:fN`bK ^·r’–ˆke0mbh”`bKNO š

:xg]f_`yK 70<9œjkguw¥“Íe0m

Q

½Üjtmy\jt’–ˆke0mbh”`bKuOž^x™&h”`bKª…u’”e_d)­kQcw¨jtmymbh–Ÿ~jt’i^šV©j~`yK]\]^!jtguw =’ij~noek’–Q` 7A9š^bQQ SCe0mbe0’”’ijtmyh–QP^„‡P€Rjtguw%‡c_˜

=ufNma`yKNQmyOžekmyQkš™&KNQg¥`yKNQmyQ jtmyQ jk^b\]Ožs_`bek`bhidžsvQPmbh–e_w_h–d e0^ydh–’”’ij~`yh”e0gu^šº`bKNQEsvQPmbh–e_w_h–d“ÍfNgud`yh”e0g

F

h–g]Ÿke0’”Ÿ0QPw hi^LQ§_sNmyQP^y^bQPww_h”myQPd`b’–\Ejtgvw¬guet`&h”g`yQmyOž^ret“h”`y^e=Nekfumbh–Qmrde:Q dh–Qg:`y^&j0^Lh•`rhi^Lfu^bfujt’–’”\E`bKNQ

dPjk^bQk˜JLKNQrQ§_sNmyQP^y^bh”e0gTek“

F

ˆkQPgNQm)jt’–h–—QP^`yKNQrmyQsNmyQP^bQg:`)j~`bh–ekgRet“®p&e0…vQPma` 7† 9vek…N`yjth–gNQcw!“Íe0m€:xg]f_`bK ^

·q’”ˆ0ekmyh•`yKNO ˜

JLKNQ ,}, et“6`yKNQT^bQP›:fNQPgudQ

(R n )

KÎjkguw±gNek`qekgu’”\ h”`y^WjŸkQm)jtˆ0Q<MChi^qh–g]ŸkQP^a`bh–ˆ0jt`bQcw¬h–gZ]Qcd½

`yh”e0g¬}u˜r=Nekmr^bh–ORsu’”hidh”`o\kš_e0gN’”\ `bKNQdj0^aQW™&KNQPmbQW`bKuQWŸ~jkmbhijt…N’–Q

G

hi^&de0gu^o`)jtg:`&jtguwE`bKNQxŸ~jtmyhijt…N’–QP^

V _·,G

jkmbQWQP›:fujk’5`be

1/G

hi^&dekgv^ahiw_QmyQPw®˜JLKNQsufNmbs.e0^bQWet“`bKNhi^&^aQcd`yh”e0g¬hi^œ`be^bKNe~™¡`yKuj~`L`bKuQw_h–^a`bmyh–…Nf_`bh–ekg ek“&`bKNQlekhi^b^bekg¥`ymyjkgu^a“ÍekmyO ek“&`bKNQ^aQc›0fuQgudQ¬jtgvw¨OžekmyQ ˆkQPgNQm)jt’–’”\0š`bKNQ w_hi^o`ymbh–…Nf_`yh”e0gªet“rOže0^a`Tek“

`yKNQ¬“ÍfNgud`yh”e0gujt’i^Tet“&`bKuQj0^b^be_dhij~`bQcw%`bmyQQ0š=djtg8…vQQ§_sNmyQP^y^aQcwª›:fNh”`bQ¬^bh–ORsu’”\¨h–g¨`bQPmbO^Ret“rle0h–^y^ae0g

sumbe_dQc^b^bQP^LjkguwEfugNh•“Íe0mbOž’–\Ew_hi^o`ymbh–…Nf_`yQPwm)jtguw_e0O Ÿ~jtmyh–jk…N’–QP^P˜

J™CeqmbQPsNmbQc^aQPg:`yj~`yh”e0gu^5et“]`yKNQCw_hi^a`bmyh”…NfN`bh–ekget“]`yKNQGlekhi^y^ae0g„`ymyjkgu^a“ÍekmyOjk^jL“ÍfNgud`bh–ekgujk’kek“]le0h–^y^ae0g

sumbe_dQc^b^bQP^GjtmyQqw_Qmyh–ŸkQPw®˜·„^CjTdekgv^aQc›0fuQgudQkš]j’–j™Õek“®’–jkmbˆ0Qrg]fNO!….Qm)^=hi^Gsumbe~Ÿ0QPwž™&KNQPg`bKNQ„g]fNO!….Qm

ek“œh–gNh”`bhijt’=h”`bQPO^Kuj0^j le0h–^y^bekg¥w_h–^a`bmyh–…Nf_`bh–ekgiKÎle0h–^y^bekg%`bm)jtgu^a“ÍekmyO M˜ V±e0mbQPe~ŸkQPmPš5`yKNQEjk^b\]Ožs_`bek`bhid

e:^bdh”’–’ij~`bh–gNˆ ….QKvjŸ:h–ekmWet“6`yKNQžjt’–ˆkekmyh”`bKNO'hi^qsNmye~ŸkQcw±jk^Wj¬dekgu^bQP›:fNQPgudQ!et“Cj , ’–j™ ek“=’–jkmbˆ0Q

g]fNOT…vQPmy^P˜JLKNQP^bQrfNg]fu^bfujt’.’ij™r^6et“®’ijtmyˆkQ&g]fNOT…vQPmy^CjtmyQkš:h”g`yKNQqQPguwš:h”g `bKNQ„mbQcjt’–O ek“ºd’ijk^y^ahidjk’N’–j™r^

ek“’ijtmyˆkQWg]fNO!….Qm)^˜

JLKNQdQg:`bm)jt’6’–h”Ožh”`x`bKNQPekmyQOYhi^jt’i^aesNmye~ŸkQcw©™&h”`bK¥j^bh–ORh–’ijtmxOžQ`bKue]w%h–g£`yKNhi^dj0^aQ0˜TJLKNhi^xh–^j

d’–j0^b^bhidjt’ºmbQc^afN’”`Pš^bQQRV©jtKNOže0fuw 7?@9 š5h•`xh–^„fu^bfujt’–’–\¬sNmye~ŸkQcw ™&h•`yKdekOžsN’–Q§±jtgujk’”\_^bh–^qORQ`bKNe_wN^„Ÿ]h–j

›:fNh”`bQW`yQPd)KNgNhidjk’QP^a`bh–Oj~`bh–ekgv^˜ ‘`rhi^LsNmbe~Ÿ0QPwEKNQmyQWj0^&jždekgv^aQc›0fuQgudQ„ek“º`bKuQ

,

dQPg0`ymyjk’5’”h–Ožh•`

`yKNQe0mbQPO“Íe0mqh–guw_QPsvQPguw_Qg:`Wmyjkguw_ekOŸ~jtmyh–jk…N’–QP^P˜L·œ`„`bKNQR^bjkOžQ`bh–OžQkšjgNQP™ myQsumbQc^aQPg0`)j~`yh”e0gek“6`bKNQ

j0^a\]Ožs_`yet`bhidWŸ~jtmyh–jkgudQWh–^rek…_`)jth–gNQPw˜

Z]Qcd`yh”e0g±|TmyQPdPjt’–’–^œ…Nmyh”Qu\E^bekOžQxmbQc^afu’•`)^&jtguw w_Q«ugNh”`bh–ekgu^&dekgudQmygNh”guˆT`yKNQmyQgNQP™œjk’v`yKNQe0mbQPO ˜

"&"L¨ "L"&

JLKumbe0fNˆkKNe0f_`L`bKNhi^&sujksvQPmPš

(N ([0, x]))

wNQgNek`bQP^&jžle0h–^y^ae0g¬sumbe_dQc^b^L™&h”`bK±h–g:`bQPgu^ah”`o\

1

šuQP›:fNh–Ÿjk’”QPg:`b’–\

h”`Gdjkgjt’i^aex….Qqw_QP^ydmyh–…vQcwTj0^=jxgNekgN½w_QcdmyQPj0^ah–gNˆ„^bQP›:fNQPgudQ

(t n )

^bfud)KR`bKujt`

(t n+1 − t n )

h–^Gj^aQc›0fuQgudQ ek“.hA˜h@˜w˜myjkguw_ekOzŸjkmbhijt…u’”Qc^Q§_sve0gNQg:`bhijt’–’–\!w_hi^o`ymbh–…Nf_`yQPwž™&h”`bKžsujtm)jtOžQ`bQm

1

^{˜^=uekm}

x ≥ 0

št`yKNQ&Ÿ~jtmyh–jk…N’”Q

N ([0, x])

h–^6^bh–ORsu’”\x`yKNQLg:fuO!….Qm6et“

t n

^h–gT`bKNQLh–g0`yQmyŸ~jt’

[0, x]

^˜Z_QQ :xh”guˆkOjtg 7‡<A9_“Íekm…uj0^ahidCmbQc^afN’”`y^

e0g¬le0h–^y^ae0g¬sumbe_dQc^b^bQP^P˜

DG›:fujt`bh–ekg K@MCjkguw`yKNQx…ve0fNguwNjkmb\ dekguwNh•`yh”e0gu^C“Íekmœ`bKNQ^bQP›:fNQPgudQ

(R n )

jtmyQW^bfNOžOjtmyh”—PQPwEh”g¬`bKNQ

“Íe0’”’–e~™&h–gNˆTmyQ’ij~`yh”e0g®š]“Íekm

n ≥ 0

^š

R n

= 1 + R 1,N ₁ ⁿ + · · · + R G,N _G ⁿ − G

{n<D} .

`yKNQmyQ“Íe0mbQ0š

R n − 1

=

G

X

i=1

R i,N _i ⁿ − 1

+ G

{n≥D}

K M

° ͌µ '-n% le0h–^y^bekg `ymyjkgu^a“ÍekmyO #,$# #%* &%,%"I%n%I%

(a n )

^{% €%E ,}

X

n≥0

a n

x ⁿ

n! e ^−x = E a _N([0,x])

.

^K?M

JLKNQž“Íe0’”’–e~™&h–gNˆ sNmbe0sve:^ah”`bh–ekg%ˆkh–ŸkQc^xfu^bQ“ÍfN’=myQsumbQc^aQPg0`)j~`yh”e0gu^Wet“œ`bKNQlekhi^y^ae0g%Jmyjkgu^a“ÍekmyOYet“œ`bKNQ

^bQP›:fNQPgudQxek“

( E (R n ))

^˜

 µ µ ‹ÎŒ µ qµ ΋c‹ µ ~Ž ‹ Aµ  µ Œ ° ‹ ° °  °

(R n )

^"

x > 0

^!

E R N([0,x])

= 1 + E (G) E

+∞

X

i=0

1 Q i

k=1 W k

{ ^t D ≤x Q i k=1 W k }

!

.

^K>M

g-n% %

(W i )

^{0 55&5LI% n%# %0{O ,,$ % I6% - ,},}

W

⁵

J@ 5 ‘“

n

hi^Cjle0h–^y^bekgžmyjkguw_e0O Ÿ~jtmyhijt…N’–Q&™&h•`yK sujkmyjkORQ`bQPm

x

š:`bKuQq^asu’”h”`a`yh”gNˆTsNmyeks.Qmb`o\Tek“5le0h–^y^ae0g

Ÿ~jkmbhijt…N’–QP^ K@^aQPQ :xh”guˆkOjtg 7‡<A9=“Íe0mQ§NjtOžsN’–Q<Mx^bKNe~™r^W`yKuj~`cšdekguw_h”`bh–ekgvjt’–’”\©e0g%`bKNQ QPŸkQPg0`

{G = `}

jkguwe0g `bKuQ„Ÿ~jtmyhijt…N’–QP^

V 1,`

š5˜P˜˜š

V `,`

š]`bKNQWŸ~jtmyh–jk…N’–QP^

N _i ⁿ

^š

1 ≤ i ≤ l

jtmyQqh–guw_QPsvQPguw_Qg:`rjtgvw

N _i ⁿ

^{Kujk^}

jRlekhi^y^ae0g¬w_hi^o`ymbh–…Nf_`yh”e0g ™&h•`yKsvjtm)jtOžQ`yQm

xV i,l

˜CSCekgu^bQP›:fNQPg:`b’–\kš:“Íekm

x > 0

^š_h”“

Φ(x) = E R N([0,x])

− 1

x E (G) ,

^KAM

h”`rhi^LQPjk^bh–’”\Ed)KuQPd)­kQcw`yKuj~`

E (G)Φ(x) → R 1 − R 0 = 0

^{jk^}

x & 0

^˜

Z]h–gudQ

{N([0, x]) ≥ D} = {t D ≤ x}

š)DC›0fvj~`bh–ekg K Mœˆkh–ŸkQc^C`bKNQmyQ’ij~`yh”e0g

Φ(x) =

+∞

X

`=2

P (G = `) E

`

X

i=1

V i,` Φ(xV i,` )

! + 1

x P (t D ≤ x).

^Ko‡c€M

DC›0fvj~`bh–ekg Ko‡c€MLdPjtg¬`bKNQPg….QmyQ™&myh•`b`bQg jk^

Φ(x) = E (Φ(xW 1 )) + E 1

x

^{{t D ≤x}}

.

^Ka‡k‡<M

JLKuQWh”`bQPmyjt`bh–ekget“ DG›:fuj~`yh”e0g Ka‡k‡ML^aKNe~™r^C`yKuj~`cš_“Íekm

n ≥ 1

^š

Φ(x) = E Φ x

n

Y

k=1

W k

!!

+ E

n−1

X

i=0

1 x Q i

k=1 W k

{ ^{t D ≤x Q i} k=1 W k }

! .

JLKuQEjk^y^afuORsN`bh–ekg¨ekgª`bKNQ Ÿ~jkmbhijt…N’–Q

G

jkguw%`bKNQ^aQc›:fNQgudQ et“rŸ0QPd`bekm)^

(V n )

h”OžsN’–h”Qc^`yKuj~`cš=jt’–Ože0^a`

^bfNmyQ’–\kš]`yKNQ^aQc›:fNQgudQ

( Q n

k=1 W k )

dekg]ŸkQPmbˆ0QP^G`be

0

˜6JLKNQx“ÍfNgvd`bh–ekg

Φ

djtg`yK:fv^L…vQmyQsNmyQP^bQg:`yQPwEj0^

Φ(x) = E

+∞

X

i=0

1 x Q i

k=1 W k

{ ^{t D ≤x Q i} k=1 W k }

! .

JLKuQWsumbe0sve:^ah”`bh–ekgKujk^L….QQPg sNmbe~Ÿ0QPw˜

=umbe0OzgNe~™¤ekgºšt`bKumbe0fNˆkKNe0f_`6`bKuQrsujts.Qmcš

(W i )

™&h–’”’.w_Qguet`bQ„jtgžh@˜hA˜w˜6^bQP›:fNQPgudQ&ek“5m)jtgvw_ekO Ÿ~jkmbhijt…N’–QP^

e0g

[0, 1]

™&h”`bK±w_hi^o`ymbh–…Nf_`yh”e0g

W

^˜

 µ µ ‹ÎŒ µ  µ ŠŽ_Š @‹ŒÎ¶ °0³ qµ ΋c‹ µ 0Ž_Œµ "

n ≥ D

^!r-n%

E (R n ) = 1 + E (G) E







T ( ^U (n) ^D ) ⁻¹ X

i=0

1 Q i

k=1 W k

,







Ka‡cM

g-n% % !k

0 < y < 1

^!

T (y) = inf (

i ≥ 1 :

i

Y

k=1

W k < y )

U _(n) ^D

^-n%

D

^{- $& 6% %, , I6%}

n

^{%)%I%k!}n,$\ , }% E $ % &#}

% ,

[0, 1]

^{%n%%In {}

(W i )

⁵

J@ 5 =uekm

x > 0

šC…]\8wNQPde0Ožsve:^ah–gNˆ¥™&h•`yKÕmbQc^as.QPd`R`ye¥`bKNQ±g]fNOT…vQPmet“xsve0h”g:`)^žet“W`bKNQ±lekhi^y^ae0g sumbe_dQc^b^

(N (t))

h–g`bKNQh–g:`bQPmbŸ~jt’

[0, x]

šNekgNQxˆ0Q`y^Pš]“Íekm

0 < α ≤ 1

^š

P (t D ≤ xα) =

+∞

X

n=D

P (t D ≤ xα, N ([0, x]) = n)

=

+∞

X

n=D

P (t D ≤ xα | N([0, x]) = n) P (N ([0, x]) = n).

=uekm

n ≥ D

šWde0guw_h”`bh–ekgujk’”’–\Õekg `yKNQQPŸkQPg0`

{N ([0, x]) = n}

šq`bKNQŸjkmbhijt…u’”Q

t D

Kvjk^E`bKNQ¥^bjkOžQ

wNh–^a`bmyh”…uf_`bh–ekgjk^G`bKNQ

D

^bOjt’–’”Qc^o`œmyjkguw_ekO Ÿ~jtmyhijt…N’–Qqek“

n

fNgNh”“ÍekmyOž’”\w_hi^a`bmyh”…NfN`bQPwEmyjkguw_e0O Ÿ~jkmbhijt…N’–QP^

e0g

[0, x]

^{˜ N KNQPg}

x = 1

š_w_Qguet`bQW…]\

U _(n) ^D

jTŸjkmbhijt…u’”Q„™&h•`yK `bKuh–^Lde0guw_h”`bh–ekgujk’w_h–^a`bmyh–…Nf_`bh–ekgº˜=SC’–QPjkmb’–\kš

…]\KNe0OžekˆkQPgNQh”`o\kš`yKNQLŸ~jtmyh–jk…N’”Q

(t D | N ([0, x]) = n)

Kvjk^`bKuQL^yjtOžQLw_hi^o`ymbh–…Nf_`yh”e0gRj0^

xU _(n) ^D

^˜ =h–gujk’”’–\kš

e0gNQxˆkQ`y^œ`bKuQh–w_QPg:`bh”`o\

P (t D ≤ xα) =

+∞

X

n=D

P

U _(n) ^D ≤ α x ⁿ

n! e ^−x = E

+∞

X

n=D

{U ^D _(n) ≤α}

x ⁿ n! e ^−x

! .

¼œ\ fu^bh”guˆE`yKNQTh–guw_QPsvQPguw_QPgudQžet“G`bKNQž^aQc›:fNQgudQ

(W i )

^jtguw

t D

h–g DG›:fujt`bh–ekg K>Mš5`bKNQR’–j0^o`xhiw_Qg:`bh”`o\

ˆ0h”Ÿ0QP^C`bKuQmbQP’–jt`bh–ekg

E R _N ([0,x])

= 1 + E (G) E

+∞

X

i=0

1 Q i

k=1 W k +∞

X

n=D

{U ^D _(n) ≤ Q i k=1 W k }

x ⁿ n! e ^−x

! .

¼œ\E=ufN…Nh–gNh^rJLKuQekmyQO jtguw¬™&mbh”`bh–gNˆ

1 = exp(x) exp(−x)

š_`bKuh–^&Q§_sNmbQc^b^bh–ekg¬dPjtg ….QWmyQ™&myh”`a`bQPg±jk^

D−1

X

n=0

x ⁿ n! e ^−x +

+∞

X

n=D

1 + E (G) E

+∞

X

i=0

1 Q i

k=1 W k

{U ^D _(n) ≤ Q i k=1 W k }

!! x ⁿ n! e ^−x .

JLKuQ©h–wNQg:`bh”«vdjt`bh–ekg ek“prQsNmyQP^bQg:`yjt`bh–ekg K?Met“

E R _N([0,x])

jkguw¤`bKNQ£’–j0^o`Ehiw_QPg0`yh•`o\¤ˆkh–ŸkQc^ =Ne0ma½ OTfN’ij4Ka‡cM

Tµ

 µ Ύ_

²

² °

ŒP"@kŽ

Q

^{³ ŽN ² ‰ µ  ͌ " -)%I}

P (G = Q) = 1

^-),\

V i,Q ≡

1/Q

^!),

i = 1

^! 5<5<5

Q

^!r.-)%I0

n ≥ D

^!

E (R n ) = 1 + Q Q − 1

E

Q ^{d− log Q U (n) D e}

− 1

Ka‡P†M

.-!,

0 ≤ x ≤ 1

^!

P

U _(n) ^D > x

=

D−1

X

k=0

n k

x ^k (1 − x) ^n−k .

=umbe0O DG›:fuj~`yh”e0g[Ka‡P†Mš®…]\©fu^bh”guˆ¬`yKNQR“Îjkd`W`yKuj~`

nU _(n) ^D

dekg]ŸkQPmbˆ0QP^qh”g¨w_hi^o`ymbh–…Nf_`yh”e0g%j0^

n

^{`bQPguwN^}

`ye£h–g_«ugNh”`o\kš6h”`Thi^TgNet`žw_h dfN’•`T`beˆkQ`T`bKNQjk^b\]ORsN`bet`yh–d …vQPKujŸ]h”e0m!et“

E (R n )

˜JLKNQ¬ˆkQPgNQm)jt’œdjk^bQkš

DC›0fvj~`bh–ekg Ko‡Mš_hi^L^a’–h–ˆkK:`b’–\OžekmyQWdekOžsN’–h–dPj~`bQcw˜GU„gNQWKujk^C`bež^a`bfuw_\ `bKNQjk^b\]Ožs_`bek`bhid^œet“®`yKNQ^aQPmbh–QP^

h–gu^bh–wNQ„`yKNQQ§_s.QPd`yj~`yh”e0g®˜

ew|)y{~€)  |}t #rx ^|)y{~€

‘“

R(x) = E (R _N (]0,x]) )

w_QPgNet`yQP^œ`yKNQQ§_s.QPd`yQPwŸ~jt’–fNQet“`bKuQle0h–^y^bekg¬`bm)jtgu^a“ÍekmyO et“`bKNQT^aQc›0fuQgudQ

(R n )

š_`bKuQg(DG›:fuj~`yh”e0g K Mœˆ0h”Ÿ0QP^C`bKuQmbQP’–jt`bh–ekg

R(x) = 1 +

+∞

X

`=2

P (G = `) E

`

X

i=1

R(xV i,` )

!

− E (G) P (t D ≥ x),

…]\EwNQgNek`bh–gNˆ

h(x) = 1 − E (G) Z +∞

x

u ^D−1 (D − 1)! du,

h”`rhi^LQPjk^b\`yež^bQQW`yKuj~`&`yKNQjt….e~ŸkQWh–wNQg:`bh”`o\¬dPjtg…vQ™&myh•`b`bQPg j0^œ`bKNQx“Íe0’”’–e~™&h–gNˆTh–g:`bQPˆkm)jt’QP›:fujt`bh–ekg

R(x) = Z +∞

0

R(xu) W(du)

u + h(x).

^Ko‡P}+M

prQPdjk’”’v`bKvj~`

W

h–^L^bekOžQqsumbe0…ujt…Nh–’–h•`o\w_hi^o`ymbh–…Nf_`yh”e0gEe0g `bKuQ„h–g:`bQmyŸ~jt’

[0, 1]

^˜ =Ne0mC`yKNQ

Q

^½jkmb\RsNmyet`be_dek’

dekgu^bhiw_QmyQPwE…:\V©j~`yK]\]^Ljkguw(=’ij~noe0’”Q`

7

A9AšN`bKNhi^&QP›:fujt`bh–ekgh–^

R(x) =

Q

X

i=1

R(xp i ) + h(x).

‘`Lhi^Cjkgujt’–\]—QPwž…]\de0gu^bh–w_QPmbh–gNˆ`yKNQxV±Q’–’”h–g `ymyjkgu^a“ÍekmyO

R ^∗ (s)

^ek“

R(x)

ekgE^bekOžQ„ŸkQmb`bhidjk’v^a`bmyh”s

S

^et“

C

^š

R ^∗ (s) = Z +∞

0

R(u)u ^s−1 du, s ∈ S

™&Kuh–d)K®š_h–g`bKNhi^rdPjk^bQkš_hi^Lˆkh–ŸkQg…]\

R ^∗ (s) = h ^∗ (s) ,

1 −

Q

X

i=1

1 p ^s _i

! .

JLKuQTjtgvjt’–\0`yh–dPjt’jtsNsumbe:jkd)K±de0gu^ahi^a`y^„h”g%jtgujk’”\]—h–gNˆ`yKNQRsve0’”Qc^qek“

R ^∗ (s)

ekg©`yKNQ!myh–ˆkK:`„Kujkguw£^bhiw_Q!ek“

S

šN…ujk^bhidjt’–’–\ž`bKNQ^bek’–f_`bh–ekgv^L™&h•`yKs.e0^bh”`bh–ŸkQxmbQcjt’sujkma`&ek“º`bKuQQP›:fuj~`yh”e0g

p ^−s ₁ + p ^−s ₂ + · · · + p ^−s _Q = 1.

JLKuQg®š]…]\žh”g]ŸkQPma`yh”guˆ`bKNQxV±QP’”’–h”g `bm)jtgu^a“ÍekmyO jtguwfv^ah–gNˆTdekOžsN’–Q§ jtgvjt’–\]^bhi^6`bQcd)KNgNhi›:fNQP^Pšk`yKNQxjk^b\:Ožs_½

`yet`yh–d!…vQPKujŸ]h”e0m„ek“

(R(x))

j~`„h–g_«uguh•`o\±hi^xw_QP^ydmyh”….QPw©h”g£`bQmyO^qek“=`bKuQP^bQ!s.ek’–QP^P˜xJLKuQ«ugujk’6^o`yQs®šjtg jkgujt’–\:`bhidxh”g]Ÿ0Qm)^ah–ekg et“`yKNQlekhi^b^bekg`ymyjkgu^o“Íe0mbO`be0ˆkQ`yKNQmq™&h•`yK±`bQcd)KNgNhidjt’QP^a`bh–Oj~`bQc^švQP^a`yjk…N’”hi^bKNQP^rj

myQ’ij~`yh”e0g….Q`o™œQQPg%`bKNQ j0^a\]Ožs_`bek`bhidR…vQPKujŸ]h”e0my^xek“G`bKuQ“ÍfNgud`bh–ekg

x → R(x)

jkguwet“œ`bKNQE^bQP›:fNQPgudQ

(R n )

^˜

‘ g¬`bKNQˆ0QgNQPmyjk’.dPjk^bQWdekgu^bhiw_QmyQPwEKNQmyQkš}DG›:fujt`bh–ekg Ko‡}/MCˆkh–ŸkQP^C`yKNQW“Íek’–’”e~™&h–gNˆRQ§_sNmyQP^y^ah–ekg“Íe0mL`bKNQ

V©Q’–’”h–g`bm)jtgu^a“ÍekmyO et“

(R(x))

R ^∗ (s) = h ^∗ (s)

1 − Z +∞

0

1

u ^s+1 W(du)

·qgjtgujk’”e0ˆkfNQ6ek“]`bKNQGjkgujt’–\:`bhid=jtsusNmbe:jkd)KW™Ce0fN’iwW^a`yjkma`™&h•`yK`bKNQC^o`yfuw_\Wet“]`bKNQCmbe]ek`y^

s ∈ C

^š

<(s) ≥ 0

^š

ek“º`yKNQQP›:fujt`bh–ekg

Z +∞

0

1

u ^s+1 W(du) = 1,

^Ko‡|M

jkguwšNh”“6sve:^b^bh–…N’”Q0šusNmye_dQPQPwN^L™&h”`bK©^bfuddQP^y^ah–ŸkQxh–g]ŸkQm)^bh”e0gu^Let“6V±Q’–’–h”g `ymyjkgu^o“Íe0mbO jtguw lekhi^y^ae0gE`ymyjkgu^o½

“Íe0mbO ˜

·„^&h•`r™&h–’–’®…vQ^bQQPg®š_ekfumrw_h”myQPd`rjtsNsNmye0j0d)KEmyQPw_fvdQP^œ`yeR`bKNQOžh–gNh”OTfNO `yKNQx`bQcd)KNgNhidjk’®jtsNsujkmyjt`bfu^

myQP›:fNh–myQPw±“Íe0m^afvd)K%jkg%jkgujt’–\_^ahi^˜RJLKNQžlekhi^b^bekg`bm)jtgu^a“ÍekmyO et“

(R n )

h–^jk’–^befu^aQcw£h–g¥ekfNmxOžQ`yKNe_wš

…uf_`œh”`Chi^Lde0g:Ÿ0QgNh–Qg:`y’”\RmbQPsNmyQP^bQg:`bQcwš0^bQQDG›:fuj~`yh”e0g K>+Mš_^ae!`bKujt`Ch”`LdPjtg ….QWmbh–ˆkK:`Cj™Lj\Th–g]ŸkQPma`yQPwR`be

ˆ0h”Ÿ0Qjtg±Q§]su’”hidh”`qQ§_sNmyQP^y^bh”e0g Ka‡cM&“Íekm

E (R n )

™&KNhid)K±™&h”’–’ˆkh–ŸkQw_h–mbQcd`y’”\¬`bKNQRjk^b\]ORsN`bet`yh–dx….QKvjŸ:h–ekm ek“º`yKNQ^aQc›:fNQgudQ

( E (R n ))

^˜

‘ g:`bQPmbQc^o`yh”guˆk’–\kš_h”“

(L n )

w_Qguet`bQc^œ`bKNQxgNe0g_½ h”gudmbQcjk^bh”guˆT^bQP›:fNQgvdQxet“`bKNQx’–QgNˆk`bKu^Lek“º`bKuQ^afN…uh”g:`bQPma½

Ÿ~jk’–^Get“

[0, 1]

jk^y^ae_dhij~`yQPwT`be`bKNQW^asN’–h”`a`bh–gNˆsumbe_dQcw_fNmyQ KÎ^bQQWZ]QPd`bh–ekg±‡0˜†M˜6JLKNQ %I }#, ek“`bKNQ

^a`bmyh–gNˆ

(L n )

hi^&w_Q«ugNQPw j0^œ`bKNQOžQmyekOže0mbsNKuh–dq“ÍfNgud`bh–ekg

ζ(s) = X

n≥1

L ^s _n , s ∈ C ,

^bQQLFqjkO!…N’–\ jtguw¬¹ºjksNh–wNfu^

7

‡P},9®jtguw¬¹ºjtsuh–w_fv^œjkguw Ÿ~jkgG=Nm)jtgu­kQguK:fu\]^bQg

7

t†,9A˜

‘

`Lh–^œgNek`LwNh dfu’•`L`be

^bQQW`yKuj~`&`yKNQmbQP’–jt`bh–ekg

E (ζ(s)) = Z +∞

0

u ^s W(du)

1 − Z +∞

0

u ^s W(du)

Kuek’iwN^˜ ‘gsujkma`yh–dfN’–jkmPšt`bKNQrs.ek’–QP^Get“5`bKuQ.Q`)jW“ÍfNgud`bh–ekget“5`yKNQqjk^y^be]dh–jt`bQcwTm)jtgvw_ekO mbQcdfNm)^ah–ŸkQ&^a`bmyh”guˆ

dPjtg…vQQ§_sNmbQc^b^bQPwEh”g`yQmyOž^Lek“`yKNQ^bek’–f_`bh–ekgu^Lek“gDG›:fujt`bh–ekg Ko‡|M˜

'¬ž!8*¬"'®  ¬x "L(!"z

LI@I ,% e ,$ , ˜ ‘“

(W i )

hi^!jtg¨hA˜h@˜w˜©^bQP›:fNQgvdQ ™&h”`bK8dekOžOžekgªw_hi^a`bmyh”…NfN`bh–ekg

W

wNQ«ugNQcwՅ]\;DG›:fujt`bh–ekg Ko‡MšL`bKuQ£^aQc›:fNQgudQ

(B i ) = (− log(W i ))

hi^EjtgÕhA˜hA˜w˜ ^bQP›:fNQPgudQ±ek“Wguekg_½ guQˆ0jt`bh–ŸkQWmyjkguw_e0O Ÿ~jtmyh–jk…N’”Qc^˜JLKNQm)jtguwNekO ™Ljt’–­

(S n )

j0^b^be_dhij~`yQPwE`be

(B i )

^š

S n = B 1 + B 2 + · · · + B n , n ≥ 0.

·„^&h”`r™&h”’–’®….Q^aQPQg®š]`yKNQ!jk^b\]Ožs_`bek`bhid„….QKvjŸ:h–ekmLek“`yKNQ^asu’”h”`a`yh”gNˆ jk’”ˆ0ekmyh•`yKNO w_QPsvQPguwN^rOTfud)Kekg`bKNQ

wNh–^a`bmyh”…uf_`bh–ekget“

(B i )

^˜r=Ne0m

x > 0

š]`yKNQdmye0^y^bh”gNˆT`bh–OžQ

ν x

ek“’–QŸkQP’

x

^]\

(S n )

h–^&w_Q«ugNQcwj0^

ν x = inf{n : S n > x}.

=uekm

0 < y < 1

š5`bKNQRŸ~jtmyh–jk…N’”Q

T (y)

et“Gl=myeks.e0^bh”`bh–ekg hi^x^ah–OžsN’”\

ν − log(y)

˜!Z]QPd`bh–ekg¥| myQPdPjt’–’–^q`bKNQ

Ojkh”gmbQc^afu’•`)^&dekgvdQmygNh–gNˆp&QguQ™Ljt’JLKNQe0mb\Efu^aQcwEh–g`bKNQx“Íe0’”’–e~™&h–gNˆu˜

‘ “

Ψ

hi^&w_Q«ugNQPw j0^

Ψ(x) = E

ν x −1

X

i=0

exp

i

X

k=1

B k

!!

, x > 0

`yKNQg …]\ DG›:fujt`bh–ekg Ko‡Mš

E (R n ) = 1 + E (G) E h Ψ

− log

U _(n) ^D i

.

^{Ka‡ M}

‘`h–^d’–QPjkm`bKvj~`

− log(U _(n) ^D )

dekg]ŸkQPmbˆ0QP^žh–gw_hi^o`ymbh–…Nf_`yh”e0gÕ`be

+∞

^{jk^}

n

ˆke]QP^ `be8h–g_«uguh•`o\0˜ JLKNQ j0^a\]Ožs_`yet`bhidW…vQPKujŸ]h”e0mLet“

Ψ

j~`&h–g_«ugNh”`o\Ehi^œ«um)^o`qjkgujt’–\]—Qcwš:`bKNhi^L“ÍfNgud`bh–ekg±djkg….QxmbQP™&mbh”`a`yQg±jk^

Ψ(x)e ^−x = E

ν x −1

X

i=0

e ^{S i −x}

!

= E

ν x

X

i=1

e ^{S νx −i −x}

!

.

^Ka‡?M

q p t&% ~€ xny| ( tb|}yw vt

‘g `yKNhi^qsujkma`cšuh”`„hi^qj0^b^bfNOžQPw`yKuj~`r`yKNQTwNh–^a`bmyh”…uf_`bh–ekg±ek“

W 1

hi^rguet`„Q§_sve0gNQg:`bhijt’–’–\¬jtmyh”`bKNOžQ`yh–dk˜qZ]QQ

‚„Q«vgNh•`yh”e0g©]˜

C° ªŽ %IO.-)%\

E

| log(W 1 )|

W 1

= Z 1

0

| log(x)|

x W(dx) < +∞,

.-)% %I6 ,,

sup

x≥0

E e ^{S νx −x}

< +∞

-n,,I5

J@ 5

¹ºekm)w_Qg ^ ‘

gNQc›0fvjt’–h•`yh”Qc^šu^bQQ¹ºekm)w_Qg

7

0|<9jtgvw±SCKvjtgNˆ

7

|,9Ašv^aKNe~™¡`bKujt`Pš_“Íe0mrjtg]\

p ≥ 0

^š

sup

x≥0

E ((S ν x − x) ^p ) ≤ p + 2 (p + 1) E (B 1 ) E

B ₁ ^p+1 ,

`yK]fu^š_e0gNQˆkQ`y^œ`yKNQmbQP’–jt`bh–ekg

sup

x≥0

E e ^{S νx −x}

≤ 1 E (B 1 )

Z +∞

0

(u + 2)e ^u P (B 1 ≥ u) du

= E (B 1 + 1)e ^{B 1}

− 1 = E

− log(W 1 ) + 1 W 1

− 1 < +∞.

=uekm

i > 1

šrJLKuQekmyQO#t€8^aKNe~™r^`bKujt`Pšr™&KNQPg

x

ˆ0e:Qc^ž`be¦h”gN«ugNh”`o\kšL`bKuQ£Ÿ~jtmyh–jk…N’–Q

S ν x −i − x

dekg]ŸkQPmbˆ0QP^œh–g©w_h–^a`bmyh–…Nf_`bh–ekg `ye

−(τ ^∗ + τ 1 + τ 2 + · · · + τ i−1 )

šu™&KNQPmbQ`bKNQŸ~jkmbhijt…N’–QP^

(τ n )

^{jtmyQWhA˜h@˜w˜}

wNh–^a`bmyh”…uf_`bQcwj0^

B 1

jkguwh”guwNQs.Qguw_QPg:`ret“

τ ^∗

™&KNe0^bQw_hi^o`ymbh–…Nf_`yh”e0ghi^Lˆkh–ŸkQPg¬…]\

E (f (τ ^∗ )) = 1 E (B 1 )

Z +∞

0

f (u) P (B 1 ≥ u) du,

“Íe0mTjkg:\gNekgN½AgNQPˆ0jt`bh–ŸkQ ¼œekmyQ’–h–jkg%“ÍfNgud`yh”e0gªekg

R

˜¼C\¥·q^y^bfNOžs_`bh–ekguK@·LMš`yKNQEh–gudmbQPORQPg:`y^ek“&`bKNQ m)jtgvw_ekO'™Ljt’–­

(S n )

jtmyQ!….ekfNguwNQPw©….Q’–e~™ …]\

− log(δ)

š5`yKNQmyQ“Íe0mbQTekgNQRˆkQ`y^q`bKNQRmyQ’ij~`bh–ekgºšv“Íe0m

1 <

K ≤ ν x

š

ν x

X

i=K

e ^{S νx −i −x} = e ^{S νx −x}

ν x

X

i=K

e ^{S νx −i −S νx} ≤ e ^{S νx −x} δ ^K 1 − δ .

=umbe0O¹®QPOžOžj&>žjtgvwGDC›0fvj~`bh–ekg Ko‡,?MšNekgNQw_Qcw_fudQP^œ`bKuQg

x→+∞ lim Ψ(x)e ^−x = E

+∞

X

i=1

exp (−τ ^∗ − τ 1 − τ 2 − · · · − τ i−1 )

!

= 1 − E (exp(−τ 1 ))

E (τ 1 ) × 1

1 − E (exp(−τ 1 )) = 1

− E (log(W 1 )) ,

^Ka‡@>M

^bh–gudQ0š_…:\EDG›:fuj~`yh”e0g K@†N‡@Mš]`bKuQw_Qgu^bh”`o\Eet“

τ ^∗

^e0g

R ₊

h–^Lˆ0h”Ÿ0Qg…]\

P (τ 1 ≥ x)/ E (τ 1 ), x ≥ 0.

 µ µ ‹ÎŒ µ !µ °  °  °Rµ ‰ ° ~Ž ° ‹ €.-)% , },G

W 1

,^%n,%In ,

-}$&%I\ n-(.-)

E

| log(W 1 )|

W 1

< +∞,

.-)%I .-)% ,6 +* , %%I*+%#%L-n,,

n→+∞ lim E (R n )

n = E (G)

(D − 1) E (− log W 1 ) .

J@ 5

DC›0fvj~`bh–ekg Ko‡ Mœˆ0h”Ÿ0QP^C`bKujt`Pš_“Íe0m

n ≥ 1

^š

E (R n ) n = 1

n + E (G) E Ψ h

− log U _(n) ^D i

exp log

U _(n) ^D 1 nU _(n) ^D

! .

·„^

n

ˆke]QP^q`beh–g_«ugNh”`o\±`bKuQTŸ~jtmyhijt…N’–Q

nU _(n) ^D

dekg]ŸkQPmbˆ0QP^&h–g%wNh–^a`bmyh”…uf_`bh–ekg£`be jEm)jtguwNekO'Ÿ~jtmyh–jk…N’”Q

t D

™&Kuh–d)Kh–^qjR^bfNO et“

D

^hA˜hA˜w˜6Q§_s.ekgNQPg0`yh–jk’myjkguw_e0O Ÿ~jtmyh–jk…N’”Qc^œ™&h•`yKsvjtm)jtOžQ`yQm

1

š]“ÍfNmb`bKNQPmbOže0mbQ0š

n→+∞ lim E 1 nU _(n) ^D

!

= E (1/t D ) = 1

D − 1 .

=uekm

ε > 0

šº`yKNQmyQ Q§_hi^o`)^

K

^bfud)K%`bKujt`Pš“Íekm

x > K

^š

|Ψ(x) exp(−x) + 1/ E (log W 1 )| < ε

^šh”“

C

wNQgNek`bQP^œ`yKNQ^bfNsNmyQO!fuO et“

x → Ψ(x) exp(−x)

^ekg

R ₊

^šN`bKNQPg

E Ψ h

− log U _(n) ^D i

exp log

U _(n) ^D 1 nU _(n) ^D

!

− 1

(D − 1) E (− log W 1 )

≤ ε E 1 nU _(n) ^D

! +

C + 1

E (− log W 1 )

E {U _(n) ^D >exp(−K)}

1 nU _(n) ^D

!

+ 1

E (− log W 1 )

E 1 nU _(n) ^D

!

− 1 D − 1

.

^Ka‡@AM

=uekm

K 2 > 0

^š

lim sup

n→+∞

E

{U _(n) ^D > exp(−K)}

1 nU _(n) ^D

!

≤ lim sup

n→+∞

E

{nU _(n) ^D > K 2 exp(−K)}

1 nU _(n) ^D

!

= E

{t D >K 2 exp(−K)}

1 t D

,

jkguw`bKuh–^„`bQPmbO'ˆ0e:Qc^L`be

0

^{j0^}

K 2

`bQgvwN^r`yeEh–g_«ugNh”`o\k˜U„gNQ!dekgud’”fuwNQP^q`bKujt`q`bKuQ!myh”ˆ0K0`„Kujtguw£^ahiw_Q!ek“

prQ’ij~`bh–ekg Ko‡<AMœhi^rjtmy…Nh”`bm)jtmyh”’–\^aOjt’–’®jk^

n

ˆke]Qc^G`yeh”gN«ugNh”`o\k˜=JLKNQsNmyeks.e0^bh”`bh–ekgEhi^&sNmye~ŸkQcw˜

Tµ

 µ Ύ_

²

5

Q

^{# ,(#,< .- I6< %} 4 ,&%, ,5

-n%

D = 2

^!

G ≡ Q V i,Q = p i

,

1 ≤ i ≤ Q

^-n%$!eE 6% ,O,#%({&.-)%G% ,

$}$ %

log p i / log p 1

!

2 ≤ i ≤ Q

^{O !€.-)% $%,%*+%I%}

n→+∞ lim E (R n )

n = Q

P Q

i=1 −p i log p i

.

-)I5

5O!k,$\$&%I ^ I%<5

G

^{&L %*%I%I ,% ,$ % I6%En- .-)}

E (log G) < +∞

^!g,

` ≥ 2

,

1 ≤ i ≤ `

^!

V i,` = 1/`

^{! .-)%I}

n→+∞ lim E (R n )

n = E (G)

(D − 1) E (log G) .

q t xny{|)( tV|)y w v+t

‘`&hi^&jk^y^afuORQcw`yKuj~`L`bKuQxwNh–^a`bmyh”…uf_`bh–ekget“

W 1

hi^LQ§_sve0gNQg:`yh–jk’”’–\ jtmyh•`yKNOžQ`yh–dW™&h”`bKQ§_sve0gNQg:`yh–jk’5^bsujkg

λ > 0

˜RJLKNQž’–j™ ek“

− log(W 1 )/λ

h–^xjsNmyek…ujk…Nh–’”h”`o\±w_h–^a`bmyh–…Nf_`bh–ekg%ekg

N

^˜0=uekm

i ≥ 1

š®e0gNQRwNQ«ugNQc^

C i = B i /λ = − log(W i )/λ

^{˜ ‘}gx`bKNQGjkmbh”`bKuORQ`bhid=dj0^aQ0š`yKNQ=h–g:`bQˆ0Qm®Ÿ~jk’”fNQcwxmyjkguw_e0O¢™Ljt’–­„jk^y^be]dh–jt`bQcw

`ye

(C i )

sN’ij\]^G`bKuQ„­0Q\žmbe0’”Q0š0OTfud)K h–g `yKNQW^yjtOžQW™œj\žjk^G“Íekm

(S n )

h–g `yKNQWgNekg_½Üjtmyh”`bKNOžQ`yh–d„dj0^aQ0˜¼œ\

wNQgNek`bh–gNˆ

τ n = inf (

k ≥ 1 :

k

X

i=1

C i ≥ n )

,

DC›0fvj~`bh–ekg Ko‡,?MLdPjtg…vQmyQ™&myh•`b`bQPgj0^š]“Íe0m

x ≥ 0

^š

Ψ(x)e ^−λdx/λe = E





τ dx/λe

X

i=1

exp



λ





τ dx/λe −i

X

k=1

C k − dx/λe











 ,

™&KuQmyQ

dye = inf{n ∈ N : n > y}

^“Íekm

y ≥ 0

˜±¼œ\£fu^bh–gNˆ©JLKNQe0mbQPON‡ jtguw¥h”`y^!gNet`)j~`yh”e0gu^šº“Íekm

i ≥ 1

^{š®jk^}

n

ˆ0e]QP^q`beh”g_«vgNh•`o\0š`bKuQRŸ~jkmbhijt…N’–Q

C 1 + · · · + C τ n −i − n

dekg]ŸkQPmbˆ0QP^rh–g¥w_h–^a`bmyh–…Nf_`bh–ekg£`be

−(C ₁ ^∗ + C 2 + · · · + C i )

^{˜ N} h”`bKE`bKNQx^bjkOžQqOžQ`yKNe_w¬j0^Gh–g `yKNQ„guekg_½Üjtmyh•`yKNOžQ`yh–d„djk^bQkš:h”“®`bKNQWŸ~jtmyh–jk…N’–Q

| log(W 1 )|/W 1

h–^Lh–g:`bQPˆkm)jt…N’–Qkš]`yKNQg

x→+∞ lim Ψ(x)e ^−λdx/λe = 1 E (| log(W 1 )|)

λe ^−λ 1 − e ^−λ .

 µ µ ‹ÎŒ µ o‰¬‹ ² Œ µ Œ@ „°  µ ¶@ ‹P Ύ_Œµ ‹ "

\.-)% } {

W 1

&% #

n,#%k G -}$&%I .- % )%In ,Vn ,

λ > 0

^{! n-(.-)}

E

| log(W 1 )|

W 1

< +∞,

.-)%I ! ,

n

^*+%IO6 *%!€.-)%\%}&%, ,6%I%

E (R n ) n ∼ F

log n λ

-n,,! g-)%I%

F

^{".-)% n%,J}#, - )%I}

1

% r#%EI"!k,

x ≥ 0

^!

F(x) = E (G) E (| log(W 1 )|)

λ 1 − e ^−λ

Z +∞

0

exp

−λ

x − log y λ

y ^D−2

(D − 1)! e ^−y dy {x} = x − bxc

⁵

J@ 5 =uekm

n ≥ 1

^šuh•“

dxe = bxc + 1

^š

1

n E h Ψ

− log(U _(n) ^D ) i

= E

"

Ψ

− log U _(n) ^D

e ^{−λd− log(U (n) D /λ)e} e ^λ exp −λ

( − log(U _(n) ^D ) λ

)! 1 nU _(n) ^D

#

,