Tails in Generalized Jackson Networks with Subexponential Service Distributions

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Tails in Generalized Jackson Networks with Subexponential Service Distributions

François Baccelli, Serguei Foss, Marc Lelarge

To cite this version:

François Baccelli, Serguei Foss, Marc Lelarge. Tails in Generalized Jackson Networks with Subexpo-

nential Service Distributions. [Research Report] RR-5081, INRIA. 2004. �inria-00071502�

ISRN INRIA/RR--5081--FR+ENG

a p p o r t

d e r e c h e r c h e

THÈME 1

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Tails in Generalized Jackson Networks with Subexponential Service Distributions

François Baccelli — Serguei Foss — Marc Lelarge

N° 5081

January 2004

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f ^ln„a gª°qu_ƒcc[¬Cmƒœ —l}¢ c„ac

f(x)∼dg(x)

¦œžqk_

d >0^bJcUl}„aj f(x)/g(x)→d ^lnj x→ ∞©>$s³xM„CMc„Cqkœ xn„:ªpqk_ƒcc[¬Cmƒœ —l}¢ c„ac

f(x)∼dg(x)

¦œžqk_

d= 0^bJc[ln„aj f(x)/g(x)→0^lnj x→ ∞ª?qk_ƒœ¨jW¦œž¢ ¢p§&c$¦wuœ²qkqkc„

f(x) =o(g(x))^{© š} c¦œ ¢ ¢Œln¢ jkx m&j~c­qk_ƒc€„ƒxnqul}qkœ xn„

f(x) =O(g(x))quxŸbJcUl}„

lim supf(x)/g(x)<∞ ^l}„a lim inff(x)/g(x)>0^©

Dð„Jqu_ƒœ¨j=|al}|LcwUª

(x)pc[„ƒx}quc[jZlf¡´ma„a+quœžxM„Ij~ma‚_Jqu_al—q

(x)−−−−→^x→∞ 0©=^_ƒc¡´ma„a+quœžxM„

bl?s—lnwks(¡´wuxnb

|a¢ lMc­qkxJ|ƒ¢¨lnc OŒ¡´xnwcoƒlnbŸ|a¢žcMª

(x) +(x) =(x)^ª (x)(1 +(x)) =(x)ªƒcqun©(QpœžbJœ ¢ lnwk¢ snªp¦$c€¦œž¢ ¢

¦wuœžqkc

(x, y)^¡´xMw (x) +(y)^ªƒxMw (x)(y)^{ªƒcq‚}©}

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š

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K ^{„ƒxppcUj©}

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jkœ xn„ajl}wuc­lnjujkxŒœ l}qkcUJ¦œ²qu_0jtq‚l—quœžxM„ajln„a„ƒxnq¦œžqk_4maj~qkxMbŸc[wuj[©^_ƒœ j$bJc[ln„aj=qu_al—q$qk_ac

jPqk_0jkcwuŒœ c xM„0j~qul—quœžxM„

k^q‚l}¥Mc[j σ_j^(k)ma„ƒœ²q‚j$x}¡9quœžbJcnªŒ¦_ƒc[wkc

{σ^(k)_j }j≥1

œ¨jl|ƒwuc[pc a„ƒcU0jkc[¬Cmƒc„&cn© Dð„qu_ƒc(jklnbJc

¦l?snª?¦_ƒc[„qu_ƒœ¨jWjkcwuŒœ cSœ¨jxnbJ|ƒ¢ cquc[°ª}qk_ƒc$¢žcUl?Œœž„ƒfmaj~qkxMbJcwœ jjkc„CqWqkx€jtq‚l—qkœ xn„

ν_j^(k) F´xMwW¢ c[l?Mc[j:qk_ƒc

„acqt¦$xnwu¥œž¡

ν_j^(k)=K+ 1HZln„aœ¨jS|ƒmƒq$l—q$qk_acrc„&Ix}¡°qu_ƒc­¬Cmƒc[mƒcrxM„qk_aœ jj~qul—quœžxM„9ªM¦_acwuc

{ν^(k)_j }j≥1

œ¨jJl}¢¨j~x3l¯|ƒwuc[ƒc a„ƒcU jkc[¬Cmƒc[„acMªZl}¢ ¢ c[«qu_ƒc4wkxMmpqkœ „ƒ3jkc[¬Cmƒc[„acM© ^_ƒc³j~cU¬Mmac„ac[j

{σ_j^(k)}j≥1

l}„&

{ν_j^(k)}j≥1

ª:¦_ƒc[wkc

k wuln„ƒncUjfx—ncwfqu_ƒcIj~cq(x}¡jtq‚l—qkœ xn„&jª:lnwkcJln¢ž¢ c[3qu_ƒcIpwuœžŒœž„a4jkc[¬Cmƒc[„acUj€x}¡$qk_ƒc

„acq[© xŒƒc

0bJxppc¢¨jqu_ƒccoCqucwu„al}¢l}wuwkœ —l}¢°x}¡ZmajtquxnbJcw‚jœ „£qk_ac„ƒcqt¦$xnwu¥Eªpqk_ac„£qu_ƒcl}wuwuœž—l}¢EquœžbJc xn¡=qk_ƒc

jGqu_¯maj~qkxMbJcwrœ „¼qu_ƒc„acqt¦$xnwu¥Iquln¥ncUjX|ƒ¢¨lncl—q

σ₁⁽⁰⁾+· · ·+σ⁽⁰⁾_j l}„a¼œ²q$ytxnœ „ajqu_ƒcŸc„a£xn¡

qu_ƒc¬Cmƒcmacfxn¡j~qul}qkœ xn„

ν_j^{(0)©RiXc[„ac} σ_j^{(0) œ jqu_ƒc} jGqu_œ „Cqkc[wðl}wuwkœ —l}¢&qkœ bJcn©

‘“[“ A@”fºhg7BDCWºhg ”FE ¶ fg » ¶ @ ¶ g » ¶ g ˜ ¶£ºDG • º :”fg>= – g »î“ ¶ •H;cfG˜ ¶ ”f ¶ “

U

¢ž¢Lqu_ƒcfjkc[¬Cmƒc[„acUj

{ν^(k)}^l}„& {σ^(k0)}l}wucrbmpqkmaln¢ž¢ sœ „apc|Lc„&pc„Cq$¡´xnw

k, k⁰ w‚l}„anœ „ƒx—nc[wZqu_ƒcfjkcq xn¡Wj~qul}qkœ xn„aj[©

‘“[“ A@”f

ºhgJICWºhg

”FE

¶ fg » ¶ @ ¶ g » ¶ g ˜

¶£ºDG

“ ¶

•H;cfG˜

¶

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š

cI¦œ ¢ž¢$lnjuj~mabŸcŸqu_ƒc0jkcwuCœ¨cJquœžbJcUjl}wucJœž„&pc|Lc„aƒc„Cq¡´xMwpœWLc[wkc[„Mqj~qul}qkœ xn„ajl}„aœP©œP©°©œ „«c[lM‚_

j~qul}qkœ xn„4¦œ²qu_a„ƒœžqkc(bJc[ln„K

E(σ^(j)) = _µ¹(j) >0¡´xMwXl}¢ ¢

1≤j≤K^©

‘“[“ A@”fºhgMLCWºhg • º ”7fgN=

š c€lnjuj~mƒbJcrqk_al}qc[lM‚_Ix}¡:qk_ƒc€j~m&cUjkj~¡´mƒ¢LwkxMmpqkcUjSm&j~cUJqkxŸ§ƒmƒœ ¢ 

ν œ¨jSxM§pqulnœž„ac[I§Œsl T¯l}wu¥nx—‚_&l}œ „

xM„0qu_ƒcj~qul—quc(j~|alMc

{0,1, . . . , K, K+ 1}¦œžqk_4qkw‚l}„ajkœžqkœ xn„4bJl}qkwuœ²o

R=

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0 p0,1 . . . p0,K 0

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p1,1 p1,2 . . . p1,K p1,K+1

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p2,1 p2,2 . . . p2,K p2,K+1

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{ν_j^(k)} ^œž„ j~qul}qkœ xn„

k ^{l}wucœP©œP©9©¼œž„}

jªLœž„&pc|Lc„aƒc„Cqfxn¡ZcMcwusMqu_ƒœ „ƒIc¢¨jkcnªEl}„a£jkma‚_£qu_al—q­qk_ƒcwkxMmpqkœ „ƒƒc[œ¨jkœžxM„¼j~c[¢žcU+quj­jtq‚l—quœžxM„

i^¦œžqk_

|awkxM§al}§ƒœ ¢ œ²qts

P[ν^(k)=i] =pk,i.

^_acZ¡Ìlnqqu_al—qqk_ƒc$wuxnmpquc[j§ƒmƒœ ¢²q¦œ²qu_qu_ƒœ j T¼lnwk¥Mx—Cœ¨l}„€|ƒwuxpcUpmƒwucl}wucSjkmac[jujt¡´ma¢Mœ bJ|ƒ¢žœ c[jqu_al—qjtq‚l—qkc

K+ 1 œ¨j­qk_ƒcJxM„ƒ¢žs¼ln§ajkxnwu§ƒœž„aj~qul—qucxn¡Squ_ƒœ j(‚_&l}œ „3l}„&¯l}¢ ¢x}qu_ƒcw(j~qul—quc[j€l}wucqkw‚l}„ajkœžc[„Cq7O°¦$cqu_ƒc„

_&l?nc(qu_ƒcMcwus4jul}bJc"T¯l}wu¥nx—Œœ ln„wuxnmƒqkœ „ƒ0lnjujkmƒbJ|pqkœ xn„ajflnjrœž„ F´cop|Lxn„ƒc[„Cqkœ¨l}¢HXŠCln‚¥pj~xM„

cqt¦$xnwu¥pj©

T¼xnwucnc[„ƒcw‚l}¢ ¢žsMªa¦_ƒc[„3pc[„ƒx}quœž„aI§Œs

Ek

qk_ƒc¢ l?¦ xn¡=qk_ƒcŸ‚_al}œ „¼¦œžqk_3œž„ƒœžqkœ¨l}¢xM„apœžqkœ xn„

k^ªLln„a Vj

qu_ƒc(„Œmƒb§Lcwxn¡WŒœ¨j~œžqujxn¡:qu_ƒœ jrl}§ajkxnwu§ƒœ „ƒŸ‚_&l}œ „œ „£jtq‚l—qkc

j^{ªƒ¦$c(pc a„ac K}

E0[Vk] =πk, P0[Vk≥1] =pk, Ek[Vj] =πk,j, Pk[Vj ≥1] =xk,j. ^FtdH

š c(¦œ ¢ž¢9m&j~cfqk_ac€¡´xn¢ ¢žx—¦œ „ƒŸ„ƒx}q‚l—qkœ xn„K

bj= πj

µ^(j), bj,i= πj,i

µ⁽ⁱ⁾, Bj = max

i bj,i.

š œžqk_ qk_ƒœ¨j„ƒx}q‚l—qkœ xn„:ª¦c pc„ƒxnqkc3§Cs

b = maxiπi/µ⁽ⁱ⁾ = maxibi

©9cq λ⁻¹ = E[σ0] = a^©

^_awkxMmƒn_ƒxMmpq$qk_ƒœ¨j|aln|&c[w¦c€¦œ ¢ž¢:lMjkjkmƒbJcfqk_al}qFK

λb <1. ^FPH

š c(wuc[ln¢ž¢E_ƒc[wkc(jkxnbJc(pc &„ƒœ²quœžxM„aj

4¶ g f´”f

ºhg B :2#%24-,#%4,h5#34

F R+

: 8I#"

y >0 F(x+y)∼F(x) ^S x→ ∞.

š c­œ „Cqkwuxppmacfl|ƒwuxn|Lcwjkmƒ§ajkcqxn¡°qk_ƒc€¢¨lnjujx}¡9¢ xn„ƒqul}œ ¢ c[Ipœ¨j~qkwuœž§ƒmƒqkœ xn„aj[ªCqk_ƒcf¢¨lnjujSxn¡:j~mƒ§Lco

|Lxn„ac„Cqkœ¨l}¢9pœ¨j~qkwuœž§ƒmƒqkœ xn„ajpc[„ƒx}quc[4§Œs

S^K

4¶ g f´”fºhgMI :2#324-5,#%4,#%4

F R+

:' 2,.-0/21.#34

F^∗2(x)∼2F(x)

4¶ g f´”fºhgJL 1 #%6GS2, -,h5#%4

f [0,+∞) ^{: ,2G68I$(#)}

0/ α∈R f ∈ R(α)

x→∞lim f(tx)

f(x) =t^α t >0

4¶ g f´”f

ºhg 1 #%6 GS2,-5 ,#%4

h [0,+∞) ^{:G0!1h4 +68} h∈

R(−∞)

x→∞lim h(tx)

h(x) = 0 ^" t >1

axnwcoƒl}bJ|ƒ¢ cnª š c[œž§amƒ¢ž¢ExMwS¢ xnn„axnwubJln¢ƒw‚l}„apxMb'—lnwkœ¨l}§ƒ¢ c[jZ_al?ncq‚l}œ ¢Lƒœ j~qkwuœž§ampqkœ xn„ajSqk_&l—q$lnwkcrwuln|ƒœ ƒ¢žs

—lnwksŒœ „ƒa©

‘“[“ A@”f

ºhgNCWºhg

”FE

¶ “ :’

¶

@

ºhgW¶ g

”f̖ 3f´”

· ºDG

“ ¶

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¶

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š

c¦œ ¢ ¢pc[„ƒx}quc§Œs

S qk_ƒcŸ¢¨lnjujx}¡Sjkmƒ§Lcop|&xM„ƒc„Cquœ ln¢pœ¨jtquwkœ §ƒmpquœžxM„³¡´mƒ„aqkœ xn„aj­xn„³qu_ƒc|LxMjkœ²quœžMcwuc[l}¢

¢ œ „ƒcn©ƒxnw(l4pœ¨jtquwkœ §ƒmpquœžxM„¯¡´mƒ„&+qkœ xn„

F xM„¼qu_ƒcJ|&xCj~œžqkœ ncŸwkcUl}¢¢ œž„ƒcJ¦œžqk_ a„ƒœžqkc aw‚jtq€bJxnbJc„Cq

M =

R∞

0 F(u)du^ª F(u) = 1−F(u)pc[„ƒx}quc[j$qu_ƒc€qulnœž¢9xn¡

F ^l}„a F^s qu_ƒc(œ „MqucMwul}qkc[0qulnœž¢9ƒœ j~qkwuœž§ampqkœ xn„K

F^s(x) = 1−min

1, Z ∞

x

F(u)du def

= 1−F^s(x).

^_ƒcJlnjujkmƒbJ|pqkœ xn„aj­xM„ac[wk„ƒœ „ƒ0jkcwuŒœ c(qkœ bJc[j€l}wuc(qu_ƒc¡´xM¢ž¢ x—¦œ „ƒ K$qk_acwuccopœ¨jtq‚j­lpœ¨jtquwkœ §ƒmpquœžxM„

¡´ma„a+quœžxM„

F ^xn„ R+

jkma‚_4qk_&l—qFK

dn©

F œ¨jjkmƒ§&cop|&xM„ƒc„Cqkœ¨l}¢Pªƒ¦œžqk_a„ƒœžqkc &wuj~qbŸxMbJc„Cq

M^©

p©^_ƒc(œ „MqucMwul}qkc[pœ¨jtquwkœ §ƒmpquœžxM„

F^s œ¨jXj~ma§&coŒ|Lxn„ac„Cqkœ¨l}¢P©

;ƒ©^_ƒc€¡´xn¢ ¢ x—¦œž„ƒŸc[¬Cmƒœ —l}¢ c„ac€_ƒxn¢¨ƒj¦_ƒc[„

xquc„aajqkx

∞^K

P(σ^(k)₁ > x)∼c^(k)F(x),

¡´xnwXln¢ž¢

k= 1, . . . , K ^¦œžqk_ PK

k=1c^(k)=c >0^©

5 JN3 9 ,

:cq f^j(σ, n) §Lcqu_ƒcI¡´xM¢ž¢ x—¦œž„a£|ƒœžcUc P¦œ¨j~c¢ œ „ƒc[lnw¡´mƒ„aqkœ xn„«xn¡

(σ, n)^ª=¦_ƒcwuc σ ^l}„a n ^{l}wuc„ƒxM„}

„acMl}qkœ ncfwkcUl}¢°„Œmƒb§&c[wujFK

f^j(σ, n) = 11{σ>na}{σ−na+npjBj}+ 11{σ≤na}max

k

pjbj,k

σ a+

bk

a −1

(na−σ) +

F4; H

ln„aJ¡´xMwl}¢ ¢L|&xCj~œžqkœ ncXwuc[ln¢a„Œmƒb§&c[wuj

xªpl}„a0l}¢ ¢

j = 1, . . . , K^ªn¢ cq ∆^j(x)§&crqk_ƒc­¡´xn¢ ¢ x—¦œž„ƒƒxnbl}œ „K

∆^j(x) = {(σ, t)∈R²+, f^j(σ, t)> x}. ^{F̎ H}

E

¶Œº

• ¶ B R 24 CE2# -5 5h50L #%$&B$(#)@2,.-0/21.#34552674

#%G :2#%24-,#%4. 5#3: _S2, `1h#%4 D 5#

Z ^{# #3 2# #%4 G5/ #P5 #}

5, 2#P 5 226

5'55#34 G R#34

Z ^G). x→ ∞ P[Z > x]∼

K

X

j=1

πj

Z Z

{(σ,t)∈∆^j(x)}

Ph

σ^(j)∈dσi

dt. ^FP‡H

):9,.#34 G -9$(2#%#_$(#)#) .2#P.#3

{α^j_i, β_i^j, γ_i^j}0≤i≤l

#).# $(c-9,# 0

G !-05$ S $

P[Z > x] ∼

K

X

j=1

πj







l

X

i=0

X

{α^j_ix≤n<α^j_i+1x}

P

"

σ^(j)> x β_i^j +nγ_i^j

#







, ^{F H}

$(#)

δ_i^j= 1/β^j_i +α^j_iγ_i^{j h} d^(j)=πjc^(j) P[Z > x]∼

K

X

j=1

d^(j) ( _l

X

i=0

1 γ_i^j

hF^s(δ_i^jx)−F^s(δ_i+1^j x)i )

. ^FH

F^s∈ R(−α)^&$(#) α >0^R$& 0M5$(2#P 5,.#%4

S

P[Z > x]

F^s(x) →

K

X

j=1

d^(j) ( _l

X

i=0

1 γ_i^j

h(δ_i^j)^−α−(δ_i+1^j )^−αi )

.

F^s∈ R(−∞)^&#). R$&9).6

P[Z > x]

F^s(x) →

K

X

j=1

d^(j) a−pjBj

.

*/" : N,0 % 34,819.D,5% % ,-0 25 " >,-

^_acjklnbŸ|a¢žc(|al}qk_£xM„aj~qkwuma+quœžxM„4¦$c(œ „Mquwkxppm&c(_ƒc[wkc(œ¨jqk_&l—qXx}¡ ?ŽA]©$^_ƒc(blnœž„³œ „Mqucwuc[j~qXx}¡=j~m&‚_³l

xn„aj~qkwumaqkœ xn„¼œ¨j­qk_al}q€jkxnbJcbJxM„ƒx}quœž„ƒœ¨œžqts£|ƒwuxn|Lcwkqkœ c[jfl}wuc|ƒwuc[jkcwuncU¼lnj­cop|ƒ¢¨l}œ „ƒc[¼œž„L?ŽAP©^_ƒcUj~c

bJxM„ƒx}quxn„ƒœ¨œžqts|ƒwuxn|Lcwkqkœ c[jlnjjk_ƒx—¦„4œž„ ?ӇSAlnwkc€wumaœ¨l}¢E¡´xMwxnmƒwXlMj~sŒbJ|pqux}qkœ¨€ln¢ mƒ¢ l}qkœ xn„9©

U Mc„ƒc[wuln¢žœ ¤cU0ŠCln‚¥pjkxn„I„ƒcqt¦xMwk¥¦œ ¢ ¢9§&cƒc a„ƒcU4§Cs

JN=n

{σ^(k)_j }j≥1, {ν_j^(k)}j≥0, n^(k), 0≤k≤Ko ,

¦_acwuc

N = (n⁽⁰⁾, n⁽¹⁾, . . . , n^(K)) pcUjkwkœ §&cUjqk_ƒc£œ „ƒœžqkœ¨l}¢XxM„apœžqkœ xn„9©®^_ƒc³œž„Cqucwu|ƒwkcqul}qkœ xn„ œ¨jlnj

¡´xM¢ž¢ x—¦XjFKJ¡´xnw

i 6= 0ªSl}qquœžbJc

t = 0^{ªZœž„ „ƒxppc} iªqu_ƒcwuc³l}wuc

n⁽ⁱ⁾ maj~qkxnbJc[wuj¦œ²qu_ jkcwuŒœ cIqkœ bJc[j

σ₁⁽ⁱ⁾, . . . , σ⁽ⁱ⁾_n(i)

F´œž¡ln|ƒ|ƒwuxn|ƒwuœ l}qkcMª

σ⁽ⁱ⁾₁ bl?s§&c(œ „Cqkc[wk|ƒwucquc[4lnjlŸwuc[jkœ pm&l}¢9jkcwuCœ¨cfquœžbJcH©

^_acfœ „Cqkc[wk|ƒwucq‚l—quœžxM„xn¡

n⁽⁰⁾ œ¨jlnj$¡´xM¢ž¢ x—¦XjFK

J

œ²¡ n⁽⁰⁾= 0ªŒqu_ƒcwuc(œ j„axŸcoŒqkcwu„aln¢9l}wuwkœ —l}¢P©

J

œ²¡ ∞> n⁽⁰⁾≥1ªŒqk_ƒc[„4¡´xnwl}¢ ¢

1≤j≤n⁽⁰⁾ªŒqu_ƒcl}wuwkœ —l}¢&qkœ bJc(x}¡:qu_ƒc

jGqu_4maj~qkxnbJc[w$œ „4qk_ƒc

„ƒcqt¦$xnwu¥Iq‚l}¥Mc[j|ƒ¢¨lncl—q

σ₁⁽⁰⁾+· · ·+σ⁽⁰⁾_j ^© x}quc€qk_al}qrœž„³qu_ƒœ¨jrlMj~cMªpqk_acwucbl?s0§Lcl a„ƒœžqkc

„Œmƒb§Lcwfx}¡SmajtquxnbJcw‚jX|&lnjuj~œ „ƒqu_ƒwkxMmƒn_3lnœ nc[„¯jtq‚l—quœžxM„¯j~xIqk_al}qrqk_ac„ƒcqt¦xMwk¥4œ¨j­lM+qkm&l}¢ ¢žs

¦c[¢ž¢Zƒc a„ƒcU¯xM„acJl a„ƒœžqkcj~cU¬Cmƒc„acJx}¡Swuxnmpquœž„a4pcUœ¨j~œ xn„&jfln„a¯jkcwuŒœ cquœžbJcUj€œ jfnœ nc[„¯xM„

qk_ƒœ¨jXj~qul}qkœ xn„9©

J

œ²¡

n⁽⁰⁾ = ∞ª9qu_ƒc„¦_ƒc[„ qul}¥Œœ „ƒ4¡´xnwœ „aj~quln„acŸqk_acIj~cU¬Mmac„ac

{σ_j⁽⁰⁾}j≥1

œG©œP©°© ª9qu_ƒc0lnwkwuœž—ln¢

|ƒwuxŒc[jujœ jlŸwuc„ƒc[¦$ln¢°|ƒwuxŒc[juj$cqun©

b ¶ •• º ” ¶DC b ¶ • gW¶ ” ¹º •

{xM„ajkœ pc[wŸl¯wkxMmpqkc

p = (p1, . . . , pφ) ^¦œ²qu_ 1 ≤ pi ≤ K ^¡´xnw i = 2. . . φ−1^©QŒma‚_ l3wuxnmpquc0œ¨j 2,.2,:œž¡

p1 = 0 ^ln„a pφ = K+ 1^© ^x3jkma‚_®l¯wuxnmpqucnª=¦$c4lMjkjkxpœ¨l—qucl¼wuxnmƒqkœ „ƒ jkc[¬Cmƒc[„ac

ν= (ν⁽⁰⁾, . . . , ν^(K))lnj$¡´xn¢ ¢ x—¦Xj F

⊕bJc[ln„aj_ƒc[wkc(xM„al}qkc[„al—quœžxM„4ln„a

∅qu_ƒc(cbJ|pqts4j~cU¬Mmac„acH K

Procedure(p) :

1 for k= 0. . . K do ν^(k):=∅;

φ^(k):= 0;

od

2 for i= 1. . . φ−1 do ν^(pi):=ν^(pi)⊕pi+1; φ^(pi):=φ^(pi)+ 1;

od

x}qucZqk_&l—q

φ^(j)œ j:qk_ac„Œmƒb§&c[wxn¡ƒCœ¨jkœ²q‚j9quxr„ƒxppc

jœž„jkma‚_lrwkxMmpqkcM© Dð„|&l}wkqkœ¨mƒ¢¨l}wœž„xMmƒwjtquxŒ‚_&lnj~qkœ¨

¡´w‚l}bJc[¦xMwk¥EªC¦c(_al?Mc

E[φ^(j)] =πj

©

U jkœ bŸ|a¢žc€zZmƒ¢ cwr„ƒcqt¦$xnwu¥œ jrlMc„ƒc[wuln¢žœ ¤cU0ŠCln‚¥pjkxn„I„ƒcqt¦xMwk¥

E={σ, ν, N},

¦œžqk_

N = (1,0, . . . ,0) = 1ªj~ma‚_3qk_&l—q€qk_ƒcwuxnmƒqkœ „ƒ4jkc[¬Cmƒc[„ac

ν ={ν_i^(k)}^φ_i=1^(k) œ¨j€nc„acw‚l—qkcU¼§Œs l¯j~ma[cUjkj~¡´mƒ¢Zwuxnmpquc0l}„&«j~ma‚_qu_al—q

σ ={σ_i^(k)}^φ_i=1^(k) œ¨jl£jkc[¬Cmƒc[„ac0x}¡wkcUl}¢P—ln¢žmƒcU „ƒxn„ ]„ƒc[Ml—quœžMc

„Œmƒb§&c[wuj[ªpwuc|ƒwuc[jkc„Cqkœ „ƒJj~c[wkŒœ¨c­qkœ bJc[j[©

{xM„aj~œ¨pc[w:lj~cU¬Cmƒc„acx}¡ƒj~œ bJ|ƒ¢ c=zZmƒ¢ cw:„acqt¦$xnwu¥Œj[ªjul?s

{E(n)}⁰_n=−∞^{ª[¦_ƒc[wkc} E(n) ={σ(n), ν(n),1}^©

axnw m≤ n≤0^{ª:¦$cJƒc a„ƒc} σ[m,n]

l}„&

ν[m,n]

qux³§&cJqu_ƒcIxM„al}qkc„&l—qkœ xn„3x}¡

{σ(k)}m≤k≤n

l}„&

{ν(k)}m≤k≤n

l}„a0qk_ac„£pc &„ƒc€qk_ƒc 1.S5€nc[„ƒcw‚l}¢ œ ¤c[ŠMlM‚¥pj~xM„I„ƒcqt¦$xnwu¥ K

JN_[m,n]={σ[m,n], ν[m,n], N[m,n]}, ^¦œžqk_ N[m,n]= (m−n+ 1,0, . . . ,0).

– f – S»W–p” ¶ •

U jI|ƒwux—ncU®œž„N?ŽA]ª¡´xnw4l}¢ ¢r|LxMjuj~œ §ƒ¢ c¼—l}¢ mƒc[jIx}¡

ν(p) ^ln„a σ(p) ^{œž„ qk_ƒc} j~œ bJ|ƒ¢žc¯zZmƒ¢ cw4„ƒcqt¦$xnwu¥pjª

¡´xMw4ln¢ž¢€œ „CqkcMcw‚j

m ≤ n^ªXqk_ac xnbJ|&xCj~cU „ƒcqt¦xMwk¥

JN_[m,n] j~qul?spjcbJ|pqts›¡´xMwkc[ncw4l—¡Ãqucw£j~xMbJc

&„ƒœ²quc¯quœžbJcM© š cpc„ax}qkc §Œs

X[m,n]

qk_ƒc3qkœ bJc¯qux c[bJ|pqts

JN_[m,n] ¡´xnwucMcw4l}„a §Cs

Z[m,n] =

X[m,n] −Pm−n+1

i=1 σ_[m,n],i⁽⁰⁾ qk_ac¯lnjuj~xpœ l}qkc[ bl—opœ bJln¢rƒl}qkcwU© ^_ac¯j~cU¬Cmƒc„ac

Z[−n,0]

œ¨j0ln„®œ „

wkcUlnjkœž„aJjkc[¬Cmƒc[„acM© š cŸpc a„ac(qk_ƒcbl—opœžbln¢ƒl}qkcwrx}¡qk_ƒcnc[„ƒcw‚l}¢ œž¤[c[³ŠCln‚¥pj~xM„„acqt¦$xnwu¥

JN= {σ, ν, N} ^¦_ƒcwuc σ ^ln„a ν lnwkc4qu_ƒc£œž„ a„ƒœžqkc¼xM„al}qkc[„al—quœžxM„ xn¡­qu_ƒc

{σ(n)}n

ln„a

{ν(n)}n

l}„&

N = (+∞,0, . . . ,0)^ªa§Cs

Z= lim

n→∞Z[−n,0]. ^{F̉ H}

^_acxnwucb"d7;Jxn¡?ŽAln|ƒ|ƒ¢ œžcUjrjkxqk_al}q­œž¡

λb <1^qu_ƒc„ Z <∞ ^lƒ©j©OExn„ŒMcw‚j~c[¢žsMªƒœ²¡

λb >1^ª Z =∞

la©j[©

^xIl}¢ ¢:Mc„ƒc[wuln¢žœ ¤cUŠCln‚¥pjkxn„4„ƒcqt¦$xnwu¥

JN_[m,n]ªL¦cln¢ jkxIlnjujkxŒœ l}qkcfqk_acnc[„ƒcw‚l}¢ œž¤[c[4ŠMlM‚¥pj~xM„„ƒcq

¦$xnwu¥

JN_[m,n](Q)œ „¦_ƒœ¨‚_ƒwkœ Œœž„ƒJjkc[¬Cmƒc„&c[jlnwkc­qk_ƒc€jul}bJcflMj$œž„qk_ƒc€xMwkœ nœ „al}¢L„ƒcqt¦xMwk¥Jcoƒc|pq

¡´xMwZqu_ƒcrjkc[¬Cmƒc„&c

{σ⁽⁰⁾_j }qk_al}qœ¨jZ„ƒx—¦

σ_j⁽⁰⁾= 0¡´xnw$l}¢ ¢

j©RQŒœ bŸœ ¢¨l}wu¢žs¦crƒc a„ƒc

Z[m,n](Q)qu_ƒcquœžbJc quxJcbJ|pqtsIqu_ƒc(nc[„ƒcw‚l}¢ œ ¤c[ŠMlM‚¥pj~xM„I„ƒcqt¦$xnwu¥

JN_[m,n](Q)^©

:cq

Y_i^(k)=

φ^(k)(i)

X

j=1

σ^(k)_j (i) ^{F H}

§Lc€qk_ac€qkx}q‚l}¢°¢ xMlM0§ƒwuxnman_Cq$§Œs F´coŒqucwu„al}¢H$maj~qkxMbŸc[w

iqkxj~qul}qkœ xn„

k^© x}quc€qk_al}q

Zi = Z[i,i]=Y_i⁽¹⁾+· · ·+Y_i^(K), ∀i Z[n,0](Q) ≥ max

j=1,...,K 0

X

i=n

Y_i^(j), ∀n≤0.

:cbJblŸŽŸxn¡ ?;SAln¢ jkxŸœžbJ|ƒ¢ œžcUjqk_&l—q

n→∞lim

Z[−n,0](Q)

n =b ^lƒ©j[©. ^{FtdUˆ H}

.10 5. &%) 5,5 % 3

r„apc[w

U

juj~mƒbJ|pquœžxM„d ;ƒªCqk_ƒc(|awkxM|&c[w~quœžcUj Z]\ ‘!^ l}„aEZ ‘‘_^ xn¡ ?ŽA]ªƒ¦_ƒœ¨‚_wuc[lM

J

Z]\ ‘!^ qk_ƒc(jkc[¬Cmƒc[„acfx}¡j~œ bJ|ƒ¢žcfzZmƒ¢ cw„acqt¦$xnwu¥Œj

{E(n)}^−∞_n=0 xM„ajkœ j~quj$x}¡œG©œG©°©=w‚l}„aƒxnb)—l}wuœ!

l}§ƒ¢ c[j[©

J Z‘‘_^ qk_ƒc$w‚l}„apxMb ?lnwkœ¨l}§a¢žcUj

{Y_i^(k)}l}wucSœ „apc[|&c[„apc„Cq=x}¡aqk_ƒc$œž„Cqucw ]lnwkwuœž—ln¢}qkœ bJc[j[ªnl}„&j~m&‚_

qk_al}qqu_ƒc4jkc[¬Cmƒc[„ac0x}¡wuln„apxMb Mc[+quxnw‚j

(Y_i⁽¹⁾, . . . , Y_i^(K)) ^{œ jœG©œG©}°©LF´nc[„ƒcw‚l}¢$pc|Lc„&pc„ac[j

§&cqt¦c[c„4qk_acxnbJ|Lxn„ƒc[„Cqujx}¡qk_ƒc(Mc[qkxnw

(Y_i⁽¹⁾, . . . , Y_i^(K)) lnwkc(l}¢ ¢ x—¦cU.H+ª lnwkcf§Lx}qk_£jul—quœ j ac[9©

r„apcw U juj~mƒbJ|pquœžxM„dMª—qk_ac?lnwkœ¨l}§a¢žc

Z lnjujkxŒœ l}qkcU(qkx

JN={σ, ν, N}wuc|ƒwuc[jkc„Cq‚j:qu_ƒcXj~qul—quœžxM„

lnwks0bJl}opœžbl}¢ƒl—qucwXxn¡qu_ƒcnc[„ƒcw‚l}¢ œ ¤c[4ŠCln‚¥pj~xM„0„ƒcqt¦xMwk¥Eªp„alnbJc¢ sIqk_ac(qkœ bJc(qk_al}qrœžqr¦xMmƒ¢¨0q‚l}¥Mc

œ „¼j~qkc[lMpsj~qul}qkc(qux0¢ c[lnwqk_ac¦$xnwu¢ž¥MxMln0x}¡Zl}¢ ¢maj~qkxnbJc[wuj|awkcUj~c[„MqXœ „³qk_acjksŒj~qkc[b ¦_ƒc[„¼jtquxn|ƒ|aœž„ƒ

¡´mƒqkmƒwucl}wuwkœ —l}¢¨j©

r„apc[w U juj~mƒbJ|pquœžxM„³ŽaªŒqk_aclnjuj~mƒbJ|pquœžxM„aj Za b ^ l}„aCZ]d ^ xn¡ ?ŽA:l}wuc(jkl}qkœ¨j&c[K

J

Za b ^ axnwXln¢ž¢

k= 1, . . . , K

P(Y₁^(k)> x)∼π^kP(σ^(k)> x)∼d^(k)F(x),

¦œ²qu_

d^(k)=c^(k)πk

l}„&Iqu_ƒc„

d^def= P

kd^(k)>0^©

J

Z]d ^

P(

K

X

k=1

Y₁^(k)> x)∼P( max

1≤k≤KY₁^(k)> x)∼

K

X

k=1

P(Y₁^(k)> x)∼dF(x).

Qpcc QŒc[qkœ xn„ajXŽa©Ža©Óln„a p©xn¡ ?ӇA]©

r„apc[w U jujkmƒbJ|pqkœ xn„«Ž&ª:qu_ƒcwucIcopœ¨jtq‚jl³„ƒxM„ ðpc[wkcUlnjkœž„ƒ³œ „CqkcMcw P—ln¢žmƒcU¼¡´ma„a+quœžxM„

Nx → ∞ ^l}„a

jkma‚_qk_al}q[ªp¡´xMwXl}¢ ¢ a„ƒœžqkc(wuc[ln¢°„Cmab§Lcw‚j

b^ª

Nx

X

n=0

F(x+nb) =o F^s(x)

, x→ ∞ ^FtdMdH

FGj~c[c QŒc[qkœ xn„³Ža© dn©Óxn¡ ?ӇAH+©

1IS# 2#€#W

*5 A5N, 0% "

U jZl}¢ wuc[lnƒsbJc„Cqkœ xn„ƒcU°ªnxM„ƒcx}¡Equ_ƒcqkxŒxM¢ j=¦$c¦œ ¢ž¢ƒm&j~cX¦œžqk_aœž„Jqu_ƒœ jSj~cq~qkœ „ƒœ¨jqu_ƒc"Kjkœ „ƒn¢ c§ƒœž(c[nc[„Mq

qu_ƒcxMwkc[bMK¡´xMwnc[„ƒcw‚l}¢ œ ¤c[ ŠCln‚¥pj~xM„¯„ƒcqt¦xMwk¥pj[©T£xMwkcJ|ƒwuc[œ jkc¢ snªW^_ƒcxMwkc[bJj 4ln„a«‰³x}¡'?ӇAMœžMc

qu_ƒc€¡´xn¢ ¢ x—¦œž„ƒŸwuc[jkmƒ¢²qHK

• º @ ¶ •?” · B 5#

Z ^{-!#) _2##%4h 7/G` # MV#) 5h57E 05 #%$&}

&hG

x j= 1, . . . , r, ^5# {K_n,x^j } ^{-0 55,.0} 56#4'2,.0)!#).#

I

n ^h j ^#).I56S5.# K_n,x^{j _#).Gh 624-5} Y_−n^(j) =Pφ^(j)(−n)

k=1 σ^(j)_k (−n)

h P1.5h5.#

"

j P(K_n,x^j )→1 ^{,. G} n≥Nx

S x→ ∞

5,.5h

n→0^{($& 5h#} xn=x+n(a−b+n)^{G) 5XS} x→ ∞

P[Z > x]∼

K

X

j=1

X

n≥Nx

P[Z > x, Y_−n^(j)> xn, K_n,x^j ],

h

P[Z > x] =O(F^s(x)).

^_ƒœ¨j|ƒwuxn|Lcwkqts¢ c[lnaj$qkxJqk_ac€¡´xn¢ ¢žx—¦œ „ƒJl}„abJxnwucf_&l}„aps0wkcUj~mƒ¢žqFK

º • º ̖p• ·B V. 5,. '6#4

{K_n^j}^{2,.0)#) # '} j P(K_n^j)→1 ^,."

n≥Nx

S x→ ∞ ⁷ zx→ ∞ zx=o(x)^{(,.)_#) #} F^s(x±zx)∼F^s(x)^{RhI #} G(x) =

K

X

j=1

X

n≥Nx

Ph

Z_[−n,0] > x, K_n^j, Y_−n^(j)> xn, φ^(j)(−n)≤Li .

). &$&9).6

(1 +(x))G(x)≤P[Z > x]≤(1 +(x))G(x−zx) +(L, x)F^s(x). ^Ftd?H

G ^{: 8I#} &$&9).6 S

x→ ∞

P[Z > x]∼G(x).

^_acf|awkxŒx}¡Wœ j¡´xnwu¦$lnwupcUquxqu_ƒcl}|a|&c[„apœžoE©

95 ) 5" 5 ,

š c(_al?Mcrqux a„a³j~cU¬Cmƒc„ac[j$x}¡cMc„Cquj

{K_n^j}l}¢ ¢žx—¦œ „ƒŸxn„ƒcfqkxI[l}¢¨mƒ¢¨l—qucfqk_ƒc(jkmƒb

X

n≥Nx

Ph

Z[−n,0]> x, K_n^j, Y_−n^(j)> xn, φ^(j)(−n)≤Li F~d7; H

¦_acwuc€lMjl}§Lx—ncMª

xn =x+n(a−b+n)^©

^_acc[nc„Cq‚j:œ „¬CmƒcUjtquœžxM„¦œ ¢ž¢Œ§Lc§alMj~cUxM„(qk_ƒc|ƒœ c[c ]¦œ jkcS¢ œ „ƒc[lnw:¡´mƒ„&+qkœ xn„&j

f^j(σ, n)^pc &„ƒc[œž„_F4; H+©

:cqmaj(ƒc[juwuœž§LcJqk_ƒcœ „Cqkmƒœžqkœ ncwuc[lMj~xM„¼¡´xMw(œž„Cqkwuxppmaœž„ƒ4qu_ƒœ j¡´mƒ„aqkœ xn„9© U jkjkmƒbJcŸqk_ƒc0§ƒœž£jkcwuŒœ c

quœžbJcœ¨j(c[¬Cmaln¢qkx

σ l}„&¯q‚l}¥Mc[jf|ƒ¢¨lncxM„«jtq‚l—qkœ xn„

j ^ln„a ¦œ²qu_ƒœ „3qu_ƒc0jkcqxn¡jkcwuŒœ cquœžbJc[jx}¡$qk_ƒc jkœ bŸ|a¢žczZma¢žc[w„ƒcqt¦xMwk¥

E(−n)^© 9cqm&j(¢žxŒxM¥3l—q(qu_ƒcIbl—opœ bl}¢Zƒl}qkcw

Z[−n,0]

œ „ qk_ƒcI±amƒœ¨«jk[l}¢ c

jkmƒMnc[j~qkcUI§Œsqu_ƒclƒ©j©=¢ œ bŸœžqXx}¡XFtdUˆ H K

J

œ²¡

σ > naªŒqk_ƒc[„0qu_ƒc€„Cmab§Lcwx}¡maj~qkxnbJc[wuj$§ƒ¢ xp‚¥nc[Iœž„³jtq‚l—quœžxM„

j ^l}qquœžbJc σ œ¨j$x}¡:qu_ƒcfxnw‚pcw x}¡ npj

ªp¦_ƒc[wkcUlnj$qu_ƒc(„Œmƒb§Lcwxn¡majtquxnbJcw‚j$œž„4qu_ƒc(x}qu_ƒcwXj~qul}qkœ xn„ajœ¨jXjkbJln¢ž¢P©RQŒxaª&lnxnw‚pœ „ƒ

qkxMF~d[ˆ HªŒqk_ƒcfquœžbJc€qkxc[bJ|pqtsqk_ac(„ƒcqt¦$xnwu¥J¡´wkxMb)qkœ bJc

σ xM„4jk_ƒxnma¢ §&c(xn¡qu_ƒc€xnw‚pcw

npjBj

O

_ƒc„&cnª°œ „3qk_ƒœ¨j€lnjkcnªLqu_ƒcJbJl}opœžbl}¢=ƒl}qkc[w­œ „3¬CmƒcUjtquœžxM„3jk_ƒxnmƒ¢¨¼§LcJx}¡Zqu_ƒcŸxnw‚pcwfx}¡

f^j(σ, n)

œž„&pccU O

J

œ²¡ σ < naª=qk_ƒc[„®l}qJqkœ bŸc

σªqk_ƒc4„Œmƒb§Lcwxn¡­maj~qkxMbJcw‚j§ƒ¢ xp‚¥ncU œ „ j~qul—quœžxM„

j œ¨jŸx}¡­qk_ƒc xnw‚pcwfx}¡

pjσ a

ª°l}„a¼qk_ƒcŸx}qu_ƒcw€j~qul—quœžxM„aj­_al?nc(¡´c[¦…m&jtquxnbJcw‚j5OL¡´wuxnb quœžbJc

σ qkxqk_ƒcqkœ bJcxn¡

qk_ƒc(¢¨lnj~ql}wuwkœ —l}¢ F´¦_ƒœ¨‚_œ¨jx}¡qk_ƒc€xMwupc[wx}¡

naH+ªƒj~qul}qkœ xn„

k_alMjqkxj~c[wkMcfln|ƒ|ƒwux?opœžbl—quc¢ sŸqk_ƒc

¢žxCln

pjσ abj,k

nc„acw‚l—qkcUJ§Œsqu_ƒc[jkcr§a¢žxp‚¥ncUmaj~qkxMbJcw‚jZ|ƒ¢ majqk_ƒc­¢žxCln

(na−σ)^b_a^k nc„acw‚l—qkcU

§Œsqu_ƒcŸcoŒqkc[wk„aln¢lnwkwuœž—ln¢ jrxn„£qu_ƒcquœžbJcœž„Cqkc[wk—ln¢:¡´wuxnb

σ quxIqk_ac¢¨lnj~qfl}wuwuœž—l}¢P©­„¼qk_ƒœ¨jXquœžbJc

œž„Cqucwu?ln¢GªWqu_ƒcjkcwuŒœ cIl}|&lnœžqts œ jx}¡qk_ƒcxnw‚pcwx}¡

(na−σ)^© iXc„&cIqu_ƒc0bl}oŒœ bl}¢$ƒl—qucw j~_axnmƒ¢¨³l}Mlnœž„§&c(xn¡:qu_ƒc(xnw‚pcwxn¡

f^j(σ, n)^©

v=lnw~quœ mƒ¢ lnwk¢ s4¡´xnw­qk_ƒcJ¢¨lnj~q€[lnjkcnª9lbJxnwucqk_axnmƒwuxnmƒM_¼mƒ„aƒcw‚jtq‚l}„apœ „ƒ4x}¡Zqu_ƒcŸ±amƒœ¨¯¢ œ bŸœžq(œ j€¢žcUl}wu¢žs

„ac[cUjkjul}wusn©^_ƒœ¨j¦œž¢ ¢9§Lc(|ƒwux—Cœ¨pcU0§Œsqu_ƒc(wkcUj~ma¢²q‚jœž„C? d[ˆSA]© š c(„ax—¦ wucqkmawk„qkxJwuœžMxnwU©

axnwXln¢ž¢9jkœžbJ|ƒ¢ c(zZmƒ¢ cw„acqt¦$xnwu¥Œj

E= (σ, ν,1)^ªƒ¢žcq Y^(j)(E) =Pφ^(j) u=1σu^(j)

©

{xM„ajkœ pc[w­lMc„ƒc[wuln¢žœ ¤cU£ŠMlM‚¥pj~xM„4„acqt¦$xnwu¥§ƒmaœž¢žqr¡´wuxnb qk_acœP©œP©9©fjkc[¬Cmƒc[„acx}¡SjkœžbJ|ƒ¢ czZma¢žc[w­„ƒcq

¦$xnwu¥pj

{E(k)}©$^:xIln¢ž¢:jkœ bŸ|a¢žc(zZma¢žc[wr„ƒcqt¦xMwk¥pj

E l}„a³l}¢ ¢9|LxMjkœ²quœžMc€œž„Cqkc[nc[wuj

nª&¦clMjkjkxpœ¨l—qkcfqu_ƒc

„acqt¦$xnwu¥

JNⁿ(E)¦œ²qu_œ „ƒ|ƒmpq

{E(k)}˜ ^∞_k=−n^ªp¦_acwuc E(k) =˜ E(k)^{¡´xnwl}¢ ¢} k >−n ^ln„a E(−n) =˜ E©^_&l—qœ¨j[ªZœ²¡­¦c4pc„ƒxnqkc§Cs

σ^{(k),n ln„a} ν^(k),n qu_ƒc4xn„a[l—quc„al}qkœ xn„aj

({σ^(k)(E)},{σ^(k)(−n+ 1)}, . . . ,{σ^(k)(0)}, . . .)^l}„a ({ν^(k)(E)},{ν^(k)(−n+ 1)}, . . . ,{ν^(k)(0)}, . . .)wkcUj~|Lc[qkœ nc¢ snªMqk_ac„

JNⁿ(E) = {σⁿ(E), νⁿ(E),0, Nⁿ}, ^¦œ²qu_ Nⁿ = (n,0, . . . ,0).

^_acIbl—opœžbln¢ƒl}qkcwx}¡xnw‚pc[w

[−n,0] œ „qk_aœ j„ƒc¦$xnwu¥¯¦œ ¢ ¢§Lc0pc[„ƒx}quc[«§Œs

Z˜ⁿ(E)©r¡Xxnmƒw‚j~c

Z˜ⁿ(E(n)) =Z[−n,0]

©VT£xnwucJnc„acw‚l}¢ ¢žsMª9¦c¦œ ¢ ¢Slnƒ3qu_ƒcIj~ma|&c[wujuwuœž|ƒq

n qkx¼ln„Œs¼x}qu_ƒcw€¡´mƒ„&+qkœ xn„

lMjkjkxpœ¨l—quc[quxlJ„ƒcqt¦$xnwu¥JqkxJbJc[ln„0qu_al—qXxn¡W„ƒcqt¦xMwk¥

JNⁿ(E)^©

axnwXln¢ž¢9jkœžbJ|ƒ¢ c(zZmƒ¢ cw„acqt¦$xnwu¥Œj

E= (σ, ν,1)^ªƒ¢žcq Y^(j)(E) =Pφ^(j) u=1σu^(j)

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š c­lnwkc­„ƒx—¦›œž„4l|LxMjkœ²quœžxM„JqkxJjtq‚l—qucXqk_acrbl}œ „Iwuc[jkmƒ¢²q$|&c[w~q‚l}œ „ƒœ „ƒquxqk_acr±amaœ ¢ œ bŸœžq[© :cq

n, zn

§&c

jkxnbJc(jkc[¬Cmƒc„&c[j$xn¡W|LxMjkœ²quœžMc€wkcUl}¢°„Œmƒb§Lcw‚j7Op¦c(pc a„ƒc K

U^j(n) = {Eœ jrlŸjkœ bŸ|a¢žc(zZma¢žc[w„ƒcqt¦$xnwu¥Ij~m&‚_qu_al—q

Y^(k)(E)≤zn∀k6=j}, V^j(n) = {E∈U^j(n), Y^(j)(E)≥n(a−b), φ^(j)≤L},

K_n^j = (

sup

{E∈Vj(n)}

Z˜ⁿ(E)−f^j(Y^(j)(E), n) n

≤n

)

∩

E(−n)∈U^j(n) . ^{F~dŽ H}

š caw‚j~qrwkcUln¢ž¢:lIwkcUj~mƒ¢žqXqu_al—q­ƒcwuœžMc[jXpœ wuc[+qu¢žs¡´wuxnb vZwkxM|&c[w~qts l}„&4qk_ƒcwucblnwk¥0¡´xn¢ ¢žx—¦œ „ƒJqk_aœ j

|awkxM|&c[w~qtsœ „C?ždUˆSAP©

• º @ ¶ •?” ·AI 5 #) 1h564, `S2, 1#34 c#) 5`0/ :#395 5,.5h

zn→ ∞ ^{$(#) z}_nⁿ →0

2,.)!#) #&$&') 6

sup

{E∈Vj(n)}

Z˜ⁿ(E)−f^j(Y^(j)(E), n) n

−−−−→n→∞ 0

¶ – B #

{K_n^j} ^{-9#) 5,.5h '56S5.#3 R0G -6F ) 5 0/:2# 5,.5h} n→0

h zn→ ∞ ^{$(#) z}_nⁿ →0^{(2,.)!#) # $&') 6} P[K_n^j]→1 ^{, I} n≥Nx

x→ ∞