Stabilization Time for a Type of Evolution on Binary Strings

Transcription

1 J Theo Pobab 05 8: DOI 0.007/s y Stabilizatio Time fo a Type of Evolutio o Biay Stigs Jacob Fuk Mihai Nica Michael Noyes Received: 4 Febuay 03 / Revised: 30 Mach 03 / Published olie: Septembe 03 Spige Sciece+Busiess Media New Yok 03 Abstact We coside a type of evolutio o {0, } which occus i discete steps wheeby at each step, we eplace evey occuece of the substig 0 by 0. Afte at most steps, we will each a stig of the fom 00 00, which we will call a stabilized stig, ad we call the umbe of steps equied the stabilizatio time. If we choose each bit of the stig idepedetly to be a with pobability p ad a 0 with pobability p, the the stabilizatio time of a stig i {0, } is a adom vaiable with values i {0,,... }. We study the asymptotic behavio of this adom vaiable as, ad we detemie its limit distibutio i the weak sese afte suitable ceteig ad scalig. Whe p =, the limit distibutio is Gaussia. Whe p =, the limit distibutio is a χ 3 distibutio. We also explicitly compute the limit distibutio i a theshold settig whee p = p vaies with give by p = + λ/ fo λ>0afixed paamete. This aalysis gives ise to a oe paamete family of distibutios that fit betwee a χ 3 ad a Gaussia distibutio. The tools used i ou agumets ae a atual itepetatio of stigs i {0, } as Youg diagams, ad a coectio with the kow distibutio fo the maximal height of a Bowia path o [0, ]. J. Fuk Depatmet of Opeatios Reseach ad Fiacial Egieeig, Piceto Uivesity, Sheed Hall, Chalto Steet, Piceto, NJ 08544, USA jjfuk@piceto.edu M. Nica B Couat Istitute of Mathematical Scieces, 5 Mece Steet, New Yok, NY 00-85, USA ica@cims.yu.edu M. Noyes Depatmet of Mathematics, Bad High School Ealy College, 55 E Housto, New Yok, NY, USA moyes@bhsec.bad.edu 3

2 J Theo Pobab 05 8: Keywods Iteactig paticle systems ad thei scalig limits Weak limit theoems Combiatoial pobability Mathematics Subject Classificatio 00 60F05 60K99 60C05 Itoductio Fo N ad p 0,, letω p deote the pobability space cosistig of stigs i {0, }, whee each bit is chose idepedetly to be a with pobability p o a 0 with pobability p. We coside the followig kid of evolutio fo ω Ω p : Replace evey occuece of the substig 0 with 0. By doig so, ew istaces of 0 may be ceated fo istace, ceates a 0 i the middle. We epeat this pocess util we each a stig of the fom A cocete example: Say we have = 8 ad we stat with the stig ω = 000. The, ou evolutio poduces the followig stigs befoe stabilizig: , ad stabilizes afte five iteatios because thee ae o moe istaces of 0 to be foud. This evolutio has a cute itepetatio as a lie of cofused soldies. Thee is also a way to view the poblem i tems of paticles whose motio is esticted to oe dimesio. Imagie that each i the stig is a paticle that would like to move to the left-had side of the stig ad that each 0 is a empty space. At evey iteatio, if a paticle has a empty space to its left, it will move ito that space. This is exactly the eplacemet ule 0 0. Equally well, oe ca thik of the 0 s as the paticles which ae tyig to move as fa ight as they ca ad the s as ope spaces. The pocess will stabilize whe all of the paticles have moved as fa as they ca go. I this guise, we have a kid of detemiistic aalog of cetai exclusio pocesses see fo istace [4, Chapte 8] except i ou case the iitial coditio is adom, but the evolutio is detemiistic. It is elemetay to show that this pocess must stabilize afte at most iteatios ad that aftewad, we will obtai a stig of the fom The umbe of iteatios util we each such a fial cofiguatio is a adom vaiable o the pobability space Ω p. We will call this adom vaiable the stabilizatio time ad Uivesity of Michiga Udegaduate Mathematics Competitio Poblem. umich.edu/~hdekse/umumc/umumc3.pdf. A lage but fiite umbe of soldies ae aaged i a east west lie, ad all the soldies ae facig oth. The commade shouts Right face! Oe secod late, all the soldies ought to be facig east, but they have ot completely masteed ight ad left so some ae facig east ad some west. Ay soldie who is face-to-face with his eighbo ealizes that thee was a mistake ad tus 80 degees disegadig the possibility that the mistake might have bee the eighbo s. Oe secod late, whe all these 80 degee tus have bee completed, ay soldie who is ow face-to-face with a eighbo tus 80 degees eve if he had just tued at the pevious step. The pocess epeats i the same mae. Pove that it stops afte fiitely may steps.. 3

3 850 J Theo Pobab 05 8: deote it by T p : Ω p {0,,..., }. Goig back to ou cocete example above, we have T p = 5. I this pape, we will examie the limit distibutio fo the adom vaiable T p i the limit ad fo vayig values of p. Because of symmety betwee the 0 s ad s i the stig, we will coside oly the case that p. The case p is completely complemetay by the eplacemet p p. Oe of the poits of iteest is the fact that the limit distibutio i the case p = is qualitatively diffeet tha fo p =. Motivated by this, we will also coside the case whee p depeds o ad is give by the followig: p = + λ/, whee λ>0 is a positive paamete. We call this the theshold settig because it is somewhee betwee the case p = ad p =. Theoem We have the followig weak limits fo the distibutio of the adom vaiable T p i the limit : I the case p > : T p p N0, p p. I the case p = : T I the theshold settig p = + λ/ : T p χ 3. ν λ. I the above limits, N 0, p p is a mea zeo Gaussia vaiable with vaiace p p, ad χ 3 Z + Z + Z 3 is half of the Euclidea om of a vecto of thee idepedet stadad N0, Gaussia vaiables. This has desity: d χ 3 x = 8 x e x dx fo x > 0. π Fially, ν λ is a adom vaiable suppoted o the positive eal axis that depeds o the paamete λ, whose desity we fid explicitly: 3 dν λ x = 4 λ π e λ sihλxxe x dx fo x > 0.

4 J Theo Pobab 05 8: Remak The distibutio ν λ aises as the law of max 0 s B λ s Bλ whee Bλ t is a Bowia motio with dift λ. It is kow that the pocess Y t = max 0 s t B 0 s B0 t is a thee-dimesioal Bessel pocess see [5]. Hece, the distibutio ν λ is the distibutio at time t = of oe half times this paticula geealizatio of the thee-dimesioal Bessel pocess. Remak The distibutio ν λ is somewhee betwee a χ 3 ad a Gaussia distibutio. Oe ca easily veify that as λ 0, the desity fuctio fo ν λ coveges poitwise to the desity fuctio fo χ 3. O the othe had, if we examie T p p, we see fom the theoem that T p p = T p λ ν λ λ. The adom vaiable ν λ λ has desity fuctio: 4 λ π e λ sih λ x + λ x + λ e x+ λ fo x > λ/. A easy calculatio shows that as λ, this desity fuctio coveges poitwise to π e x. This is exactly the desity of a N0, /4 adom vaiable. Remak 3 The poof of the theoem comes though coectig seveal ideas fom combiatoics ad pobability theoy. Fist, we otice a atual coectio betwee stigs i Ω p ad Youg diagams. Usig Youg diagams as a tool, we ca aalyze the special case of stigs that begis with a 0 ad ed with a ad elates the stabilizatio time to a simple adom walk i oe dimesio. Usig this coectio to the adom walk, we ca fid the limit distibutio fo the stabilizatio time i the special case metioed above usig the cetal limit theoem ad Doske s theoem. Fially, we show that the limit distibutio fo the special case of stigs is actually the same as the limit distibutio fo geeal stigs. We have divided the poof ito thee lemmas which ae stated below, ad each was discussed ad poved i thei ow sectios. Lemma Give a stig ω {0, } thee is a atual adom walk associated with ω which takes a step up fo evey 0 i ω ad takes a step dow fo evey i ω. To be pecise, fo 0 k lets k = k i= ω i. I the special case whee ω = 0 ad ω =, we have the followig explicit elatioship betwee the adom walk S k ad the stabilizatio time T p ω: T p ω = + max k S k S. Remak 4 The poof of this lemma comes by mappig each stig to a Youg diagam i a atual way. The stabilizatio time of stig tus out to be equal to a quatity called the depth of the Youg diagam. I the special case, ω = 0 ad ω =, the 3

5 85 J Theo Pobab 05 8: depth of the Youg diagam is also foud to be equal to the above expessio fo ou adom walk. The poof of this lemma is discussed i Sect.. Lemma Let Ω p ={ω {0, } : ω = 0,ω = } be the pobability space whee each bit, except fo the fist ad last, is chose idepedetly at adom to be a with pobability p ad to be 0 with pobability p. The stabilizatio time of a stig is a adom vaiable T p : Ω p {,,..., }. The we have covegece of T p aalogous to the statemet i Theoem : T p p/ N0, p p fo p >, T p / χ 3 fo p = ad T p / ν λ i the theshold settig p = p. Remak 5 Oce T p ω = + max k S k S is established i Lemma, the poof of Lemma is a execise usig the cetal limit theoem ad esults about covegece of adom walks to diffusio pocesses. Fo p >, some simple aalysis shows that the tem max k S k does ot cotibute to the limit distibutio, ad the cetal limit theoem is eough to pove the limit. I the settig p =, the adom vaiable χ 3 aises fom a coectio to Bowia motio. By Doske s theoem, we kow that whe scaled coectly the adom walk S k coveges to a Bowia motio. The χ 3 aises fom the followig fact: If B t is a Bowia motio ad M t = max s t B s is its uig maximum, the M B = d χ 3. I the theshold settig, we aalogously obseve that the adom walk S k coveges to a Bowia motio, but this time with costat positive dift λ. We the use the Gisaov theoem to calculate M λ Bλ d = ν λ whe Bt λ is a Bowia motio with dift λ. The full details of the poof of this lemma ae displayed i Sect. 3. Lemma 3 Let μ p be the law of the special case adom vaiable T p ad let μ p be the law of the adom vaiable T p. Let δ 0 be the uit mass at 0. Fo p = these measues ae elated by: μ = + δ 0 + μ +. =0 Fo p = these measues ae elated by: μ p = p+ p + p δ 0 + =0 pp p p p μ p +. Moeove, fom this elatio, we ca show that the adom vaiable T p will have the same limit distibutio as the special case adom vaiable T p. The same agumet holds i the theshold settig p = p. Remak 6 The fomulas stated i Lemma 3 ae obtaied by caefully coutig the umbe of ways of addig leadig s o the left ad tailig 0 s o the ight of a stig of special type ω Ω p, ad by exploitig the obvious fact that such additios of leadig s ad tailig 0 s does ot affect stabilizatio time. Hece, the stabilizatio time fo 3

6 J Theo Pobab 05 8: a geeal stig ω is equal to the stabilizatio time fo the special case substig ω oe obtais fom ω by deletig ay leadig s o tailig 0 s. This gives the elatioship betwee μ ad μ i the lemma. Oe ca see that this is a covex combiatio of the pobability measues μ, which is heavily weighted towad highe values of. Some elemetay aalysis is used to show that this weightig is such that μ ad μ have the same limit distibutio. The details of this lemma ae give i Sect. 4. Coectio to Radom Walk i the Special Case I this sectio, we aim to pove Lemma which gives the followig explicit fomula fo the stabilizatio time of ω {0, } satisfyig ω = 0 ad ω =. T p ω = + max k S k S. This fomula, oce kow, ca be poved diectly by iductio. The poof by iductio, howeve, obscues the actual mechaics of the evolutio. A moe illumiatig solutio, which is how the fomula was fist discoveed, comes fom mappig biay stigs to Youg diagams i a paticula way. Thikig of paticle systems as Youg diagams o moe geeally as subsets of R is a classical idea i iteactig paticle systems see e.g., [6] o[] fo some famous esults. I most cases, oe obtais a egio of iteest which is gowig as the pocess evolves fowad i time. I ou poblem, howeve, the egio of iteest is shikig as the pocess evolves fowad i time. The stabilizatio time we ae iteested is pecisely the waitig time util this shikig egio vaishes. We will begi with some basic defiitios ad ceate a map fom {0, } to the space of Youg diagams. Defiitio Let Y deote the collectio of all fiite sets Y N N which have the followig popety. i, j Y i, j Y fo all i, j N such that i i ad j j. The empty set is also couted i Y. It is customay to epeset a set Y Y as a pictue cotaiig a collectio of boxes squae boxes of side, whee fo evey i, j Y, we take i ou pictue the box which has i, j as its top ight coe. I this guise, the set Y is called a Youg diagam, see e.g., [7]. The followig example illustates this epesetatio. Defiitio Fo evey ω {0, } with ω = 0 ad ω =, let U = Uω be the umbe of s i the stig ω. LetB be a U U gid o the x y plae. 3

7 854 J Theo Pobab 05 8: Readig ω fom left to ight, we costuct a path statig fom the top-left coe of B by dawig a lie hoizotally to the ight wheeve we ecoute a 0 i ω ad a lie vetically dowwad wheeve we ecoute a i ω. Sice thee ae U s ad U 0 s to be foud i ω we will get a path fom the top-left coe i B to the bottom-ight coe i B. Hee is a example of the path geeated by the stig ω = 0,,, 0,, 0,, : Defie πω N N to be the uique path costucted i this mae. The set of boxes ude this path defies a Youg diagam, which we deote Y ω. I ou example, this is the followig diagam. Defiitio 3 Let Y Y be a Youg diagam ad let i, j Y. We say that i, j is a exposed coe of Y if i +, j / Y ad i, j + / Y.Thecoe cuttig map K : Y Y is the map that emoves all the exposed coes of a Youg diagam as follows. K Y := Y \ {i, j Y i, j is a exposed coe of Y }. It is easily veified by the defiitio that emovig the exposed coes does ideed yield aothe Youg diagam. That is to say, K is a well-defied fuctio fom Y Y. Below is a example of a Youg diagam with its exposed coes shaded. The Youg diagam K Y cosists of the ushaded squaes. Popositio Let ω {0, } be a stig ad let ω be the stig obtaied fom ω afte oe stage of the evolutio, that is afte eplacig istaces of 0 with 0 oce. The the Youg diagam Y ω is obtaied by cuttig the coes of the Youg 3

8 J Theo Pobab 05 8: diagam Y ω. Y ω = K Y ω. Poof Coside the path πω defied i Defiitio. Ay istaces of 0 i ω will coespod to a hoizotal segmet followed by a vetical segmet, while istaces of 0 coespod to a vetical segmet followed by a hoizotal segmet. As such, the evolutio 0 0 will taslate ito the followig pictoial evolutio fo the path πω. These istaces of 0 coespod exactly to the exposed coes of Y ω sice they have o eighbos above them o to thei ight. We theefoe see that the substitutio ule 0 0 amouts pecisely to emovig the exposed coes of Y ω. Defiitio 4 Let Y Y, Y = be a o-empty Youg diagam. We defie: DepthY = max {i + j i, j Y }. We also set the covetio Depth = 0. I ou illustative example, ω = 0,,, 0,, 0,,, oe ca check that DepthY ω = 5. The maximum i + j is achieved at the top most exposed coe of this Youg diagam, amely, 5 Y ω. Fo a geeal Youg diagam, the maximum is always achieved at oe of the exposed coes. Hece, the coe cuttig map K always educes the depth of a Youg diagam by at least. This is the ituitio behid the followig popositio: Popositio Let Y Y, Y =. The DepthK Y = DepthY. Poof Suppose i, j Y such that i + j = DepthY. Clealy i +, j / Y ad i, j + / Y, sice assumig othewise cotadicts i + j = DepthY is maximal. The i, j is a exposed coe, so i, j / K Y. Hece DepthK Y < DepthY. Now, if eithe i, j K Y o i, j K Y the DepthK Y i + j = DepthY. I this case, we coclude DepthK Y = DepthY. Othewise, both i, j / K Y ad i, j / K Y. But this oly happes i the case Y ={, }, i which case K Y = ad the popositio holds by Depth = 0. Popositio 3 Fo evey ω {0, } we have that: T p ω = Depth Y ω. As i the itoductio, T p ω deotes the stabilizatio time of the stig ω. 3

9 856 J Theo Pobab 05 8: Poof The poof follows by iductio o DepthY ω usig the pevious two popositios. Fo the base case, if DepthY ω = 0, the Y = so ω is aleady stable, ad the stabilizatio time is 0 ad the esult holds. Now, assume that fo all ω with DepthY ω = k, T p ω = k. Give ay ω with DepthY ω = k, let ω be the evolutio of ω by oe step. The, by the last two popositios, we have that DepthY ω = DepthK Y ω = DepthY ω = k, ad so by the iductio hypothesis, T p ω = k. Sice we have applied oe step to get fom ω to ω, T p ω = T p ω + = k. Popositio 4 Let ω {0, }, ad fo 0 k lets k = k i= ω i. Let U = Uω be the umbe of s i the stig ω. I the case that ω = 0 ad ω = we have the followig explicit elatioship betwee S k ad the depth of the Youg diagam Y ω: DepthY ω = U + max 0 k S k Poof Let πω be the path associated with ω i Defiitio. A elemetay computatio shows that S is elated to πω by the followig fomula: {k, S k : 0 k } = {i + U j, i U j : i, j πω}. This fomula meely says that because of the specifics of how each of S ad πω ae costucted fom ω the set of lattice poits Sω := {k, S k : 0 k } is obtaied out of πω via 45 otatio ad dilatio by. The veificatio of the fomula is left as execise to the eade. A illustatio of how πω ad Sω look i a cocete case is show i the ext pictue, daw fo ω = 0, 0,,, 0,,,, 0, {0, } 0. Now, if ω = 0 ad ω =, we kow that the bouday of the Youg diagam Y ω is pecisely the path πω The fact that ω = 0 ad ω = is eeded hee because othewise thee ae some poits i πω which ae ot icluded i Y ω. Sice the DepthY ω is achieved somewhee o its bouday, we have that: DepthY ω = max {i + j : i, j Y ω} = max {i + j : i, j πω} = U + max {i U j : i, j πω} = U + max {S k : 0 k }, whee the last equality follows fom the map betwee S ad πω descibed above. 3

10 J Theo Pobab 05 8: Poof of Lemma This follows immediately whe we combie Popositio 3 with Popositio 4 ad also use the elemetay elatio U = S. 3 Limit Distibutio i the Special Case I this sectio, we aim to pove the afoemetioed weak limits fo the adom vaiable T p, the stabilizatio time fo stigs fom the pobability space Ω p ={ω {0, } : ω = 0,ω = }. We will use the esult fom Lemma coectig T p to the adom walk associated with ω, S k = k i= ω i. Lemma 4 Let X,...,X be i.i.d adom vaiables which take the value with pobability p ad with pobability p. Fo 0 k,letw k = k i= X i be the adom walk which takes the X i s as its steps. The: T p d = + max 0 k W k W. d Poof Make the idetificatio that X i = ωi+. Sice ω = 0, ω =, we d have that S k+ = + Wk fo 0 k ad S = + W. Fom this, it is clea that max k S k = + max 0 k W k, ad the esult is the immediate fom Lemma. Sice W k is the sum of may i.i.d. adom vaiables, we ae i a positio to use tools like the cetal limit theoem ad Doske s theoem to fid the limit distibutio. We divide the emaiig esults ito ou thee settigs, whe p >, whe p =, ad the theshold settig p = + λ/. Popositio 5 Fo p >, we have that, as : max 0 k W k P 0. Poof max k N W k is the maximum height achieved at ay time by a weighted adom walk, which takes steps upwad with pobability p < ad steps dowwad with pobability p >. It is a esult fom elemetay pobability that this maximum is distibuted like a geometic adom vaiable elated to the paamete q = 4p p d p <, amely max W k = Geom q. Sice this adom vaiable is fiite almost suely, the esult of the popositio is immediate fom Makov s iequality. Lemma 5 I the settig p >, T p has the followig weak covegece to a Gaussia as : T p p N0, p p. 3

11 858 J Theo Pobab 05 8: Poof We have, usig the esults of Lemma 4 ad Popositio 5, that: T p p = T p p p max d 0 k W k W = = max 0 k W k W p 0 + N0, p p. Sice max 0 k W k 0 by Popositio 5 ad sice W is the sum of the i.i.d. adom vaiables X i with mea p ad vaiace 4p p, so by the cetal limit theoem, we have weak covegece to a Gaussia: W p N0, 4p p. Lemma 6 Let B t, t [0, ] be a Bowia motio ad let M t = max s t B s be its uig maximum. The, i the case that p = we have the followig weak limit: T M B. d Poof We kow that T p = + max 0 k W k W, so it suffices by eplacig with + to show that: max 0 k W k W M B. This is a diect applicatio of Doske s theoem, which stipulates weak covegece of the adom walk to a Bowia motio whe teated as a piecewise liea fuctio ude the coect scalig. I paticula, this says that whe X k = ± with pobability, the the adom walk L : [0, ] R by [ L t = t ] k= X k + t t X t + has L B as i the sese of weak covegece o C[0, ] see e.g., Sectio 8 of []. This is exactly the settig we ae i with L k = ki= X i = W k.now,let h : C[0, ] R by h f = sup t [0,] f t f. This is a cotiuous fuctio o C[0, ] with the sup om, ad so it espects weak limits. The covegece max 0 k W k W = hl hb = M B is exactly what is desied. Lemma 7 Let B λ t, t [0, ] be a Bowia motio with dift λ ad let Mλ t = max s t B λ s be its uig maximum. The, i the theshold settig that p = + λ/ 3

12 J Theo Pobab 05 8: we have the followig weak limit: T p M λ Bλ. Poof Followig the agumet of the pevious lemma, it suffices to show that the piecewise liea adom walk L :[0, ] R defied by the followig has the popety that L B λ as : L t = t k= X k + t t X t + This is a execise i the theoy of diffusio pocesses. I the theshold settig, X j = with pobability + λ/ ad X j = with pobability λ/ so Doske s theoem does ot diectly apply. Istead, oe ca examie the geeato of this Makov chai, A f x = f y f x Π x, dy whee Π x, is the pobability desity of L t + give that L t = x. I ou case, sice L t + = L t + X k,we see fom the distibutio of X k that: A f x = + λ/ f x + + λ/ f x f x. It is the easily veified usig the defiitio of a deivative that A f x f x + λf x as. This is exactly the geeato fo a Bowia motio with dift λ! This is eough to coclude that L B λ,see[8, Sectio.]. Lemma 8 M B = d χ 3 with pobability desity fuctio: d χ 3 = 8 x exp x dx fo x > 0. π Poof We veify this by computig the desity of M B. This is just a computatio usig the joit desity fo Bowia motio ad its maximum, which is eadily calculated usig the eflectio piciple see fo istace [3, p. 95]. The joit desity fuctio fo Bowia motio ad its maximum is as follows: ρ M T = b, B T = a = b a πt 3 exp. b a T fo b > a, b > 0, ad 0, othewise. Now, to get the desity fuctio fo M B, oe just itegates the joit desity fo B t ad M t alog a lie: ρ M B = x = x x ρ M = y + x, B = y dy, 3

13 860 J Theo Pobab 05 8: = π x x y + x y y + x y exp dy = 8 x exp x. π Lemma 9 M λ Bλ d = ν λ with pobability desity fuctio: dν λ x = 4 λ π e λ sihλxxe x dxfox> 0. Poof By the Gisaov theoem, the Bowia motio with dift Bt λ is absolutely cotiuous with espect to the dift fee Bowia motio B t, ad the measues ae elated by the likelihood fuctio exp λb λ see fo istace [3, p. 90]. Sice this likelihood fuctio depeds oly o the fial positio B, the calculatio poceeds i exactly the same way as i Lemma 8, with this additioal facto ude the itegal sig: ρ M λ Bλ = x = x x = π ρ M = y + x, B = y exp λy λ dy x x x exp x exp λy λ dy = 4 λ π e λ sihλxxe x Poof of Lemma I the settig p >, the equied covegece follows fom Lemma 5. Likewise, the case p = is coveed by Lemmas 6 ad 8, while the theshold settig p = + λ/ is coveed by Lemmas 7 ad 9. 4 Limit Distibutio i the Geeal Case So fa, we have esults i the special case that the stig ω has ω = 0 ad ω =. I this sectio, we will bootstap off these esults to see that we have the same limit i the geeal case whee thee is o estictio o ω o ω. 3

14 J Theo Pobab 05 8: Lemma 0 Let μ p be the law of the special case adom vaiable T p ad let μ p be the law of the adom vaiable T p.fop = the measues ae elated by: μ p = p+ p + p δ 0 + =0 pp p p p μ p +. Poof We begi by splittig the space {0, } ito disjoit subsets. Fo stable ω {0, }, all s i ω must lie to the left of all 0 s i ω. Cosequetly, thee ae + stable stigs i {0, } fo which the time to stabilizatio is zeo. Fo each i, thee is pecisely oe such stig with exactly i s. These stigs cotibute the followig value to μ p i=0 p i p i δ 0 = p+ p + δ 0. p Fo o-stable ω {0, }, we will itoduce, the umbe of elemets of ω lyig betwee the fist 0 ad the last, so that we may wite ω =,...,, 0, x,...,x,, 0,...,0, with x {0, }. The time util stabilizatio of ω is the time util stabilizatio of 0, x, x...x,. Fo give, the distibutio of these times is pecisely distibuted like T p + because this stig is i the special case. These stigs cotibute the followig value to μ p fo a give : i=0 p i+ p i μ p + = pp p p p By summig ove all possible values of, we get the desied esult. Lemma Whe p =, we have the followig: μ p + μ = + δ 0 + μ +. =0 Poof The poof is the same as above. The oly diffeece is that i the case p = the sums we eed to evaluate ae aithmetic sums, istead of geometic s. Defiitio 5 We itoduce the otatio C,, p a set of positive coefficiets defied by the esults of Lemmas 0 ad, so that the followig holds tue fo all values of p: μ p = C p 0, δ 0 + = C p,. μ p. The easo fo doig this is to uify the otatio fom the cases p = ad p =. To pove the esult we ae afte, we oly eed thee popeties of the coefficiets C,. p These ae pove i the ext lemma. 3

15 86 J Theo Pobab 05 8: Lemma Fo each, let θ = 4. I fact, ay θ = α whee 0 <α< will wok. Give θ as defied above, we have the followig popeties of C p,:. C p 0, + = θ. C p 0, + 3. = = θ + C p, =. C, p 0. C p,. The same popeties still hold i the theshold settig whee p = p. Poof Popety holds sice the C,s p aose i Lemmas 0 ad as the divisio of the pobability space Ω p ito disjoit pieces. Oce is established, ad 3 ae equivalet. To establish, we use the defiitios of C, p which ae foud i Lemmas 0 ad. Fop >, we have the followig: θ C p 0, + = C, p = p+ p + θ pp + p p + + p p = p+ p + p θ p + p = = pθ + 0. p The last lie holds sice θ as. This estimate also woks to see that C p 0, + θ C p, 0as i the theshold settig p = = + λ/.ithis θ + p case, p is a idetemiat 0/0 limit. Elemetay methods, howeve, show that the limit is zeo whe θ is a powe of as we have chose hee. Fially, whe p =, we have: θ C p 0, + = C, p = + θ + = + + = + + θ + θ Popositio 6 The adom vaiable T p coveges i the same way that T p coveges as established i Lemma. That is T p p/ N0, p p fo p >,T p / χ 3 fo p = ad T p / ν λ i the theshold settig whee p = p. 3

16 J Theo Pobab 05 8: Poof The poof of the thee sepaate settigs, p =, p =, ad the theshold settig p = p is all hadled by the same agumet. To be cocete, we will focus o the theshold settig ad show T p / ν λ. The agumet poceeds i the same way fo all thee settigs because Lemma holds fo all thee settigs. We use the followig chaacteizatio of weak covegece fo distibutios ρ : ρ ρ if ad oly if ρ, a] ρ, a] fo evey a R with ρ{a} = 0 see e.g., [, Example.3, p. 8]. Sice ν λ has o atoms, ou aim is to show that fo evey a R: p T P, a] ν λ, a] 0. Fo the emaide of the poof, fix a abitay a R ad fo coveiece deote A =, a]. We poceed by dividig this ito thee tems usig the tiagle iequality, the idetity μ p = C p 0, δ 0 + = C, p μ p ad the idetity A+ = A+ a +, a + ] which holds fo evey. Wehavethe followig: p T P μ A ν λ A = p A+ ν λ A θ C p 0, δ A+ 0 + C p, μ p + C p, μ p A+ ν λ A θ + + C p, μ p a +, a + ]. θ + We ow show that each tem idividually goes to zeo as. Each tem is hadled i a sepaate claim. Claim C p 0, δ 0 + θ = C p, μ p A+ 0. Poof This is immediate fom the secod popety i Lemma : θ C p 0, δ A+ 0 + C p, μ p θ C p 0, + C p, = = 0. 3

17 864 J Theo Pobab 05 8: Claim C p = θ +, μ p A+ ν λ A 0. Poof This follows fom the esults of Lemma ad fom the weak covegece fo p T T, amely ν λ.fixɛ>0. We use the weak covegece of T p applied to the set A to fid a N N so lage so that p μ A+ ν λa <ɛfo all > N θ N +. The, fo all > N, wehavethat: θ + C p, μ p A+ ν λ A θ + + C p, μ p θ C p 0, + C p, θ + A+ ν λ A C p, ɛ+ ɛ + 0 ν λ A = ɛ, ν λ A θ C p 0, + C p, ν λ A whee the limits ae fom Lemma. Sice ɛ abitay, we have the esult of the claim. Claim C p = θ +, μ p a +, a + ] 0. Poof Fix a ɛ>0. Sice the measue ν λ has o atoms, choose a δ>0sosmallso p T that ν λ a, a + δ] <ɛ. Now, sice ν λ, choose N so lage so that fo all > N θ N +, we have P T p / a, a + δ] ν λ a, a + δ] <ɛ. Now, choose M so lage, so that the followig holds wheeve > M ad θ + : 3 a + <δ.

18 J Theo Pobab 05 8: Such a M always exists because of the followig: a + a θ + θ θ a θ + θ 0 sice θ 0. Fo lage tha both N ad M, we will have the fo evey with θ + that: μ p a +, a + ] T p = P a, a + ] T p P a, a + δ] ν λ a, a + δ] + ɛ ɛ. Hece, fo such, we have the followig: = θ + C p, μ p a +, a + ] ɛ. Ackowledgmets The fist autho s eseach is suppoted by a Udegaduate Studet Reseach Awad povided by the Natual Scieces ad Egieeig Reseach Coucil of Caada. The secod autho s eseach is suppoted by the MacCacke fellowship fom New Yok Uivesity. The secod autho is also idebted to Pofesso S. R. S. Vaadha fo helpful discussios that led to the developmet ad poof of the calculatio i the theshold settig. Refeeces. Billigsley, P.: Covegece of Pobability Measues. Wiley, Lodo 999. Johasso, K.: Shape fluctuatios ad adom matices. Commu. Math. Phys. 09, Kaatzas, I., Sheve, S.E.: Bowia Motio ad Stochastic Calculus. Spige, Beli Liggett, T.M.: Iteactig Paticle Systems. Spige, Beli Pitma, J.W.: Oe-dimesioal Bowia motio ad the thee-dimesioal Bessel pocess. Adv. Appl. Pobab. 7, Rost, H.: No-equilibium behaviou of a may paticle pocess: desity pofile ad local equilibia. Zeitschift f Wahscheilichkeitstheoie ud Vewadte Gebiete 58, Staley, R.P.: Eumeative Combiatoics, vol.. Cambidge Uivesity Pess, Cambidge Stoock, D.W., Vaadha, S.R.S.: Multidimesioal Diffusio Pocesses. Spige, Beli 979 3