Table Of ContentSOME RESULTS AND PROBLEMS FOR
ANISOTROPIC RANDOM WALKS ON THE
PLANE
EndreCsáki,AntóniaFöldes,andPálRévész
5
1
0
2
n
a
J
3
1 Abstract This is an expository paper on the asymptotic results concerning path
behaviour of the anisotropic random walk on the two-dimensional square lattice
]
R Z2. In recent years Miklós and the authors of the present paper investigated the
P propertiesof thisrandomwalk concerningstrongapproximations,localtimesand
. range.Wegiveasurveyoftheseresultstogetherwithsomefurtherproblems.
h
t
a Keywords: anisotropicrandomwalk;strongapproximation;2-dimensionalWiener
m
process;localtime;lawsoftheiteratedlogarithm.
[
Mathematics Subject Classification (2010): Primary 60F17, 60G50; Secondary
1 60F15,60J10,60J65
v
6
6
9 1 Introduction
2
0
. We consider random walks on the square lattice Z2 with possibly unequal sym-
1
0 metric horizontal and vertical step probabilities, so that these probabilities can
5
1 EndreCsáki
: AlfrédRényiInstituteofMathematics,HungarianAcademyofSciences,
v
Budapest,P.O.B.127,H-1364,Hungary,
i
X e-mail:[email protected]
r AntóniaFöldes
a DepartmentofMathematics,CollegeofStatenIsland,
CUNY,2800VictoryBlvd.,StatenIsland,NewYork10314,U.S.A.
e-mail:[email protected]
PálRévész
InstitutfürStatistikundWahrscheinlichkeitstheorie,
TechnischeUniversitätWien,WiednerHauptstrasse8-10/107
A-1040Vienna,Austria,
e-mail:[email protected]
1
2 EndreCsáki,AntóniaFöldes,andPálRévész
only depend on the value of the vertical coordinate. In particular, if such a ran-
dom walk is situated at a site on the horizontal line y= j Z, then at the next
∈
step it moves with probability p to either vertical neighbor, and with probability
j
1/2 p to either horizontal neighbor. More formally, consider the random walk
j
C(N−)=(C (N),C (N));N=0,1,2,... onZ2 withthetransitionprobabilities
1 2
{ }
P(C(N+1)=(k+1,j)C(N)=(k,j))
|
1
=P(C(N+1)=(k 1,j)C(N)=(k,j))= p ,
j
− | 2−
P(C(N+1)=(k,j+1)C(N)=(k,j))
|
=P(C(N+1)=(k,j 1)C(N)=(k,j))=p , (1.1)
j
− |
for(k,j) Z2,N=0,1,2,...Weassumethroughoutthepaperthat0<p 1/2and
j
∈ ≤
minj Zpj <1/2.UnlessotherwisestatedweassumealsothatC(0)=(0,0).This
mode∈l has a number of physical applications and the topic has a broad literature.
WerefertoSilveretal.[28],Seshadrietal.[26],Shuler[27],Westcott[30],where
certainpropertiesofthisrandomwalkwerestudiedundervariousconditions.Heyde
[14]provedanalmostsureapproximationforC ()underthecondition
2
·
n n
n−1(cid:229) p−j1=2g +o(n−h ), n−1(cid:229) p−1j =2g +o(n−h ) (1.2)
j=1 j=1 −
asn ¥ forsomeconstantsg ,1<g <¥ and1/2<h <¥ .
→
Heydeetal.[16]treatedthecasewhenconditionssimilarto(1.2)areassumedbut
g canbedifferentforthetwopartsof(1.2)andobtainedalmostsureconvergenceto
theso-calledoscillatingBrownianmotion.InHeyde[15]limitingdistributionswere
givenforC()underthecondition(1.2)butwithoutremainder.DenHollander[12]
·
provedstrongapproximationsforC()inthecasewhen p -sarerandomvariables
j
·
withvalues1/4and1/2.RoerdinkandShuler[25]provedsomeasymptoticprop-
erties, including local limit theorems, under certain conditions. For more detailed
historysee[12].
Firstwegiveageneralconstructionanddiscusstheissueofrecurrenceandtran-
sience of this randomwalk. In Section 2 we discuss strong approximationsof the
randomwalk C(N),N=0,1,... .Section3treatsthelocaltimeandinSection4
{ }
someresultsontherangewillbegiven.
1.1 General construction
Suppose that in a probability space we have two independent simple symmetric
randomwalks,i.e.,
S (n),n=0,1,2,..., S (n),n=0,1,2,...,
1 2
Anisotropicrandomwalk 3
whereS (0)=S (0)=0,S()aresumsofi.i.d.randomvariableseachtakingthe
1 2 i
·
values1and 1withprobability1/2.ThelocaltimesofS aredefinedby
i
−
x (j,n)=# 0 k n:S(k)= j , j Z, n=0,1,2,...
i i
{ ≤ ≤ } ∈
Moreover,onthesameprobabilityspacewehaveadoublearrayofindependent
geometricrandomvariables
G(j),i=1,2,..., j Z
i ∈
withdistribution
P(G(j)=k)=2p (1 2p )k, k=0,1,2,...
i j − j
WenowconstructourwalkC(N)asfollows.Wewilltakeallthehorizontalsteps
consecutively from S () and all the vertical steps consecutively from S (). First
1 2
· ·
we will take some horizontalsteps fromS (), thenexactlyone verticalstep from
1
·
S (),thenagainsomehorizontalstepsfromS ()andexactlyoneverticalstepfrom
2 1
· ·
S (),andsoon.Nowweexplainhowtogetthenumberofhorizontalstepsoneach
2
·
occasion. Consider our walk starting from the origin proceedingfirst horizontally
(0) (0)
G steps(notethatG =0ispossiblewithprobability2p ),afterwhichittakes
1 1 0
(1)
exactly one vertical step, arriving either to the level 1 or 1, where it takes G
− 1
( 1)
or G1− horizontalsteps (which mightbe no steps at all) before proceedingwith
anotherverticalstep.Ifthisstepcarriesthewalktothelevel j,thenitwilltakeG(j)
1
(j)
horizontalsteps,ifthisisthefirstvisittolevel j,otherwiseittakesG horizontal
2
steps.Ingeneral,ifwefinishedthek -thverticalstepandarrivedtothelevel j for
thei-thtime,thenitwilltakeG(j) horizontalsteps.
i
In this paper N will denote the number of steps of the walk out of which H
N
denotesthenumberofhorizontalstepsandV =nthenumberofverticalsteps,i.e.,
N
H +V =N.Thenclearly
N N
C(N)=(C (N),C (N))=(S (H ),S (V )).
1 2 1 N 2 N
1.2 Recurrence, transience
Our result on recurrence is a simple application of the celebrated Nash-Williams
theorem [21]. To state this result we need some definitions. Consider a Markov
chain(X,Y,p)withcountablestatespaceX,processYandtransitionprobabilities
p(u,v).Thechainisreversibleifthereexiststrictlypositiveweightsp forallu X
u
∈
suchthat
p p(u,v)=p p(v,u). (1.3)
u v
4 EndreCsáki,AntóniaFöldes,andPálRévész
Ifthechainisreversiblewewilluse
a(u,v):=p p(u,v).
u
Obviously the above defined anisotropic walk is a Markov chain on the state
space X=Z2, with the transition probabilitiesdefined in (1.1). Furthermore, this
Markovchainisreversiblewiththestrictlypositiveweights
1
p (k,j)=
p
j
and
a((k,j),(k,j+1)) =a((k,j),(k,j 1))=1
−
1
a((k,j),(k+1,j)) =a((k,j),(k 1,j))= 1 (1.4)
− 2pj −
(andfornonnearestneighborsitesa(.,.)=0).ThisMarkovchainisalsotimeho-
mogeneous, irreducible, i.e. it is possible to get to any state from any state with
positiveprobability.Theinvariantmeasureisgivenby
1
m (k,j)=p (k,j)= , (k,j) Z2, (1.5)
pj ∈
i.e.,
m (u)=(cid:229) m (v)p(v,u),
v
wherethesummationgoesforthefournearestneighborsofu.
Theorem A (Nash-Williams [21]): Suppose that (X,Y,p) is a reversible Markov
chainandthatX= ¥ L kwhereL karedisjoint.Supposefurtherthatu L kand
a(u,v)>0togetherikm=p0lythatv L k 1 L k L k+1,andthatforeachk∈thesum
(cid:229) a(u,v)<¥ S.Let[L k,L k+∈1]de−notethesetofpairs(u,v)suchthatu L k
S S ∈
u L k,v X
a∈ndv∈ L k+1.TheMarkovchainisrecurrentif
∈
1
¥ −
(cid:229) (cid:229) a(u,v) =¥ . (1.6)
k=0 (u,v) [L k,L k+1]
∈
Toapplythistheorem,letL kbethesetof8klatticepointsonthesquareofwidth
2k, centeredat the origin.Furthermore,let [L k,L k+1] be the set of8k+4 nearest
neighborpairs(edges)betweenL k andL k+1.
Itiseasytoseeby(1.4)thatthesumin(1.6)isequalto
(cid:229)¥ (cid:229) k 1 (cid:229) k −1 (cid:229)¥ (cid:229) k 1 −1
2 1 + 1 = .
k=0 j= k(cid:18)2pj − (cid:19) j= k !! k=0 j= kpj!
− − −
Anisotropicrandomwalk 5
Sowegotthefollowingresult.
Theorem1.1 Theanisotropicwalkisrecurrentif
(cid:229)¥ (cid:229) k 1 −1=¥ . (1.7)
p
k=0 j= k j!
−
Asasimpleconsequence,ifminj Zpj>0,thentheanisotropicwalkisrecurrent.
Itisanintriguingquestionwhe∈thertheconverseofthisstatementistrueaswell.
Thatistosay,isittruethat
Conjecture1.1 If
(cid:229)¥ (cid:229) k 1 −1<¥ , (1.8)
p
k=0 j= k j!
−
thentheanisotropicwalkistransient.
Wecan’tprovethisconjecture,butasomewhatweakerresultistrue.
Theorem1.2 Assumethat
(cid:229) k 1 =Ck1+A+O(k1+A d ), k ¥ (1.9)
−
j= k pj →
−
forsomeC>0,A>0and0<d 1.Thentheanisotropicrandomwalkistransient.
≤
Proof.ConsiderthesimplesymmetricrandomwalkS ()oftheverticalstepsandlet
2
x (, )beitslocaltime,andr ()beitsreturntimetoze·ro.Considertheanisotropic
2 2
· · ·
randomwalkofNsteps,whereN=N(m)isthetimeofm-threturnofS ()tozero,
2
i.e.,letV =r (m). ·
N 2
FirstwegivealowerboundforthenumberofthehorizontalstepsH .
N
Lemma1.Forsmallenoughe >0wehavealmostsurelyforlargeenoughm
H =H m1+(1 e)(A+1). (1.10)
N N(m)≥ −
Proof.Forsimplicityintheproof,wedenotex byx andr byr .Fromthecon-
2 2
structioninSection1.1itcanbeseenthatthenumberofhorizontalstepsuptothe
m-threturntozerobytheverticalcomponentisgivenby
¥ x (j,r (m))
(cid:229) (cid:229) (j)
H = G ,
N i
j= ¥ i=1
−
whereGjareasinSection1.1.Sincer (m) m2 e forsmalle >0andlargeenough
i ≥ −
malmostsurely,itfollowsfromthestabilityoflocaltime(see[23],Theorem11.22,
p.127),thatforanye >0, j m1 e wehave
−
| |≤
6 EndreCsáki,AntóniaFöldes,andPálRévész
(1 e )m x (j,r (m)) (1+e )m
− ≤ ≤
almostsurelyforlargeenoughm.Hence
(1 e)m
(cid:229) −(cid:229) (j)
H G =:U .
N ≥ i m
j m1 e i=1
| |≤ −
WeconsidertheexpectationofU andshowthattheothertermsarenegligible.We
m
have
1 2p
(j) j
EG = − ,
i 2p
j
1 2p
(j) j
VarG = − .
i 4p2
j
Henceby(1.9)weget
EUm=m(1 e ) (cid:229) 1−2pj cm1+(1−e)(A+1),
− j m1 e 2pj ≥
| |≤ −
wherec>0isaconstant.Inwhatfollowsthevalueofsuchacmightchangefrom
linetoline.Wehave
VarUm=m(1−e ) (cid:229) 1−4p22pj.
j m1 e j
| |≤ −
Itfollowsfrom(1.9)that 1 c j1+A d ,hence
2pj ≤ | | −
VarUm cm(1 e ) (cid:229) |j|1+A−d cm1+(1−e)(2+2A−d ).
≤ − j m1 e pj ≤
| |≤ −
ByChebyshevinequality
m3+2A 2e(1+A) (1 e)d
P(U EU m(1 e)(A+2)) c − − − =cm 1 (1 e)d +2e
| m− m|≥ − ≤ m2(1 e)(A+2) − − −
−
which,bychoosinge smallenough,issummable.Hence,asm ¥ ,
→
U =EU +O(m(1 e)(A+2)) a.s.,
m m −
consequently
H U cm1+(1 e)(1+A)
N m −
≥ ≥
almostsurelyforlargem.✷
Lemma2.LetS()beasimplesymmetricrandomwalkandletr(m)beasequence
· b
of integer valued random variables independentof S() and such that r(m) m
almostsurelyforlargemwithb >2.Thenwithsmalle·noughe >0wehave≥
Anisotropicrandomwalk 7
S(r(m)) mb /2 1 e
− −
| |≥
almostsurelyforlargem.
Proof.Fromthelocalcentrallimittheorem
c
P(S(k)= j)
≤ √k
forallk 0and j Zwithanabsoluteconstantc>0.Hence
≥ ∈
S(k) (cid:229)
P | | x = P(S(k)= j) cx,
√k ≤ ≤
(cid:18) (cid:19) j x√k
| |≤
This remains true if k is replaced by a random variable, independentof S(), e.g.
·
k=r(m),i.e.wehave
S(r(m)) 1 1
P | | c ,
r(m) ≤ m1+e !≤ m1+e
p
consequentlybyBorel-CantelliLemma
|S(r(m))|≥ mr1(+me ) ≥mb /2−1−e,
p
almostsurelyforalllargeenoughm.ThiscompletestheproofoftheLemma.✷
Applyingthetwolemmaswithr(m)=H ,weget
N(m)
S (H ) mA/2 eA/2 3e/2=mg
1 N − −
| |≥
with g >0 by choosing e >0 small enough. It follows that for large N, S (H )
1 N
almostsurelycan’tbeequaltozero.
Let
A := j,r (m)< j<r (m+1) suchthat C(j)=(0,0) .
m 2 2
{∃ }
ClearlyA couldonlyhappeniffromr (m)tor (m+1)thewalkonlystepshori-
m 2 2
zontally(ifitmakesoneverticalstepthereturntotheorigincouldonlyhappenafter
oratr (m+1)).ThusbyourlemmasinorderthatA couldhappen,thewalkneeds
2 m
g
tohaveatleastm consecutivestepsonthex-axis,thus
¥ ¥
(cid:229) P(A ) (cid:229) (1/2 p )mg <¥ .
m 0
≤ −
m m
Sotheanisotropicrandomwalkcannotreturntozeroinfinitelyoftenwithprobabil-
ity1,itistransient.Thisprovesthetheorem.✷
8 EndreCsáki,AntóniaFöldes,andPálRévész
2 Strong approximations
In this section we present results concerning strong approximations of the two-
dimensional anisotropic random walks. Of course, the results are different in the
variouscases,andinsomecasestheproblemisopen.We alsomentionweakcon-
vergenceresultsavailableintheliterature.Firstwedescribethegeneralmethodhow
toobtainthesestrongapproximations.
Assumethatouranisotropicrandomwalkisconstructedonaprobabilityspaceas
describedinSection1.1,andinaccordancewithTheorems6.1and10.1ofRévész
[23]wemayassumethatonthesameprobabilityspacetherearealsotwoindepen-
dentstandardWienerprocesses(Brownianmotions)W(),i=1,2withlocaltimes
i
h (, )suchthatforalle >0,asn ¥ ,wehave ·
i
· · →
S(n)=W(n)+O(n1/4+e) a.s.
i i
and
x (j,n)=h (j,n)+O(n1/4+e) a.s.
i i
Then
1/4+e
C (N)=S (H )=W (H )+O(H ) a.s.,
1 1 N 1 N N
and
1/4+e
C (N)=S (V )=W (V )+O(V ) a.s.,
2 2 N 2 N N
ifH ¥ andV ¥ asN ¥ ,almostsurely.
N N
→ → →
SowehavetogivereasonableapproximationstoH andV ,oratleasttooneof
N N
them,anduseV +H =N.
N N
Itturnedoutthatinmanycasestreated,thefollowingisagoodapproximationof
H .
N
HN ∼(cid:229) j x2i(cid:229)(=j1,n)G(ij)∼(cid:229) j x 2(j,n)1−2p2jpj,
withn=V .H andthedoublesumabovearenotnecessarilyequal,sincethelast
N N
geometricvariablemightbetruncatedinH .Sowehavetoinvestigatetheadditive
N
functional
A(n)=(cid:229) f(j)x (j,n)= (cid:229) n f(S (k)), f(j)= 1−2pj
2 2
2p
j k=0 j
oftheverticalcomponentandapproximateitbytheadditivefunctionalofW ()
2
·
¥ t
B(t)= f(x)h (x,t)dx= f(W (s))ds,
2 2
¥ 0
Z− Z
wherebetweenintegerswedefine f(x)= f(j), j x< j+1.
≤
IncertaincasesthefollowingLemma,equivalenttoLemma2.3ofHorváth[17],
givingstrongapproximationofadditivefunctionals,maybeuseful.
Anisotropicrandomwalk 9
LemmaALetg(t)beanon-negativefunctionsuchthatg(t)=g(j), j t< j+1,
for j Zandassumethatg(t) c(t b +1)forsome0<candb 0.≤Then
∈ ≤ | | ≥
n n
(cid:229) g(S (j)) g(W (s))ds =o(nb /2+3/4+e) a.s.
2 2
|j=0 −Z0 |
asn ¥ .
→
Nowletusintroducethenotations
k k
(cid:229) (cid:229)
f(j)=b , f( j)=c
k k
−
j=1 j=1
Thenextassumptionisareasonableoneusedintheliterature:ask ¥
→
bk=(g 1)ka +o(ka −d ) (2.1)
−
c =(g 1)ka +o(ka d ) (2.2)
k −
−
withsomeg 1,a 0andd >0.Observethat(1.2)isaparticularcasewitha =1.
≥ ≥
Weconsiderthefollowingcases:
(1)a =0
(2)0<a <1
(3)a =1
(4)a >1
(5)nonsymmetriccase,i.e.theconstantsg in(2.1)and(2.2)aredifferent.
2.1 Thecasea =0
Themostinterestingandwellestablishedcaseistheso-calledcombstructure,i.e.,
p =1/4, p =1/2, j= 1, 2,....ItfollowsfromTheorem1.1thattherandom
0 j
± ±
walkinthiscaseisrecurrent.WenoteinpassingtheinterestingresultofKrishnapur-
Peres[19]:twoindependentrandomwalksonthecombmeetonlyfinitelyoftenwith
probability1.
For random walk on comb we refer to Weiss and Havlin [29], Bertacchi and
Zucca [2] and references given there. The following result on weak convergence
wasestablishedbyBertacchi[1].
TheoremB
C1(Nt),C2(Nt);t 0 Law(W (h (0,t)),W (t);t 0), N ¥ .
N1/4 N1/2 ≥ −→ 1 2 2 ≥ →
(cid:18) (cid:19)
StrongapproximationwasgiveninCsákietal.[5].
Theorem2.1 Onanappropriateprobabilityspacewehave
10 EndreCsáki,AntóniaFöldes,andPálRévész
N 1/4C (N) W (h (0,N)) +N 1/2C (N) W (N) =O(N 1/8+e) a.s.,
− 1 1 2 − 2 2 −
| − | | − |
asN ¥ .
→
Wehavethefollowingconsequences.
Corollary1.
C (N)
1
limsup =1 a.s.
N ¥ 25/43−3/4N1/4(loglogN)3/4
→
C (N)
2
limsup =1 a.s.
N ¥ (2NloglogN)1/2
→
Forgeneralresultsinthecasea =0wejustremarkthatinthiscase f¯=(cid:229) f(j)
j
is convergent,then byour assumptionsits termsare non-negativeand atleast one
of them is strictly positive, hence f¯>0. By the ratio ergodic theorem (cf., e.g.,
Theorem3.6inRevuz[24])
A(n) f¯x (0,n), f¯=(cid:229) f(j)=2(g 1)+f(0),
2
∼ −
j
almostsurely,asn ¥ ,hence
→
A(n)=O((nloglogn)1/2) a.s.,n ¥ .
→
Let
HN+=(cid:229) x2(cid:229)(j,n)G(ij), HN−=(cid:229) x2((cid:229)j,n)−1G(ij).
j i=1 j i=1
Obviously,H H H+.ConsiderH+,whichisa(random)sumofindependent
randomvariaNb−le≤s.UNnd≤ertNheconditionFN = S (k),k 0 wehave
2
{ ≥ }
E(H+ F)=(cid:229) f(j)x (j,n)=A(n)
N| 2
j
Var(H+ F)=(cid:229) f(j)x (j,n).
N| j 2pj 2
Itiseasytoseethatthesum(cid:229) f(j)/2p isalsoconvergent,hence
j j
Var(H+ F) cx (0,n)
N| ∼
with somec>0.Now applyTheorem6.17in Petrov[22] sayingthatforsumsof
independent(notnecessaryidenticallydistributed)randomvariableswehave
1/2+e
(cid:229) (cid:229) (cid:229)
X = EX +o VarX
i i i
i i i !
