On large increments of a two-parameter fractional Wiener process

Vol.44 No.9SCIENCE IN CHINA( Series A)September 2001On large increments of a two- parameter fractional WienerprocessZHANG Lixin(张立新) , LU Chuanrong(陆传荣}& WANG Yaohong(王尧弘)1. Department of Mathematics , Xixi Campus , Zhejiang University , Hangzhou 310028 , China ;2. Department of Statistics , Zhejiang University of Finance and Economics , Hangzhou 310012 , China ;3. Department of Statistics ,Tunghai University , Taizhong , ChinaCorrespondence should be adressed to Zhang Lixin( email : lxzhang@ mail. hz. zj. cn)Received February 12 ,2001AbstractIn this paper , how big the increments are and some liminf behaviors are studied of a two-parameter fractional Wiener process. The results are based on some inequalities on the suprema ofthis process , which also are of independent interest.Keywords : fractional Wiener process , increments , liminfs.1 Introduction and main resultsLet {Z(t ,s );t ,s≥0}be a mean zero Gaussian process with Z(0 0)=0 a.s. andEZ( t; si)z( t2 s2)={It|2a+ 1t2|2a- It2-t|2a }{|s|2a + Is2|2a-Is2- sμ|2a }/4. Itis called the two-parameter fractional Wiener process of order a(00 ,there exists θo= 0( ε ) such thatlimsup8δo/8d+1≤1+ε，for1<θ≤θo,(1.2)h→∞then1116SCIENCE IN CHINA( Series A)Vol.44lim supsupδr | W( R)1= lim sup supδr | W( R )|= 1 a.s.( 1.3)R∈ipT→∞R∈4Furthermore ,if( ii)lim{log T/ar)+ log1 + log br/ / ar )}loglogT=∞，thenlimsupδr|W叭R)1=limsupδr|W(R)|=1a.s.(1.4)下∞R∈LR∈年Lir 2] and Zhang'’ 41 studied the liminf's when condition( ii ) is not satisfied.Almost all results of Csorgo and Revest 11 on the increments of one- parameter W iener pro-cess were extended to other and more general one- parameter Gaussian processes( cf. refs.[ 5一8 ]). Also , some authors tried to extend Theorem A to other two- parameter Gaussian processes( cf. ref.[9]). But to the best of our knowledge , there were no pioneering results actually in-cluding Theorem A. Especially , the equalities in( 1.3 ) were not obtained for other two-parame-ter Gaussian processes. The purpose of this paper is to extend Theorem A to a two-parameterfractional Wiener process of order a. Our results read as follows.Theorem1. Let0< ar≤T and br ≥V T be two functions of T. Suppose that br isquasi-increasing ,i.e. for some co ,br, ≤cobr, for all T≤T2. Thenlim sup supδrZ( R )= lim sup supδτ | Z( R )|= 1 a.s.( 1.5)R∈lFurthermore , if condition( ii ) in Theorem A is satisfied , thenlim supδpZ( R )= lim supδr | Z( R)I= 1 a.s.( 1.6)Remark.In our Theorem 1 , conditions( i),( ii )and( 1.2 ) in Theorem A are not used.Also , the functions ar and br may be not non-decreasing.If condition( ii ) is not satisfied , we have the following result on the liminfs.Theorem 2. Let0< ar≤T and br≥V T be two non-decreasing functions of T. LetYτ = {2af( log( T/ar)+ log1 + log br/↓ar ))- logloglogT )}-1/2.Assume that Yτ and ar/ T are both non- increasing , and Yr satisfies condition( 1.2 ) , and thatlim inf.log T/ar)+ log1 + log br/V ar))= r>1.(ili')logloglogTIf r= +∞or00 , thenlim inf supYrZ( R )= lim inf supYr 1 Z( R)1= 1 a.s.R∈Take br= T and ar= T. From Theorems 1 and 2 , it follows thatCorollary 1. We havelim supZ( T,T)= lim supxy山_1a.s.V 27a loglogT一∞中国煤化工Z(x,y)CCH.CNMH G)1lim sup。sup= 1 a.s.y=T" / 472a loglogT°0∈x y≤" V 472° loglogT功≤Tlim inf sup=liminfsup.| Z(xry)II∞0σ≤ε≈°V 2 T2a loglogT0≤xvi≤T√2 T2a loglogT可=T炒y≤7The石凉整据Theorems 1 and 2 are based on the fllwing propositions.No.9ON LARGE INCREMENTS OF A TWO-PARAMETER FRACTIONAL WIENER PROCESS1117Proposition1. Let L* ={R=[x| x2]x[ y1ry2]:R∈Lj ,0≤x1≤x2≤VT }.Assume that0< ar≤T ,br≥V T. Then for any δ>0 ,there exists C= C( δ )> 0 ,such thatp{pz(?)E uais }呀}≤p{supZ( R )≤uaiy }≤exp{{-c(1v(2kexp{ --20u2}( 1.7)for any k large enough. Furthermore ,if0 0 , i.e. for some constant Co ,d(x )x≤co( y Vy" for all 0< x≤y. Then for any given ε >0 ,there exists C= C(ε )such thatPtsup | X(R)|≤ud(ar)}≥expT(1 + log,67)(1-<2+e)川}，( 1.10)ar+.01小。for all u≥1 00 ,there exists C= C( ε ) such thatP{sup)(l)≤u}≤exp{r{-c(Iv +2-2-)}+exp{-cfor all u≥1 and_ h large enough.Proot方数握ξ= Y( ik),i=0..[ b/k ]. Then1118SCIENCE IN CHINA( Series A)Vol.44EK≤d1 i-jl k)≤dh) (i≠j).Let τ ,7; ,i=0.. [ b/k ] be independent mean zero Gaussian random variables with Et2 =d k)Eη=1-p h). ThenE( τ+ η:}=1=Eξ , and .EξS≤dh)= Eτ+ η:Xτ+ n;), i≠j.By the Slepian lemma , we haveP{ sup)Y(l)≤u}≤P{max 6≤u}≤P{。max, (τ + η:)≤ujt0∈IEb0≤i≤[ 6/k]0≤i≤[ b/kj≤P{max. η:≤(1 + δ)u}+ R(τ≥δu )0≤ist b/k](1 + δ)u)1+[ b/k]+ P{N(0,l)≥δu/V [h)}={-r(M0.)>V1-d k)}(1+ δyu2{_ 2二≤exp{-a1+[b/k]xp[- x1-dh))}+ exp2p(k)} 'where the inequality 1- x≤e~x and the tail probability of normal distribution are used in thelast inequality. By noting d h )→0 , the lemma is proved.Proof of Proposition1. Let Ut )= Z(V Te-' ,V Te' )- Z((V T- aτ/vT )e-',V Te' ). ThenEUl)U(s)= ra2d(1-0){1 +e-2d(1-x)-11-e<1-s)|20}|2aar'e-(1-s)+ar ). e-(1-s){|1-(1-1TI(1_-T- |1-e(1-s)2a -(1- )-(1-")e-)|”}.(2.1)If ar/T<1 ,let d= -log1- ar/T )x=e-d=1-ar/T. Then00. Let j( t )=(1+ t }°- t2a. Then f'( t )=2a{ 1 +t尸a-1-12a-1}. It fllows that lf'( l )|≤K。l2a-3( 1>0),and f'(t )≤0( l>0)if01 and μ(1-ε )<1. DefineX,(t ,s)=J (x(()J-o中国煤化工2 }1HCNMHG{1 y-s|2-iyI2}W(dx dy)，x,(t ,s)={1x-l1_1x2}(3.11 )J 1xlg(ud)J -。{1y-s12-1y12}W(dx dy),1122SCIENCE IN CHINA( Series A)Vol.44Y,(t ,s)=x- t2-1 x陟}J1xlσAJ_∞ K&，2a-1{1 y-sI 2-1y1^2 }W(dx dy ),Then Z(1 ,s )=X,(t ,s )+ X,(t rs ),{X.C ; )}%=1 are independent , and{X,(t s);t s≥0}={e" yr,(t/e" ,s);t is ≥0}.Lemma2. Let Y>0 and p> 1. Then there exists a constant c > 0 dependingon a ,Y，p such thatEY:([t1+ h1]x[s + h2])≤c(h12)2e(1ogh1n-δ≤(h2P0(logh1h2)n-δ = 6( hyh2).uniformly in n≥1 ,0≤t≤n/2 ,0≤h] ,h2≤1 and s≥0.Proof. It follows thatEY2(t s)=x-t-| x附{1y-s-1y附}dxdy= h2a|J lxIgA.{1x-l102-1x引}dx≤d h1hn y.(10gby Lemma2.9 in ref.[ 11 ] and its proof.Proof of Step 2. From Proposition 1 , it follows that for any ε > 0 , there exists C =C( ε ) such thatP{sup8rZ(R)≤1-2e}1o br≤exp{{-c2arEexp(-x1-2e r(log。5 ar;+1og(1 + log_VT4)+loglogTr)(2-e2)}(-(ngbarT.L + log(1 + log'br)+ olog7)}NT){-c. aτ. ( logT)4(3.12 )for any0< ar≤T ,br≥V T.We first assume thatlim sup log{Tazlo1 + l8 br/V T ))}=∞.Then there exists a sequence Tn↑∞such thatlog{T,a~'( Tn )o(1 + logb( T)V T，)}loglogTn中国煤化工It follows from(3.12 ) that for n large enough ,MYHCNMHGP{sup8rZ( R )≤1 - 2ε{≤exp{- d logTn尸}+( logT)+→0,(n→∞)R∈4Thus ,liw. sup supδrZ( R )≥lim sup supδrZ( R )≥1 - 2e a.s.P RE T,R∈LNo.9ON LARGE INCREMENTS OF A TWO-PARAMETER FRACTIONAL WIENER PROCESS1123which implies(3. 10 ).Now we assume thatlog{Tarlog1 + lo8 br/VT))}< r<∞.(3.13)loglogTLet Tn=e" ,p>1 ,p( 1-ε )<1. Let dn An X(. ; ),X,( ; ), Y,( ; ) be defined asin(3.11 ). Then dn= nTn An=( dn_1Tπ1 ,dnTπ1 )and ,{x,(I ,s);t ,s ≥0}={r"Y,(tT-' ,s);t s≥0}.LetL。= {R =[x1 x2]x[y1 y2]0≤x1≤x2≤T。/T,0≤y1≤y2≤K Tn),x2y2≤1,I RI≤d( Tn)T.}.By Lenma2, EY( R )≤d(IRI )for any R∈LT ,whered(I)=cn-20(log 1)'.From the Remark of Proposition 2 , it follows thatP{ sup&( Tn)1 x,( R)1≥e}R∈L=P( sup&( T )Tq 1 Y,( R)I≥e)R∈,Tn6( Tn)V Tn/Tn≤CdT,)1+log( T,)/1 + log↓a( Tn )Tn2e2δ-2( Tn )T-20exp'l- (2+ ε )cn-8(d( T)VT,)2"(log Tn/d( Tn ))cloglogTn≤a logTn Y'exp{-n~°( lognyJ≤( logTn )3≤n-2，(3.14 )for n large enough. It follows from the Borel-Cantelli lemma thatlim sup supd( Tn )| X,( R )1≤ε a.s.(3.15 )On the other hand ,by( 3.12 ) we havesupδrZ( R)≥1-≥1- exp|.R∈行≥d logT )(1-e)_( logT )4.If follows that2rP{ supX( T,)X,( R)≥1 - 3e}之(p{;sup2( Tn)z( R)≥1中国煤化工X.(R)1≥e})n=lR∈4MHCNMHG5{{ cnμ1-ε)- cn-4p-n-2}=+∞,n=1which together with the Borel-Cantelli lemma and the independence of {X.(C ; )} implies thatlim sup sup8( Tn )X.( R)≥1 - 3e.(3.16)1124SCIENCE IN CHINA( Series A)Vol.44Putting( 3.15 ) and( 3. 16 ) together yieldslim sup supδrZ( R )≥limsup sup8( Tn )Z( R )≥1 -4ε a.s.which implies( 3.10 ).Step 3. If condition( ii ) is satisfied , thenlim inf supδrZ( R )≥1 a.s.(3.17 )R∈iProof. If condition( ii ) is satisfied , thenlogCrloglof ar + a7" )，∞,8rδ4→1,(3.18 )where Cr and δτ is defined as in Step1. Let1<θ<2. LetAk ,B; ,Tki , Tki and b; ; be de-fined as in( 3.3 ). Also defineb%n = inf{( T):T∈Ah B{},Lk,= {R =[x x2]x[yr y2]0≤x≤x2≤b%,0≤y1≤y2≤bi, ,x2y2≤T%; ,1 RI= θl/a}L'r,= {R =[x x2]x[y1 y]:0≤x≤x2≤bi,0≤y1≤yz≤bkj,x2y2≤Tk,I RI≤θh+1Ya_ θa}:Thenlim inf supδrZ( R)= lim inf supδrZ R )T→∞RE p≥lim inf inf sup supδγZ( R )1h1+1→∞ i≥l T∈AB REL≥lim inf inf supZ( R )Y1 yield(3.17 ).Acknowledgements This work was suppored by the National Natural Science Foundation of China( Grant No. 10071072 ) andby NSC 88-21 18- M029-001 of Taiwan of China.References1. Csibrgi ,M. , Revesz , P. , Strong Approximations in Probability and Statistics , New York : Academic Press , 1981 .2. Lin ,Z. Y. ,On increments of 2-parameter W iener process , Scientia Sinica , 1985 ,28 :579- 588 .3. Zhang ,L. X. ,Two diferent kinds of liminf results on the LIL of the two parameter Wiener processes , Stochastic Processes'Their Appl. ,1996 ,63 :175- -188.4. Zhang ,L. X. ,A liminf of the increments of two- parameter Wiener process , Chinese Ann. Math. ( in Chinese),1997 ,18 :235- -246 ; Chinese J. of Contemporary Math. ,1997 ,18 :185- 198 .5. Csaki ,E. ,Csxirgi ,M. , Shao Q. M. , Femique type inequalities and moduli of continuity for I2-valued Omstein-Uhlenbeckprocesses , Ann Inst. Henri. Poincare Probab. Statist. , 1992 ,28 :479- -517.6. CSs0rgo，M .. Shao,Q. M. , Strong limit theorems for large and small increments of I"-valued Gaussian processes , Ann.Probab. , 1993 ,21 : 1958- -1990.7. Monrad ,D. , Rootzen ,H. ,Small values of Gussian proces中国煤化工ed logarithm , Probab. TheoryRel. Fields , 1995 ,101 :173- -192.8. Ortega J ,On the sive of the increments of non-stationaryTYHC N M H Gsses Their Appl. ,1984 ,18:9. Kong ,F. C. ,The increments of a two parameter Gaussian process ,J. of Anhui University( in Chinese),1989 ,10 :8- -18.10. Zhang ,L. X. ,A note on liminfs for increments of a fractional Brownain motion , Acta Math. Hungar. , 1997 ,76 : 145一154.11. Zhang,L. X. ,Some liminf resuts on increments of fractional Brownian motion , Acta Math. Hungar. ,1996 ,71 :209- -234.