A remark on the range of three dimensional pinned random walks

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A remark on the range of three dimensional pinned random walks

Dedicated to Professor Shin'ichi K otani on his 60th birthday

Yuji Hamana *

(Received January 11, 2006)

Abstract. The range of a random walk means the number of distinct sites visited at least once by the random walk. In the three dimensional case, it has already known that the second term of the expectation of the range of the simple symmetric random walk under the conditional probability given the event that the last point is the origin is small in comparison with that of the original random walk. This paper claims that the second term in the pinned case is bounded.

1. Introduction

Asymptotic behavior of the expected volume of the Wiener sausage for a Brownian bridge on the time interval [0, t) associated with a non-polar compact set was supplied by van den Berg and Bolthausen [1) in the two dimensional case and by McGillivray [5) in higher dimensional cases. They conclude that the leading term is the same as that of the Wiener sausage for a Brownian motion up to time t, which is given by Spitzer [6]. In the case that the non-polar compact set is the closed ball with radius r, the second term of the expected volume of the pinned Wiener sausage is small in comparison with the non-pinned Wiener sausage if the dimension is three. McGillivray [5] showed that the former is 61rr

³,

which is already implicit in the work by Uhlenbeck and Beth [7], and Le Gall [4] showed that the latter is 4J2;rr

²

.Jt.

A discrete analogue of the volume of the Wiener sausage up to time t is the number of distinct sites entered by a random walk in the first n steps, which is called the range at time n of the random walk. Asymptotic behavior of the expectation of the range at time n of the simple random walk was first given by Dvoretzky and Erdos [2], and by Hamana [3) in the pinned case. Similarly to the Wiener sausage, their results show that each expectation has the same leading

Mathematics Subject Classification {2000): 60G50, 60A99, 60E99 Keywords: Pinned random walk, range of random walk

*Partly supported

by

the Grant-in-Aid for Scientific Research (C) (2) no. 14540126, Japan

Society for the Promotion of Science.

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84 Y・Hamana

term,andifthedimensionisthree,Hamanaprovedthatthesecondtermofthe expectationoftherangeofthenon‑pinnedsimplerandomwalkisc,/nforsome suitablepositiveconstantcandthatofthepinnedsimplerandomwalkisoforder y / n ( ¥ o g n ) * f o r a n y 6 > 0 . T h e r e s u l t f o r t h e W i e n e r s a u s a g e s h o w s t h a t t h e l a s t resultcanstandfurtherimprovement.Theconclusioninthispaperisthatthe secondtermoftheexpectationoftherangeofsimplerandomwalkisboundedfor

dimensionthree.

2．Preliminariesandnotation

Byarandomwalk{S"}経oontheddimensionalintegerlatticeZd,wemeana sequenceofrandomvariablesdefined1Ⅳ助=OaxidSn=Xi+X2+‑‑hXn, where{X"}足,isasequenceofindependentandidenticaⅡydistributedrandom v a r i a b l e s w h i c h t a k e v a l u e s i n Z * ^ . T h e s i m p l e r a n d o m w a l k m e a n s a r a n d o m w a l k suchthatP[Xi=disequaltol/2difxisaunitvectorinZandisequalto0 otherwise.Throughoutthispaperweconsidertheddimensionalsimplerandom walk.Let7^betheprobabilitythatarandomwalkneverreturnstothestarting p o i n t . I t i s w e l l ‑ k n o w n t h a t 7 d i s s t r i c t l y p o s i t i v e i f d ≧ 3 a n d e q u a l t o 0 o t h e r w i s e . SinceitwillbeconvenienttoregardtherandomwalkasaMarkovchain,we wiuusesometermnologyofgeneralMarkovchams・Fbr:rEZdletR腰{･ldenote theprobabilitymeasuresofeventsrelatedtotherandomwalkstartingatx.When z＝0,wesimplywritePI･linsteadof局{･1.Fbr泥≧0andX,iﾉeZthenotation p^{x,y)isusedforPxlSn=y¥‑Itisobviousthatp^(x,iﾉ)=P"(0,2ノーx).For xGZ*^letr‑cbethefirsthittingtimeofx,thatis,Tx=inf{≧1;Sh＝z}・If therearenopositiveintegerswithSn=Xythenr^=oc.Thetabooprobabilities aredefinedby

p^{x,y)=PJSn=y,産≧ 1．

Asimplecalculationshowsthat

Po(0,x)=p"(0,x)(2.1) forall ≧1andzEZd・

Forsimplicitywewm唾eu"fbrp"(0,0)andAfbrp8(0,0).Ⅱ isodd,both tinand/nareequalto0､Itiswell‑knownthat

沌一 1

U2u=y¥血秘2(n‑k)+fin‑(2.2)

ん＝1

Anotherusefulformulaisthat

U2n=Kan‑''l^+0[n‑'‑'''¥(2.3) w h e r e / c j = 2 ( d / 4 打 ¥ d / 2 . F o r s e q u e n c e s { α 郷 } ^ = i . { & n } ^ = i , { c n }= i o f r e a l n u m b e r s suchthatCn>0forn≧1,thenotationan＝6"＋OIclJmeansthat(α"−6")/c

remainsbounded.

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Throughoutthispaper,Ci,C2,…,Cuwilldenotesuitablepositiverealcon‑

stants.

3.Mainresultandproof

Forapositiveintegernlet

Rn=¥{Sl,S2,…,sh}￨，

wherelAldenotesthecardinalityofasetA,WecallR therangeattime of therandomwalk.Ifd=3,Hamana[3]provedthat

[i22￨S2n=0]=273"+0[n^(logn)‑'](3.1)

forany5>0.However,thisresultmustbeimprovedwhenweconsiderananalogy betweenrandomwalkandBrownianmotion.Themailresultinthispaperisthe f o l l o w i n g .

Theorem3.1.Ifd=3,wehavethat

EIR2"lS2"＝0}＝2γ3冗十OI1l． ( 3 . 2 ） Unfortunatelywehavenoideaforcalculatingtheexplicitformofthesecond termoftherighthandsideof(3.2).

WenowgiveaproofofTheorem3.1.Sincewetreatonlythethreedimensional randomwalkinthissection,wewrite7andfor73andK3,respectivelyfor simplicity.Accordingto(3.21)inHamana[3],wehavethatE[R2n¥S2n=0]is

equalto

◎ o m − 1

, γ " 十 2 " 震ん ‑ 2 "' ' 雲 ( ん ‑ " ‑ " ‑ I / * ) f 9 { n ‑ h ) + 0 [ l ¥ . ( 3 . 3 )

I t s u 田 c e s t o g i v e a s y m p t o t i c b e h a v i o r o f んf b r c a l c u l a t i o n o f ( 3 . 3 ) . T h e s e c o n d claimofLe皿na3.1inHamana[3]isthat

ん " ＝ , γ * K n ‑ * / * + 0 [ n ‑ / * ( l Q g n ) ‑ * ]

f o r a n y 8 > 0 . S i n c e t h i s f o r m u l a c a n p r o v i d e o n l y ( 3 . 1 ) , w e n e e d t o i m p r o v e t h i s e s t i m a t e i n o r d e r t o s h o w ( 3 . 2 ) . W e a c c o r d i n g l y o b t a i n t h e f o l l o w i n g l e m m a , o f whichtheproofisdeferedtoSection4.

Lemma3､2.Wehatﾉe城

/2=7W+0[n‑"/].

f o r t h e t h r ℃ e d i m e n s i o n a l s i m p l e r a n d o m t " α 雌．

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86 Y.Hamana

Invirtueofthislemma,weobtainthatthesecondtermof(3.3)is 4 γ 2 侭冗 ^ + 0 [ n ‑ i / 8 ] .

Wecancalculatethethirdtermof(3.3)inananalogousmannerto(3.23)in Hamana[31.ItfollowsfromLemma3.2that

九一 l

( * ‑ " ‑ ‑ ' ﾉ * ) f 2 { n ‑ h ) ^（ ³ ^. ⁴ ^）

ん＝l

isequalto

繕満州何半｡{売言州̲刷夫(Va+v/^J*^

S i n c e t h e f i r s t t e r m o f ( 3 . 5 ) i s t h e s a m e a s ( 3 . 2 5 ) i n H a m a n a [ 3 ] , w e h a v e t h a t i t i s 2 γ 2 施冗 . ‑ i + 0 [ n ‑ " l .

Itsu伍cesforanestimateofthesecondtermof(3.5)togiveaboundof 1 抑 − 1

扇吾伽 ‑ ん ) , , ．

S i n c e i t i s o f o r d e r n' ^ , w e h a 八 ' e t h a t ( 3 . 4 ) i s 2 7 ^ / 0 1 " * + 0 [ n^ ¥ , w h i c h i m p l i e s thatthethirdtermof(3.3)is

‑4yw"+o[i].

ThiscompletestheproofofTheorem3.1.

Remark.Inthethreedimensionalcase,Proposition2.1inHamana[3]shows

that

ERn=7"+2/V臆冗/+(V(logn)‑]

forany6>0．WiththehelpofLemma3､2,theestimateoftheerrortermcanbe easilyimproved.Wethenconcludethat

ERn=‑yn+2^V脇冗'+0[l].

4．ProofofLemma3.2

ThissectionisdevotedtoaproofofLemma3.2.Wealsoconsiderthesimple randomwaⅨmovingonZ3、

L e t N = [ n / 4 ] f o r a p o s i t i v e i n t e g e r n , w h e r e t h e n o t a t i o n [ x ] i s u s e d f o r t h e g r e a t e s t i n t e g e r w h i c h i s n o t l a r g e r t h a n a r e a l n u m b e r x . N o t e t h a t

た " ＝ E E p W V ( 0 , z ) p : " ‑ 4 N ( 懸 , ， ) p : 1 V (, 0 ）（ 4 1 ）

霊≠Oy≠0

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forapositiveintegern.Forsimplicity,weuseLfor−2N.Wefirstconsiderthe e f l f e c t o f r e p l a c i n g P o ^ ( x , y ) w i t h p ^ ^ ( x , i ﾉ ) i n ( 4 . 1 ) , f o r w h i c h t h e f o l l o w i n g e q u a l i t y

willbeuseful:

p'^(x,2ﾉ)‑Po^{x,y)=PAro≦L,S2L=y]+Px[L<To≦2L,S2L=y].(4.2)

Itiseasytoseethethefirsttermof(4.2)isequalto

L

E p S ( ^ . 0 ) p 2 ^ ( 0 , y ) .

た＝l

Classifyingtheevent{L<tq≦2L,S2L=y}bythelasthittingtimeof0,wecan obtainthatthesecondtermoftherighthandsideof(4.2)isequalto

2 L − 1 2 L − 1 L

Z剛｡o=k,S2L=y]‑y,y]Px[ro=k,ao=j,S2L=y],(4.3)

た＝L＋1 j=L+lk=l

whereao=max{α≦L;S2a=0}.Thefirsttermof(4.3)isexpressedby

2L−l

T p H x , o ) p l ' ' ‑ ' { o , y )

ん＝L＋l

andthesecondoneisexpressedby

2 L − 1 L

‑ E E P o ( ^ > 0 ) p ^ ‑ ' = ( 0 , 0 ) p S ^ ‑ ^ ( 0 . y ) .

j=L+lfc=l

RecallthatL= −2N.Wehencehavethat/znisequalto

EEpl^iO,x)p2‑'^(x,y)pl^{y,0)(4.4)

工≠Oy≠O 加−2Ⅳ

‑EEEp5''(o.^)p5(^.o)p2"‑'N‑fc(o,y)p2N(^^o)(4.5)

ん＝l工≠Oy≠0 2 泥 − 4 N − l

‑EEEpS'‑co,加騰(x,0)pJ"‑^^‑*(0,y)pS^(y,0)(4.6)

た＝ −2N＋1ェ≠Oy≠0

2 祁一 4 Ⅳ − 1 九一 2 N

+ E E E E p ' ' ( o . ^ ) p S ( ^ . o k ‑ 砺施 ‑ ' ' ' ‑ ^ ( 0 , y ) p r ( y . O ) . ( 4 . 7 )

J=丸−2Ar+ik=i垂≠Oy≠0

Wenexttrytoshowthat

I冗/211沌/211沌/21

た " ＝ " 2 " − 2 エル k U 2 n ‑ 2 k + E E ん吻迦 2 n ‑ 2 k ‑ 2 j + 0 [ r r * ﾉ * ] ( 4 ‑ 8 )

k = l k = l 7 = 1

Weneedthefollowinglemma.

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88 Y､H紐、血､a

Lemma4.1.

a

E#(0.<W)=p^(o,)‑E九膳̲jla+2ft‑9k,Q¥uq)

≠0 ^k＝l

P r o o f . H α = 1 , w e c a n e a s i l y o b t a i n ( 4 . 9 ) f r o m t h e f a c t t h a t p j U O , w ) = p ^ ( 0 , w ) foranyuﾉ.Wemayconsiderthecasewhena≧2.Foruﾉ≠O

a − l

# ( 0 , u O = p * ( 0 , u O ‑ 逗血 j * ‑ * * ( o , t i O .

k＝l

Then

; r # ( o , i i > ) p * w ) = p * w ( o , z ) ‑ p ^ ( o , o ) p ^ ( o , z )

≠O

a − 1 a − 1

‑EW^‑(0,z)+EW‑(0,0)p^(0,z).

k＝1 k ＝ l

Using(2.2),wehavethattheforthtermoftherighthandsideisequalto W(0,z)−んaP^(0,z),

w h i c h i m p l i e s ( 4 . 9 ) .

^□

Animmediateconsequenceof(2.1)and(4.9)isthat

yy(o,x)pfm)=Tp^(w,o)p^(om

垂 ≠ 0 t " ≠ 0

（ 4 . 1 0 ）

β

＝ " 2 ． + 2 β − Z 血迦 2 α + 2 β ‑ 2 肱

k＝l

Lemma4.1yieldsthat(4.4)is

N

E p 2 "2 N ( 0 , , ) p : 1 V ( y0 ) − E Z 血 p 2 耐 ‑ 2 " ‑ 2 k ( 0 , , ) p ; " ( 0 ) ．

U ≠ 0 J t = l y ≠ o

T h e r e f o r e i t f o l l o w s f r o m ( 4 . 1 0 ) t h a t ( 4 . 4 ) i s e q u a l t o

Ⅳ Ⅳ N

U 2 n ‑ 2 E f 2 k U 2 n ‑ 2 k + E E 血た , " 2 " ̲ 2 膳 ‑ 2 j

k＝1 k=l7=1

Foracalculationof(4.5),(4.6)and(4.7)wenowconsiderthecasethatn=2m.

Wehavethat(4.5)isequalto

− 2 N

‑ E / 2 ^ E p ^ " ^ " " ( o , y ) p r ( y , o ) ,

k＝l y≠0

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whichcanbeexpressedby

汀 a − N

− E 伽 * E p " " " ' " ' " 2 ' ( ' > . ! ' ) 垢』 ^ ( y , o ) .

k＝l y≠o

MakingthesubstitutionノI=TV+A;mthesummationonk,weobtainthatthis

summationcoincideswith

W E

‑EMEp"""'''"''(o,y)pr(y,o).(4.11)

h = N + l y ≠ O

Itfollowsfrom(4.10)that(4.11)andalso(4.5)areequalto

m m N

−EんiU4m‑2h+EE血血迦4m‑2/i‑2fc‑(4.12)

ん＝ N ＋ 1 九＝Ⅳ＋1k＝l

Similarlyto(4.5),wecanseethat(4.6)hasthefollowingform:

2打1−2Ⅳ−1

−ヱル‑2N‑2kTpl''{0,x)p^Hx,0).

峠＝m−N＋1 錘≠o

Substitutingん=2m一Ⅳ一kinthesummationonk,wehavethatitisequalto

アァE−l

‑EんEP;N(0,鰯)P4"2'v2伽(慾,0）（4.13）

h = N + l 垂 ≠ O

Applying(4.9),weeasilyshowthatthedifferencebetween(4.12)and(4.13)is dominatedby/2mW2mwhichisofordern^.Thereforetheleadingtermof(4.6) c o i n c i d e s w i t h ( 4 . 1 2 ) a n d i t s r e m a i n i n g t e r m i s 0 [ n ^ 1 .

I t c a n b e e a s i l y s e e n t h a t ( 4 . 7 ) i s o f o r d e r n ^ / ^ . I n d e e d , i t i s b o u n d e d b y

27n−2N−17n−N

Z E 伽 + 2 k ' a 2 j ‑ 2 k f 4 m ‑ 2 N ‑ 2 j

7=m−N＋1k＝1 2 m − 2 Ⅳ − 1 m − Ⅳ

≦OiEE(^+^)"'^'0"‑^)‑'/'(2m‑N‑3)‑'f¥

7=m−N＋1k＝l

whichisdominatedbyaconstantmultipleof

2 m − 2 N − 1

ノ V " ' ^ ' ' E ( j ‑ m + A r ) ‑ i / 2 ( 2 m ‑ ^ ‑ j ) ‑ 3 / 2 ≦ C 2 N ‑ ^ { m ̲ Ⅳ U / 2

7=Tn一N＋l

Therefore,ifn=2m,weconcludethat

8 m Ⅳ

/ 2 n = U 2 n " 2 E 血池 2 n ‑ 2 f c + E E ル偽ん似 2 n ‑ 2 f c ‑ 2 j

ん＝1 A:=l7=1

( 4 . 1 4 )

、 Ⅳ

+EEん吻泌2n‑2fc‑27+0[n‑5/2].

fc=iV+l7=l

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90 Y・Hamana

Since

m 河 1 打 3 W B

EE九曲'"2n‑2Jfc‑2j≦G*‑'EE(2m‑k‑Jr^=0[n‑^%

k = N + l j = N + l k = N + l j = N + l

theforthtermoftherighthandsideof(4.14)is

m m

E E f 2 k f 2 j U 2 n ‑ 2 k ‑ 2 j + 0 [ n ' ^ ] .

虎=N+l7=l

T h i s i m m e d i a t e i m p l i e s ( 4 . 8 ) .

Inthecasethatn=2m+1,wecanapplythesamecalculation.Detailisleft tothereader.Wethenfinishtheproofof(4.8).

BeforeprovingLemma3.2,wemustprovidetwomorelemmatta.

L昼前耐na4．2．

允冗＝ γ 2 侭冗 ‑ 3 / 2 ＋ O I 沌 ‑ 2 } ．

Proof.Wefirstcalculatethesecondtermoftherighthandsideof(4.8).Weuse m f o r [ n / 2 ] a g a i n . B y ( 2 . 3 ) ,

E 伽 … 鵬 E 帥 ‑ 鋤 ‑ ･ ' 判￨茎鵬 − ，催州 ‑ ｡腰｝側

Itisobviousthatthesecondtermoftherighthandsideof(4.15)isoforder 冗一 5 / 2 ． W e e s t i m a t e t h e e f f e c t o f r e p l a c i n g 冗一 k w i t h n i n t h e f i r s t t e r m . B y t h e meanvaluetheorem,wehavethat0≦(九一A:)‑3/2−−3/2≦Ctkt仰一k)‑***for

l≦ん≦m.Then

m

O ≦ / { ( n ‑ * ) ‑ * ﾉ 2 ‑ " ‑ 3 / 2 } ≦ 伽 ‑ s / z T > ‑ v ‑ o i ‑ ￨ .

k＝1 k＝1

whichyieldsthatthelefthandsideof(4.15)isequalto

m

‑'"/+0[n‑*].

k＝1

Remarkthat

。｡

Em‑>‑‑E

た＝1 k＝α＋1

血＝1−γ+Ola‑W].

Wethereforeobtainthatthesecondtermoftherighthandsideof(4.8)is

−2応(1−γJn‑^+0[n‑2].

( 4 . 1 6 ）

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Wenextcalculatethethirdtermoftherighthandsideof(4.8),whichis

嶋正肺伽嶋 ‑ … 陸織榊い ‑ j ) ‑ 5 / 2 . ( 4 . 1 7 )

RecallthatNhasbeenusedfor[n/4),whichisequalto[m/2].Wedividethe summationinthesecondtermof(4.17)intothefollowingtwoparts:

T,y,k‑'"r"Hn‑k‑j)‑"¥Ty‑k‑''2r"Hn‑k‑j)‑"¥

I t i s e a s y t o s e e t h a t b o t h s u m m a t i o n s a r e o f o r d e r n ^ Z ^ . I n d e e d , t h e f i r s t s u m i s boundedbyaconstantmultipleof

( n ‑ m ‑ i V ) ‑ V 2 V E ^ " ' ^ ' r ' ﾉ z

andthesecondoneisboundedbyaconstantmultipleof

Ⅳ ‑ 3 / ' E E k ‑ V 2 i i r n ‑ k ‑ i ) ‑ ' / 2 ≦ C も n ‑ 3 / 2 V i f c ‑ 3 / 2 ( ^ ̲ i t ) ‑ 3 / 2

Theestimateofthefirsttermof(4.17)canbeobtainedinthesamemanneras thefirsttermof(4.15).Weconsidertheeffectofreplacing九一j‑kwithninthe f i r s t t e r m o f ( 4 . 1 7 ) . S i n c e ( n ‑ f c ‑ 7 ) ‑ 3 / 2 ‑ n ‑ 3 / 2 ≦ C i i k + j V n − i f c ̲ j ) ‑ 5 / 2 ^ i t sufficestogiveanestimateof

J2Tk‑"'r"Hn‑k‑j)‑"¥(4.18)

Wealsodividethesummationonjin(4.18)intothecasethat1≦j≦ﾉVandthe casethatN<j≦m.Inthefirstcase,thesummationisboundedbyaconstant multipleof

( n ‑ m ‑ 7 V ) ‑ / 2 y ‑ y ‑ i c ^ ^ j ‑ ^ n = o [ n ‑ 2 ) .

Inthesecondcase,thesummationisboundedbyaconstantmultipleof

^ ‑ 3 / 2 ^ E f c ‑ ' / ' ( 2 m ‑ i f c ‑ j ) ‑ 5 / 2 ≦ 伽 ‑ 3 / 2 y ‑ ^ ‑ i ﾉ 2 ( m ‑ A ; ) ‑ 3 / 2 ,

鰐Wご撫孟綱郷欝蕊.謎鶏潔雛若隅猟雛

o f ( 4 . 1 6 ) , t h i s i m m e d i a t e l y i m p l i e s t h a t ( 4 . 1 7 ) i s

‑ 卿縄 ( 言 " 岬 I 諏 ‑ 2 ] = ( l ‑ ^ ) 2 / , ; i ‑ 3 " 半｡ I 凧一興 I

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92 Y.Hamana

A p p l y i n g ( 2 . 3 ) a g a i n , w e c o n c l u d e t h e a s s e r t i o n o f t h i s l e m m a .

^□

Thefollowinglemmaisthemaintooltocalculatetherighthandsideof(4.8).

Lemma4､3．

恥 { ( 2 " ‑ 耐， ‑ ( " ‑ ･ ' ， } = 為 ÷ ｡ [ m ‑ R / * l o g m l . ( 4 . 1 9 )

Proof.ByLemma4､2,thelefthandsideof(4.19)isequalto

γ * * ‑ ﾉ 2 { ( 2 m ‑ i f c ) ‑ 3 ′ 2 ‑ ( 2 m ) ‑ 3 / 2 } ( 4 . 2 0 )

￥｡￨か { 修州 " 3 / 2 ‑ ( 2 m ) ‑ 3 / 2 } . ( 4 . 2 1 )

Since(2m‑k)‑W‑(2m)‑W≦αA;(2m−ん)*/2t^gsummationonkin(4.21)

isdominatedby

CgYV^m‑fc)‑5/2

whichmeansthat(4.21)isoforderm‑5/21ogm・Sincez3‑y3＝(z−y池2＋

xy+y2),thesummationonkin(4.20)isexpressedby

茎赤(蒜苛‑歳)(病+而皇而荒幸夫)，

whichisthesumofthefollowingthreesummations:

志吾 , k U 2 m ‑ f c ) 3 ( v / 2

^l

m + y / 2 m ‑ k ) ' 夫茎伽榊 ̲ 偽 x 侃刑

¹

雨 = 鴇 ) ，

雨茎〃扇 i ‑ k ( y / 2 m + V 扇同 )

Herewehaveusedthefollowingformula:

読書覇志祷霜緬(侃手；京壱霧)

BythestandardargumentofRiemannianintegral,wehavethat(4.22)is

忘茎 " m ( y 2 ‑ A : / m ) 3 ( v ^ + V 冒二胴

一志 " 癖侭 ( x / 2 ^ ^ ) 3 ( x /

^dx

2 + v / 2 ^ x ) 半｡ i m ‑ n

( 4 . 2 2 ）

( 4 . 2 3 ）

( 4 . 2 4 ）

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S u b s t i t u t i n g y = ､ / 房 1 2 i n t h e i n t e g r a l o n x , w e o b t a i n t h a t ( 4 . 2 2 ) i s

索 " 二 M 言ｧ ' ) * ( i + > / ^dy r 言ｧ ) 寺｡ [ m ‑ s ' 霞 I

Wecancalculate(4.23)and(4.24)inananalogousmanner,andconcludethat ( 4 . 2 3 ) i s

索偽 l ‑ y 2 ) ( l + y r ^ 2 ) 手｡ f n , ‑ V > ]

a n d t h a t ( 4 . 2 4 ) i s

制 ; 黒､ / r = ｧ ( 害/ F ｧ ) 半伽． " '

Thereforetheleadingtermof(4.20)isequalto

奈鵬{雨÷赤寺叢声r手去言ｧ血

whichcoincideswith

蕊"祭圭{雨評‑'} 側

s i n c e x ' * + x ' * + a T * = { x ' * ‑ 1 ) / ( 1 − 勿 ) a n d t h e r e m a i n i n g t e r m o f ( 4 . 2 0 ) i s o f orderm ^．Moreoverthefactthat

d y ( y y / 令言ｧ合帯 y j y M 耐一宗 F 一 ' ｝

i m p l i e s t h a t ( 4 . 2 5 ) i s ^ k I ､ / 2 m 2 + 0 [ m ‑ V * L w h i c h y i e l d s ( 4 . 1 9 ) .

^□

WearereadytoshowLemma3.2.Weconsideronlythecasethatn=2msince wecanshowthelemmainthecasethatn=2m+1analogouslytothiscase.

L e m m a 4 ､ 3 a n d ( 4 . 1 5 ) y i e l d s t h a t t h e s e c o n d t e r m o f t h e r i g h t h a n d s i d e o f ( 4 . 8 )

E ^ ( 2 m ) ‑ 3 / 2 半券 ÷ 0 [ m ‑ ^ l o g m l .

Lemma4.2immediatelyimpUcsthat

E/2Jk=1‑7‑27^m‑^2+0[m‑*].(4.26)

Then

/ , f 2 k ' U 4 m ‑ 2 k = ( 1 ‑ γ ) / c ( 2 m ) ‑ 3 / 2 + 0 [ m ‑ ^ i o g m ] .

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94 Y・Hamana

Wehavealreadyseethatthethirdtermoftherighthandsideof(4.8)is

汀 B 7 ｱ 8

"EE允吻(2m‑k‑j)‑'+OWn‑'](4.27)

fc=l7=1

intheproofofLemma4.2.WerecallthatJV=[n/4]=[m/2]andwriteMfor [ m ^ l . T h e d o u b l e s u m i n ( 4 . 2 7 ) i s d i v i d e d i n t o t h e f o l l o w i n g t h r e e p a r t s :

M m

EEた山(2m‑‑j)‑",

fc=l7=1 w l N

EEん (2ro‑A‑i)‑/,

fc=M+l7'=l

7 ｱ 1 訂 l

EEた鵬たA2m‑k‑j)‑V*.

k=M+lj=N+l

( 4 . 2 8 ）

( 4 . 2 9 ）

( 4 . 3 0 ）

I t i s e a s y t o s e e t h a t ( 4 . 3 0 ) i s o f o r d e r m " " W . I n d e e d , b y ( 2 . 3 ) , i t i s d o m i n a t e d byaconstantmultipleof

W L w B

^‑21/8yE(2m‑fc‑j)‑".

fc=M+l7=N+1

Wefirstestimatetheeffectofreplacing2m‑k‑jwith2m‑jin(4.28).By

themeanvaluetheorem,

競血ん { ( 2 m ‑ f c ‑ j ) ‑ 3 / 2 ‑ ( 2 m ‑ . 7 ) ‑ 3 / 2 }

fc=l7=1 M m

≦C1oEE*"'^'"^(2m‑fc‑j)‑^,

Is=lj=l

whichisboundedby

m

Cio(m‑Af)"/EE^"r^=0[m‑"/].

fc=l7=1

Thismeansthat(4.28)is

M m

EE允吻(2m‑j)‑^2+0(m‑^8].

Jk=l7=1

Wenextestimatetheeffectofreplacing2m‑jwith2minthisdoublesum.It

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f o l l o w s f r o m ( 4 . 1 6 ) a n d ( 4 . 1 9 ) t h a t

正

^fc=l7=1^{M m}

仙 { 伽 ‑ " ･ ' ｡ ‑ 帥 " } 雲為茎伽･榊 l o g m ]

γ ' 器 ) 総半 0 [ m ‑ / l ,

whichyieldsthat(4.28)isequalto

M m

(2m)‑^VV血ん＋

fc=lJ=l

γ 霞綜 ^ + 0 [ m ‑ " ' 圏

Theeffectofreplacing2m‑k‑jwith2m‑kin(4.29)isoforder

7 ｧ E N

E E k − 3 / 2 j ‑ ' / 2 ( 2 m ‑ k ‑ j ) ‑ 5 / 2 ，

fc=M+lji=l

whichisboundedbyaconstantmultipleof

Ⅳ

M ‑ * ' * Y ] r V * { m ‑ J ) ‑ * ' * = 0 [ m ‑ " ' * ] .

i=i

Thereforetheleadingtermof(4.29)is

m Ⅳ

EEた騰胸(2m‑)‑/

k=M+l7=1

( 4 . 3 1 ）

andtheremainingtermof(4.29)isoforderm‑W*.Wedivide(4.31)intothe f o l l o w i n g t w o p a r t s :

m N

EEかん;(2m)‑3/2

k=M+l7=1 m N

E E 〃ん { ( 2 m ‑ A : ) ‑ 3 / 2 ‑ ( 2 m ) ‑ * / 2 '

fc=A/+l7=1

Itfollowsthat

アア B 7 y 8

EEん山(2m)‑3/2=0[m‑^],

k=M+lj=N+l

w h i c h y i e l d s t h a t ( 4 . 3 2 ) i s

W u W E

(2m)‑VTT血均+0[ro‑"'*l.

Jk=M+l7=1

( 4 . 3 2 ）

( 4 . 3 3 ）

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96 Y.Hamana

Moreoverwehavethat

M N M

Z E 九曲 { ( 2 m ‑ 俺 ) − 3 ﾉ 2 ‑ ( 2 , " ) ‑ 3 / 2 } ≦ C u * ‑ ' ﾉ 2 ( 2 m ‑ k ) ‑ * ﾉ 2

fc=l7=1 k＝l

w h i c h i s o f o r d e r m ^ ^ . T h e r e f o r e ( 4 . 3 3 ) i s

m N

E E た{ ( 2 , ‑ k ) − 3 ﾉ M 2 m ) ‑ * / * } + 0 [ m ‑ " / * ] ,

fc=l7=1

o f w h i c h t h e f i r s t t e r m i s e q u a l t o 7 2 ( 1 − γ ) 脇 / ､ / Z h 1 2 ・ H e r e W e h a V C a p p l i e d ( 4 . , 6 ） and(4.19).Wehenceobtainthat(4.31)andalso(4.29)are

（吋率膿嘉茎仙型璽綜 ) 嬢伽 ‑ 1 7 / 8 ]

Consequentlywehavethat

m m

EZ血伽4m‑2k‑2j=K(2m)‑3/2^^た

（ 4 . 3 4 ）

fc=l7=1 fc=l7=1

＋､ / 百 7 2 ， ( 2 ' j ＋ ‑ O ‑ I + m 0 − [ 1 n 7 , / ‑ 8 / ￨ 1 . ． 2

B y ( 4 . 2 6 ) , t h e f i r s t t e r m o f t h e r i g h t h a n d s i d e o f ( 4 . 3 4 ) i s

（ 1 − γ f K { 2 m ) ‑ ^ − ､ / z γ ^， ( ² 1 ^＋ ^ ^O i ^ ^I + ^m O ^‑ h ⁵ , ^/ ‑ ² ^ ^} , ^， 2

whichmeansthatthelefthandsideof(4.34)is

（ 1 − γ ) * K ( 2 r o ) ‑ V * + 0 [ m ‑ W l .

Then(4.8)yields

九 m ＝ γ 2 脇 ( 2 m ) ‑ 3 / 2 ＋ O I m − 1 7 / 8 1 ,

whichisequivalenttotheassertionofLemma3.2ifn=2m・Thiscompletethe proofofLemma3.2.

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