On the limiting law of the length of the longest common and increasing subsequences in random words

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1 O the ltg law of the legth of the logest coo ad ceasg subsequeces ado wods axv: v [ath.pr] 0 Sep 06 Jea-Chstophe Beto IRMAR, UMR 665, Uvesté de Rees, 63 Aveue du Gééal Leclec CS 7405, 3504, Rees, Face Chsta Houdé School of Matheatcs, Geoga Isttute of Techology, Atlata, GA 3033, USA À la Méoe de Mac Yo Abstact Let X = X ad Y = Y be two sequeces of depedet ad detcally dstbuted d ado vaables takg the values, ufoly, a coo totally odeed fte alphabet. Let LCI be the legth of the logest coo ad weakly ceasg subsequece of X X ad Y Y. As gows wthout boud, ad whe popely ceteed ad scaled, LCI s show to covege, dstbuto, towads a Bowa fuctoal that we detfy. Keywods: Logest Coo Subsequece, Logest Iceasg Subsequece, Rado Wods, Rado Matces, Doske s Theoe, Optal Alget, Last Passage Pecolato. Matheatcs Subect Classfcato 00: 05A05, 60C05, 60F05. Itoducto We aalyze below the asyptotc behavo of the legth of the logest coo subsequece ado wods wth a addtoal weakly ceasg equeet. Although t has bee studed fo a algothc pot of vew copute scece, bo-foatcs, o statstcal physcs see, fo stace, [CZFYZ], [DKFPWS] o [Sak], to ae but a few felds, atheatcal esults fo ths hybd poble ae vey spase. To peset ou Fal veso to appea Stochastc Pocesses ad The Applcatos E-al: ea-chstophe.beto@uv-ees.f E-al: houde@ath.gatech.edu

2 faewok, let X = X ad Y = Y be two fte sequeces whose coodates take the values A = {α < α < < α }, a fte totally odeed alphabet of cadalty. Next, LCI, the legth of the logest coo ad weakly ceasg subsequeces of the wods X X ad Y Y s the axal tege k {,..., }, such that thee exst < < k ad < < k, satsfyg the followg two codtos: X s = Y s, fo all s =,,..., k, X X X k ad Y Y Y k. Asyptotcally, the stctly ceasg case s of lttle teest, havg as a potwse ltg behavo. LCI s a easue of the slaty/dsslaty of the ado wods ofte used patte atchg, ad ts asyptotc behavo s the pupose of ou study. Ths ltg behavo dffes fo the oe of aothe bette-kow, easue of slaty/dsslaty, aely, LC, the legth of the logest coo subsequeces of two o oe ado wods. Ideed, afte eoalzato, the fst esult o LC, obtaed [HI], eveals, ude a sublea vaace lowe boud assupto, a oal ltg law. I cotast, fo LCI, we have: Theoe. Let X = X ad Y = Y be two sequeces of d ado vaables ufoly dstbuted o A = {α < α < < α }, a totally odeed fte alphabet of cadalty. Let LCI be the legth of the logest coo ad ceasg subsequeces of X X ad Y Y. The, as +, LCI / = ax / 0=t 0 t t t = B + B + B t B t, B t B t,. whee B ad B ae two -desoal stadad Bowa otos o [0, ]. The a otvato fo ou wok has ts ogs the detfcato, fst obtaed by Keov [Ke], of the ltg legth popely ceteed ad scaled of the logest ceasg subsequece of a ado wod, as the axal egevalue of a ceta Gaussa ado atx. Whe cobed wth esults of Bayshkov [Ba] o Gave, Tacy ad Wdo [GTW] see also [BGH], ths ltg law has a epesetato as a Bowa fuctoal. Moeove, the logest ceasg subsequece coespods to the fst ow of the RSK Youg dagas assocated wth the ado wod ad [Ke, Chap. 3, Sec. 3.4, Theoe ] showed that the whole oalzed ltg shape of these RSK Youg dagas s the spectu of the taceless Gaussa Utay Eseble GUE. Sce the legth of the top ow of the dagas s the legth of the logest ceasg subsequece of the ado wod, the axal egevalue esult s ecoveed. The asyptotc legth

3 esult was edscoveed by Tacy ad Wdo [TW] ad the asyptotc shape oe by Johasso [Joh]. Extesos to o-ufo lettes wee also obtaed by Its, Tacy ad Wdo [ITW, ITW]. Aothe otvato fo the peset study coes fo the tepetato of the LCI fuctoal tes of last passage te dected pecolato. Ths s detalled ou cocludg eaks. The asyptotc behavo of the legth of the logest coo ad ceasg subsequeces has actually aleady bee vestgated fo bay wods = [HLM]. Howeve, the ethods used thee do ot allow to cosde a alphabet of abtay fte sze. Whe = wth lettes α ad α, t s eough to cosde coo subsequeces ade of a ado ube of coo α s detestcally copleted by the coo α s, so that a way the coespodg study s educed to deal wth oly oe type of lette. I cotast, whe 3, the stuato s uch oe coplcated sce a sla stategy educed the poble to types of lette fo whch thee s stll, oughly speakg, too uch adoess to successfully hadle, ths way, the study of LCI. A ew ethodology based o a ew epesetato of LCI s thus equed to deal wth geeal fte alphabet of sze. Ths s acheved below whee a appopate epesetato of LCI, that allows to vestgate ts asyptotc behavo fo abtay, s obtaed. Ou esults thus exted ad ecopass the bay LCI esult of [HLM]. The depedece o depedece stuctue betwee the two sequeces of lettes X ad Y s caed ove at the lt to a sla stuctue betwee the two stadad Bowa otos B ad B. Hece, whe X = Y, ou esults ecove, wth the help of [BGH], the weak lts obtaed [Ke], [Joh], [TW], [ITW], [ITW], [HL], ad [HX], whle f X ad Y ae depedet so ae B ad B. As a by-poduct of ou appoach, we futhe fx soe loose pots peset [HLM]. As suggested to us, let us futhe put ou a theoe cotext. At fst, fo =, the ght had-sde of. becoes ax B B B t B t, B B B t B t. 0 t I case the two-desoal stadad Bowa otos ae depedet, ths last expesso has the sae law as ax B t 0 t B, B t B, whee, ow, B ad B ae two depedet oe-desoal stadad Bowa otos o [0, ]. Theefoe, ou ltg esult atches the bay oe peseted [HLM]. Next, ad stll fo futhe cotext, let us copae the asyptotc behavo of LCI to the oe of, say, L, the legth of the optal algets whch alg oly oe type of lettes. I case of a sgle wod, L could coespod to, e.g, the legth of the logest costat subsequeces. Clealy, LCI L ad ude a ufo assupto, LCI l + = l L + =,. 3

4 wth pobablty oe. Moeove, t s easy to see that, as +, L / = / B, B,.3 fo, say, two oe-desoal stadad Bowa otos B ad B. Now etug to., ote that fo =,, = B + B B t B t B + B t B t B t,.4 whee the ado vaable B B / has exactly the sae law as /B. Theefoe, the pesece of the exta tes volvg the t s o the ght had-sde of. allows to dstgush the eoalzed lt of LCI fo that of L ad esues that the latte lt s stll alost suely doated by the foe. Ths obsevato should be cotasted wth the o-ufo case whee a sgle lette s attaed wth axal pobablty p ax, ad whee L algs ths lette. Ideed, vew of 5., below, whe ceteed by p ax ad scaled by p ax, both LCI ad L covege to p ax B, p ax B. A atual questo asg fo ths study s the ado atx tepetato of ou ltg dtbuto.. Aothe atual questo s to tepet LCI tes of RSK Youg dagas ad to vestgate, oe geeally, the shape of a RSK coutepat of LCI. Both questos go actually fa beyod the scope of ths pape but wll be the subect of fothcog vestgatos. As fo the cotet of the pape, the ext secto Secto establshes a pathwse epesetato fo the legth of the logest coo ad ceasg subsequece of the two wods as a ax/ fuctoal. I Secto 3, the pobablstc faewok s tated, the epesetato becoes the axu ove a ado set of the u of ado sus of adoly stopped ado vaables. The vaous ado vaables volved ae studed ad the codtoal laws foud. I Secto 4, the ltg law s obtaed. Ths s doe pat by a deadozato pocedue of the ado sus ad of the ado costats leadg to the Bowa fuctoal. of Theoe.. I the last secto Secto 5, vaous extesos ad geealzatos ae dscussed as well as soe ope questos elated to ths poble. Fally, Appedx A. copletes the poof of soe techcal esults ad whle Appedx A. gves ssg steps the poof of the a theoe [HLM] as well as coectos to aguets peseted thee; povdg, the uch sple bay case, a athe self-cotaed poof. 4

5 Cobatocs The a of ths secto s to obta a pathwse epesetato fo the legth of the logest coo ad ceasg subsequeces of two fte wods. Thoughout the pape, X = X ad Y = Y ae two fte sequeces whose coodates take the values A = {α < α < < α }, a fte totally odeed alphabet of cadalty. Recall ext that LCI s the axal tege k {,..., }, such that thee exst < < k ad < < k, satsfyg the followg two codtos: X s = Y s, fo all s =,,..., k, X X X k ad Y Y Y k. Now that LCI has bee foally defed, let us set soe stadg otato. Let N X, =,...,, be the ube of α s X, X,..., X,.e., N X = # { =,..., : X = α } = {X =α },. ad slaly let N Y, =,...,, be the ube of α s Y, Y,..., Y. Clealy, N X = = N Y =. = Let us futhe set a coveto: Thoughout the pape whe thee s o abguty o whe a popety s vald fo both sequeces X = X ad Y = Y we ofte ot the sybol X o Y ad, e.g., wte N fo ethe N X o N Y o, below, H fo ethe H X o H Y. Cotug o ou otatoal path, fo each =,...,, let N s,t X be the ube of α s X s+, X s+,..., X t,.e., N s,t X = # { = s +,..., t : X = α } = t =s+ wth a sla defto fo N s,t Y. Aga, t s tvally vefed that = N s,t X = = N s,t Y = t s, {X =α },. ad, of couse, N 0, = N. Stll cotug wth ou otatos, let T X, =,...,, be the locato of the th α the fte sequece X, X,..., X,..., wth the coveto that T 0 X = 0. The, fo =,,..., T X ca be defed ecusvely va, T X = { } s N : s > T X, X s = α.3 5

6 whee as usual N = {0,,,...}. Aga eplacg X by Y gves the coespodg oto fo the sequece Y = Y. Next, let us beg ou fdg of a epesetato fo LCI va the ado vaables defed to date. Fst, let H X k, k,..., k be the axal ube of α s cotaed a ceasg subsequece, of X X X, cotag k α s, k α s,..., k α s pcked that ode. Replacg X = X by Y = Y, t s the clea that k + + k + H X k,..., k, k + + k + H Y k,..., k,.4 s, theefoe, the legth of the logest coo ad ceasg subsequece of X X X ad Y Y Y cotag exactly k α s, fo all =,,...,, the lettes beg pcked a ceasg ode. Hece, to fd LCI, the fucto H eeds to be detfed ad.4 eeds to be axzed ove all possble choces of k, k,..., k. Let us stat wth the axzg costats. Assue, fo a whle, that a sgle wod, say, X X, s cosdeed. Fst, ad clealy, 0 k N. Next, k s the ube of α s peset the sequece afte the k th α. Ay lette α s adssble but the oes occug befoe the k th α, attaed at the locato T k. Sce thee ae lettes, cosdeed so fa, thee ae thus N 0,T k adssble α s ad the equeet o k wtes k N N 0,T k. Slaly fo each = 3,...,, k s the ube of lettes α us the adssble α s whch occu dug the ecupeato, of the k α s, followed by the k α s, followed by the k 3 α 3 s, etc that ode. Thus the equeet o k s of the fo k N Ñ, whee Ñ s the ube of α s occug befoe the k α s,, pcked the ode ust descbed. Fo =,, ad as aleady show, Ñ = 0 ad Ñ = N 0,T k. Assue ext that, fo 3, Ñ s well defed, the Ñ s the ube of α s occug befoe, that ode, the k α s,..., the k α s. A lttle oet of eflecto akes t clea that the locato of the k th such α s T k +Ñ, fo whch t ecusvely follows that: Ñ = N k +Ñ 0,T. Reak. Note that Ñ as well as N defed below.8 actually deped o k,..., k, but ode to ot oveload ou otato we wll ot ths depedecy theeafte. Retug to two sequeces X,..., X ad Y,..., Y, the codto o k,, wtes as 0 k N X Ñ X N Y Ñ Y. Fo these choces of dces ad.4, LCI = ax C k + H X k,..., k, 6 k + H Y k,..., k,.5

7 whee the oute axu s take ove k,..., k { C = k,..., k : k C,, k C, k, k 3 C,3 k, k, k C }, k,..., k,.6 whee C {, = 0 k N X Ñ X N Y Ñ Y } ad fo =,...,, C, k,..., k = { 0 k N X Ñ X N Y Ñ Y }..7 Next, obseve that f T k +Ñ >, the N Ñ = 0. Also, sce the above axu does ot chage ude vacuous costats, oe ca eplace the defg costats, Ñ by N ecusvely gve va: N = 0 ad fo =,...,, k +N 0,T The cobatoal expesso.5 the becoes LCI = ax C N = N..8 k + H X k,..., k, k + H Y k,..., k whee the oute axu s take ove k,..., k C wth C ad C,, =,...,, espectvely defed as.6 ad.7 but wth Ñ eplaced by N, =,...,. ad, of couse, N X = N Y =. Afte ths detfcato, ecall that H s the axal ube of α afte, that ode, the k α s, k α s,..., k α s. Coutg the α s peset betwee the vaous locatos of the α, =,...,, ad afte aothe oet of eflecto, t s clea that H = N R, whee R = N +k N T,T =N +,.9 ad whee the N ae gve by.8. Recall also that accodg to Reak., R depeds actually o k,..., k but that fo the sake of eadablty ths depedecy s otted fo ou otatos. Suazg ou esults leads so fa to: Theoe. Let X = X ad Y = Y be two sequeces whose coodates take the values A = {α < α < < α }, a totally odeed fte alphabet of, 7

8 cadalty. Let LCI be the legth of the logest coo ad ceasg subsequeces of X X ad Y Y. The, LCI = ax k + N X RX, k + N Y RY,.0 C whee the oute axu s take ove k,..., k { } C = k,..., k : k C,, k C, k, k 3 C,3 k, k, k C, k,..., k, {. whee C, = 0 k N X N X N Y N Y } ad fo =,...,, ad whee C, k,..., k = { 0 k N X N X N Y N Y },. R = N +k N T,T, =N + wth the vaous N s ad T s gve above by.,.,.3 ad.8. The epesetato.0 has the geat advatage of essetally oly volvg the quattes N, N, =,,..., ad T, =,,...,, =,,..., ad N. 3 Pobablty Let us ow bg ou pobablstc faewok to the pctue by fst studyg the ado vaables N T,T, =,,..., ad =,,... ad the the ado vaables N, =,,...,, appeag R.9. Poposto 3. Let Z be a sequece of d ado vaables wth PZ = α = p, =,...,. Fo each =,,...,, let T 0 = 0, ad let T, =,,... be the locato of the th α the fte sequece Z. Let, {,..., }, wth. The, fo ay =,,..., the codtoal law of N T,T gve T, T, s boal wth paaetes T T ad p / p, whch we deote by B T T, p / p. Moeove, the codtoal law of N T,T gve T =,...,,, T, s ultoal wth paaetes T T ad p / p =,...,,, whch we deote by Mul T T T, p / p =,...,,. Fally, fo each, the ado vaables N,T, ae depedet wth ea p /p ad vaace p /p + p /p ; ad, oeove, they ae detcally dstbuted case the Z, ae ufoly dstbuted. 8

9 Poof. T, T Let us deote by L N T. Recall, see.3, that T,T T ad T the codtoal law of N T, T gve ae the espectve locatos of the th α ad the th α the fte sequece Z. Thus betwee T + ad T, thee ae T T fee spots ad each oe s equally lkely cota α,, wth pobablty p / l= l p l = p / p. Theefoe, L N T,T T, T = B T T, p p,t. 3.,T, Let us ow copute the pobablty geeatg fucto of the ado vaables N T. Fst, va 3. [ E x N T ] [ [,T = E E x N T,T ]], T = = = l= T p + p l x p p l p p p p p p + p p x p p + p p x, 3. sce T s a egatve boal Pascal ado vaable wth paaetes ad p whch we shall deote BN, p the sequel ad T T s a geoetc ado vaables wth paaete p, whch we shall deote Gp. Theefoe, ] E [N T,T = p Va N T,T p, = p p + p p. 3.3 I the ufo case,.e., p = /, =,...,, the N T,T, =,...,,, =,,... ae clealy see to be detcally dstbuted, va 3.. The ultoal pat of the stateet s poved a vey sla ae. The T T fee spots ae to cota the lettes α, {,..., },, wth espectve pobabltes p / p. Theefoe, N T L,T =,..., T, T = Mul T T p,. 3.4 p =,...,, Va 3.4, the pobablty geeatg fucto of the ado vecto N T,T s =,...,, the gve by: [ ] E x N T,T = E E x N T,T T, T =, 9 =

10 p = x p = l= = As a dect cosequece of 3.5 ad fo, s, Cov N T l p p l p =, p x. 3.5,T, N T s,t = p p s. p The poof of the poposto wll be coplete oce, fo each, the ado vaables N T,T,, ae show to be depedet. Fst, ote that gve T, T, T k, T k, the ado vaables N T,T = T l=t + {X l =α } ad N T k,t k = T k l=t k + {X l =α } +, T k ] ae dsot, ad sce the X l l ae also depedet. Moeove, ecall that codtoal dstbutos ae gve by 3., ad so, fo stace, ae depedet sce the tevals [T L N T,T +, T ] ad [T k T, T, T k, T k = L N T = B,T T T T,, T p p Theefoe, fo ay easuable fuctos f, g : R + R +, ad f E B,p deotes the expectato wth espect to a boal B, p dstbuto the E [f N T,T T g N k,t ] k [ = E E [f N T,T T g N k,t ]] k T, T, T k, T k [ = E E [f N T,T ] T, T, T k, T k E [g N T k,t k [ ] = E E B T T, p [f] E p B T k k T, p [g] p [ ] [ ] = E E B T T, p [f] E E p B T k k T, p [g] p = E [f N T,T ] E [g N T k,t ] k, T, T, T k. ]], T k whee the equalty 3.6 s due to the codtoal depedece popety, whle the oe 3.7 follows fo that E B T T, p [f] = F T T ad E p B T k k T, p [g] = G T k T k, p 0

11 fo soe fuctos F, G, ad fo the depedece of T T ad T k T k. The aguet ca the be easly adapted to ustfy the utual depedece of the ado vaables N T,T. Wth the help of the pevous poposto ad ode to pepae ou fst fluctuato esult, t s elevat to ewte the epesetato.0 as LCI = ax C { k + N X G, X, k + N Y G, Y }, 3.8 whee G, = N +k =N + N T,T p p p p + p p p + p + p, 3.9 p p p ad whee p X = PX = α ad p Y = PY = α,. Recall oce oe that G, actually depeds o k,..., k but that, fo the sake of eadablty, ths depedecy s otted fo ou otatos, see Reak.. Va 3.8 ad 3.9, LCI s ow epeseted as a ax/ ove ado costats of ado sus of adoly stopped depedet ado vaables, except fo the pesece of N X ad N Y. Ou ext esult also epesets, up to a sall eo te, both N X ad N Y va the sae ado vaables. Poposto 3. Fo each =,,...,, ad, N T N = p p N + N,T p p p p + p p p + p + S, p p, 3.0 whee l + S, / = 0, pobablty. I patcula, fo each =,,...,, N = p + p + p N p p p N T,T p p p p + p p + p S,. 3. Poof. Let us stat the poof of 3.0 by detfyg the ado vaable S, ad show that, whe scaled by, they covege pobablty to zeo. Clealy, fo =,...,,, 0 S, := N N N T,T.

12 I othe wods, S, s the ube of α the teval [T +, ], whee T s the locato of the last α [, ]. Theefoe, 0 S, T But, PT = k = p p k, k = 0,,..., ad PT fo all ɛ > 0, ad lage eough, P S, ɛ P T ɛ l=[ɛ ] = T N. 3. p p l p [ɛ ] = 0 = p. Hece, Let us cotue wth the poof of 3.. Sug ove =,...,,, both sdes of 3.0, we get p p N = N + + p p N N T,T p p p p p p + p S, p p. p + p p But, N =, ad so 3.4 becoes N = p + p + p p p N N T p,t p p p p + p p + p S,, 3.4 whch s pecsely 3.. Reak 3. Fo all, l + E [ S, / ] = 0. Ideed, E [ S, / ] = P S, x dx p x dx = + 0 p [ x] dx p l p. Retug to the epesetato 3.8, the pevous poposto allows us to ewte LCI as: k LCI = ax p X + k p X C, p X + H,X + p XS, X,

13 p Y + k p Y k p Y + H,Y + whee ottg the depedecy k,..., k see Reak., H, = p + p N p p p p + p p p N +k =N + N T p p,t p p + p p N T,T p p p p + p p p Y S, Y, We ow study soe of the popetes of the ado vaables N whch ae peset both the ado costats ad the ado sus. The ado vaables N ae defed ecusvely by.8 wth N = 0. We fx k = k,..., k whee k s the ube of lettes α peset the coo ceasg subsequeces. The ado vaables N,, deped o k, actually N = N k,..., k. We wte N = N, 3.7 whee N, = N,k s the ube of lettes α peset the step cosstg collectg the k lettes α,. I the sequel, ode ot to futhe bude the otatos, we shall skp the sybols k, =,...,, N ad N,. The followg daga ecapsulates the dawg of the lettes: T k T k +N T k 3 +N T k +N k α N, α k α N3, α 3 N3, α 3 k 3 α 3 k α T k +N... T k +N k α N, α N, α N,3 α... N, α... N, α T k +N I Step, thee ae T k +N T k +N lettes selected but k lettes ae α, N +, ae α +,..., N, ae α, fo =, thee ae also k lettes α but oe of the othes α +, etc. Moeove, thee ae T k +N T k +N k N +, N, possble spots T k +N T k +N k case = whch the pobablty of havg a α s p, := p / p p. Theefoe, codtoally o G, k = σ N +,,..., N,, T k +N, T k +N, 3

14 the σ-feld geeated by N+,,..., N,, T k +N, T k +N t follows that N, B T k +N T k +N k N +, N,, p,. 3.8 The two fothcog popostos espectvely chaacteze the laws of N, ad of N. Poposto 3.3 Fo each =,...,, the pobablty geeatg fucto of N,,, s gve by [ ] k E x N p, =. 3.9 p + p p x Theefoe, N, s dstbuted as k l= G l, whee G l l k ae depedet wth geoetc law G p /p + p ad so, E[N,] = p k ad VaN p, = + p p k. 3.0 p p Poof. Recall that, fo N B, p, E[x N ] = p + px whle, fo N Gp, E[x N ] = px/ px. Usg 3.8, we the have fo N = T k +N k N+, N,, [ ] E x N, settg y = p, + p, x, ad T k +N [ = E E [ x N, N ]] ] = E [ p, + p, x T k +N T k +N k N+, N, ] = E [y U V, 3. U := T k +N T k +N k BN k, p δ k 3. V := N, p B U,, 3.3 p =+ =+ whee fo =, we also set V = 0. The otato BN k, p above stads fo the egatve boal Pascal dstbuto wth paaetes k ad p. The paaetes of the boal ado vaables V 3.3 ste fo that V couts the ube of lettes α, +, betwee two lettes α, whle exactly k such lettes ae obtaed, so that each α has pobablty p / p to appea. Hece, ] E [y U V = E [E [ y U V U ]] = E [y U E [ y V U ]] 4

15 = E [y U =+ [ = E =+ p =+ + p p p y p y + p =+ U ] p p G ] k, sce, fo 3., U k l= G l, whee the G l, l k, ae d wth dstbuto Gp. Fally, ] E [y U V = = p p =+ k, p p + p p x p p y + =+ p p sce p, = p / = p. The expessos fo the expectato ad fo the vaace 3.0 follow fo staghtfowad coputatos. k Recall that by coveto, N = 0, ad fo, the followg poposto gves the law of N : Poposto 3.4 Fo each =,...,, the ado vaables N, ae depedet. Hece, the pobablty geeatg fucto of N s gve by [ ] E x N = k p, 3.4 p + p p x ad so, E[N ] = p p k ad VaN = + p p k. 3.5 p p Poof. I vew of Poposto 3.3 ad of 3.7, t s eough to pove the fst pat of the poposto,.e., to pove that the ado vaables N,,, ae depedet. I ode to splfy otatos, we oly show that N, ad N, ae depedet, but the aguet ca easly be exteded to pove the full depedece popety. Sce the T k s ae stoppg tes, by the stog Makov popety, obseve that σ X,..., X k T σ X k T k T +,..., X T k +N whee, aga σx,..., X deotes the σ- feld geeated by the ado vaables X,..., X, whle stads fo depedece codtoally o T k. Moeove, T k ad σ X T k +,..., X T k +N T k ae depedet, ad thus so 5

16 ae σ X,..., X k T ad σ XT k +,..., X T k +N. The depedece of N, ad N, becoes clea, sce N, s σ X,..., X k T -easuable whle N, s σ X k T +,..., X T k +N - easuable. The whole cocluso of the poposto the follows. 4 The Ufo Case I ths secto, we specalze ous esults to the case whee the lettes ae ufoly daw fo the alphabet,.e., p X = p Y = /, fo all. Hece, the fuctoal LCI 3.5 ewtes as LCI = ax C ad theefoe + H,X + LCI / = ax C S, X, + H,Y + H, X + S, X, S, Y, 4. H, Y + S, Y. 4. The followg sple equalty, a veso of whch s aleady peset [HLM], wll be of ultple use see Appedx A. fo a poof: Lea 4. Let a k, b k, c k, d k, k K, be eals. The, ax ak b k ax ak + c k b k + d k ax c k d k. 4.3 k=,...,k k=,...,k k=,...,k The pevous lea etals ax C H, X + S, X, H,Y + S, Y H, X ax, H,Y C S, X S, Y. But, fo Poposto 3., as +, both S, X/ P 0 ad S, Y / P 0, fo all see 3.3. Theefoe, the fluctuatos of LCI expessed 4. ae the sae as that of H, X ax, H,Y. C 6

17 Fo ufo daws, the fuctoal H, 3.6 ewtes as H, = = N N T,T B N B N +k T N,T =N + N + k N B, whee B s the Bowa appoxato defed fo the ado vaables N T,T,, whch ae d, by Poposto 3., ceteed ad scaled to have vaace oe,.e., B s the polygoal pocess o [0, ] defed by lea tepolato betwee the values k k Z B =, 4.4 whee Z = N T,T. 4.5 Next, we peset soe heustc aguets whch povde the ltg behavo of ax B,X N X N B,X X + k N B,X X C B,Y N Y N B,Y Y + k N B,Y Y, 4.6, kowg that, by Doske theoe, B,..., B C 0[0,] = B,..., B, +, whee B,..., B s a dft-less, -desoal, coelated Bowa oto o [0, ], whch s also zeo at the og. The coelato stuctue of ths ultvaate Bowa oto s gve by that of the Z,, whch tu s gve by Poposto 3.. Above, C 0[0,] = stads fo the covegece law the poduct space of cotuous fucto o [0, ] vashg at the og. Sce the ultvaate Doske theoe s cucal s ou aguet, we gve a pecse stateet: Theoe 4. Doske Let Z be d squae tegable ceteed ado vectos R,, wth covaace atx Σ. Let B t [0,] be the polygoal pocess defed, fo each, by B t = [t] Z k + k= t [t] Z [t]+, t [0, ]. C 0 [0,] The B = B whee B s a Bowa oto o [0, ] wth covaace atx tσ ad whee C 0[0,] = stads fo the covegece law the poduct space of cotuous fucto o [0, ] vashg at the og. 7

18 Poof. The ultvaate Doske theoe easly deves fo the classcal uvaate oe fo whch we efe, fo stace to [Bl, Th. 8.] ad fo the ultvaate CLT as C 0 [0,] follows. Recall that the covegece B = B s equvalet to the covegece of fte-desoal dstbutos of B to that of B ad to the tghtess of B C 0 [0, ]. Fst, the ultvaate CLT gves the covegece of the fte-desoal dstbutos of B t,..., B t 0 t wth a covaace stuctue gve by that of the Z,. Secod, the tghtess of B t,..., B t s obtaed 0 t fo that of ts coodates: sce B s tght fo each by the uvaate Doske theoe, fo all ε > 0, thee s a copact K of C 0 [0, ], the usual space of cotuous fuctos o [0, ] vashg at the og, such that sup P B K < ε ad we have sup P B,..., B K K sup P B K < ε, wth K K copact of C 0 [0, ] so that B,X,..., B,X s tght C 0 [0, ]. Heustcs Roughly speakg, thee ae thee lts to hadle 4.6:. The lt of the costats the axu ove C ;. The lt of the lea tes: B,X ; 3. The lt of the ceets: B,X N X N X+k B,X N X ad, slaly, fo X eplaced by Y. Below, the sybol dcates a heustc eplaceet o a heustc lt, as +. Fst Lt to be teated last, Secto 4.3: Sce C, k,..., k = {k = k,..., k : 0 k N X N X, N Y N Y }, ad, aga, wth vacuous costats case ethe N X > o N Y > ad fo the cocetato popety of the N, we expect wth aga k 0 = 0, ad t 0 = 0, below: { } C, k,..., k k = k,..., k : 0 k E[N X] k E[N Y ] k = { k = k,..., k : k ; } k E[N ], =,...,. 8

19 Hece, fo C defed.: C V,...,, whee Vp,..., p = { t = t,..., t : t 0, =,...,, t p, t + t p,..., t + + t p }. Secod Lt see Secto 4.: Fo each =,...,, the ado vaables N ae cocetated aoud the espectve ea E[N ] = /, ad so N E[N ] whee the lt B ad B N C 0 [0,] === B s take sultaeously. B E[N ] = B Thd Lt see Secto 4.: Fo each =,...,, the ado vaables N ae also cocetated aoud the ea E[N ] = k, ad so N k. Theefoe, B,X N X + k B,X N X B,X B,X k t B,X, B,X k t, ad slaly fo X eplaced by Y. Hece, LCI / ax B,X B,X t B,X t, V/,...,/ B,Y B,Y t B,Y t L = ax B,X B,X u B,X u, 0=u 0 u u B,Y B,Y u B,Y u, by Bowa scalg ad the epaaetzato t = u /, =,..., u 0 = t 0 = 0. I othe wods, LCI / / ax 0=u 0 u u 9 B,X B,X u B,X u,

20 B,Y B,Y u B,Y u. Fally, a lea tasfoato ad Bowa popetes allow to tasfo the paaete space to the Weyl chabe W := { t = t 0, t,..., t, t : 0 = t 0 t t t = }, ad to eplace the -desoal coelated Bowa oto B X esp. B Y, by a -desoal stadad oe B esp. B. Cobg these facts, the expesso o the ght-had sde above, becoes equal, law, to: ax t W B + B t B u, B + B t B u, whch s the fal fo of ou esult, Theoe.. I the sequel, we ake pecse the pevous heustc aguets. All alog, we use dffeet sets costats. Fo easy efeeces, we gathe hee the efeeces to these otatos: C s defed.6, C, k,..., k.7, C., C, k,..., k., C, 4., C 4.3, C, above 4.9, C #, 4.9, C± The Lea Tes Set RX = B,X N + k B,X N, whee aga the depedecy of RX k,..., k s otted see Reak., so that wth the help of 4.6, 4. ewtes as: LCI / = ax C N X RX, B,Y N Y RY + o P, 4.7 B,X whee, thoughout, o P dcates a te, whch ght be dffeet fo a expesso to aothe, covegg to zeo, pobablty, as coveges to fty. 0

21 Next, by Lea 4., ax C ax C ax C ax C ax B,X B,X N X RX, B,X B,Y N Y RY E[N X] RX, B,Y E[N Y ] RY N X RX, B,Y N Y RY B,X B,X E[N X] RX, N X B,X B,Y B,Y E[N X], N Y B,Y E[N Y ] RY E[N Y ]. 4.8 We ow wsh to show that the ght-had sde of 4.8 coveges to zeo, pobablty. Fst ote that fo each, C, k..., k { k = k,..., k : 0 k N X, N Y } { k = k,..., k : 0 k }, ad the sae holds tue fo C,, see.. But, B N / B E[N ]/, whee we have dopped X ad Y, does ot deped o k. Theefoe, the axu ca be skpped ad the poble educes to showg that, fo all : N B B E[N ] P 0, 4.9 as +. Ths follows fo the fothcog lea appled, fo each =,...,, to the ado vaables Z = N T,T /, peset both 4.4 ad 4.5 ad whch, by Poposto 3., ae d wth ea zeo ad vaace oe. Note that the lea below see Appedx A. fo a poof ca deed be bought to play sce Hoeffdg s equalty, appled to the ado vaables N, esues that fo x = l, l P N E[N ] x l + + e x / = Lea 4. Let Z be d ceteed ado vaables wth ut vaace, ad fo each N, let N be a N-valued ado vaable such that l + P N E[N ] x = 0, whee x 0 s such that l + x / = 0. The, [N,E[N ]] Z P 0, whee [N, E[N ]] s shot fo [N, E[N ], axn, E[N ]].

22 At ths stage, 4.9 s poved ad theefoe, LCI / = ax C B,Y B,X E[N X] RX, E[N Y ] RY + o P, 4. fshg the fst pat of the poof of Theoe.. Ideed, N,..., N s ultoal wth paaetes ad p,..., p. So, fo ufo daws, E[N X] = E[N Y ] = p = /. The, by the ultvaate Doske theoe, see Th. 4., ad scalg, B,X E[N X] = B,X, +, 4. whee B,X t,..., B,X t s a -desoal Bowa oto ad 0 t slaly fo Y. As show ext, the covaace atx of ths Bowa oto at te t s tσ = tσ k,l k,l, whee /... / / /. Σ =.... /. 4.3 /... / Ideed, Σ 4.3 s obtaed as follows: Fst, sce B,X,..., B,X C0 [0,] = B,X,..., B,X, whle ufo tegablty see Lea 4.3, below etals l Cov B k,x, B l,x = Cov B k,x, B l,x = σ k,l. + Next, the ufo case, Poposto 3. wtes, fo =,...,, as N = N + B,X N + o P, so that usg also Reak 3. Cov B k,x Nk, B l,x Nl = Cov N N k, N N l + o. 4.4 But N,..., N Mult,,...,, N / /, VaN = / ad whe,, CovN, N = /. Theefoe, Cov N N k, N N l = + =. 4.5

23 Sce by Lea 4.3, l + E [ B k Nk / B k ] / = 0, t follows l Cov + B k,x Nk, B l,x Nl = l Cov B k,x, B + l,x = Cov B k,x, B l,x = B Cov k,x, B l,x. 4.6 Fally, 4.4, 4.5, 4.6 esue the expesso 4.3 fo the covaace. To fsh, let us state a lea, ust used above ad, whose poof s peseted Appedx A.. Lea 4.3 The sequeces B k ufoly tegable ad 4. The Iceets [ l E + N k / ad B k B k Nk /, k =,...,, ae ] B k = I ths secto, we copae the axu of two dffeet quattes ove the sae set of costats ode to splfy the quattes to be axzed befoe splfyg the costats C theselves, the ext secto. The quattes to copae ae: ax k C ad ax k C { { B,X p X B,Y p Y B,X p X B,Y p Y B,X B,Y B,X B,Y N X + k N Y + k k k B,X B,Y B,X B,Y N X N Y k k Usg 4.3 Lea 4., the absolute dffeece s uppe-bouded by { N ax B,X X + k N B,X X k C B,X k B,X k 3 } 4.8, }.4.9

24 ax k C + ax k C N B,Y Y + k N B,Y Y B,Y k B,Y k } { N B,X X + k B,X k N B,Y Y + k B,Y k } { N B,X X B,X k N B,Y Y B,Y k }. Recall that N X = N Y = 0. Hece, fo =, N B,X X + k B,X k = B,X ax k C B,X N X B,X k = 0, wth the sae popety fo fuctoals elatve to Y. Theefoe, we ae left wth vestgatg tes of the fo { N X + k B,X k ad ax k C { B,X N X B,X B,Y k N Y + k B,Y B,Y N Y k }, 4.0 B,Y k }, 4. fo. Above, all the quattes cosdeed oly deped o a sgle sequece, say X o Y, except fo the costats C whch deped o both X ad Y. Howeve, C, k,..., k C,X := { k = k,..., k : 0 k N X N X } 4. esp. C, k,..., k C,Y, ad the sae fo C, ad so uppe-boudg, 4.0 ad 4., the e axa by sus ad the axa ove C by axa ove C X := 4 C,X, 4.3

25 esp. CY, we ae left wth vestgatg, fo, the covegece pobablty of tes of the fo { N ax X + k k C X B,X B,X k }, 4.4 ad ax k C X { B,X N X B,X k }, 4.5 ad, slaly wth X eplaced by Y. Ottg the efeece to ethe X o Y, the tes to cotol ae, fo 4.4 ad fo each,, of the fo: N +k Z ax k C, 4.6 =k + +k + ad ax k C N =k + +k + Z, 4.7 whee the Z,, ae defed 4.5 ad whee C = C,, wth C, = { k = k,..., k : 0 k N N }. I 4.6, 4.7 ad hecefoth, we wte = egadeless of the ode of ad,.e., by coveto ths su s = whe <. Sce 4.7 s sla, but ease to tackle tha 4.6, we oly deal wth 4.6. Aga, as Secto 4., let D = { N E[N ] l } fo =,,...,, ad, thus, fo ε > 0, N +k P ax k C P ax k C =k + +k + Z ε 4.8 Z ε D + P D c. N +k =k + +k + Let C C #, = C = { k E[N ] + l } ad let C #, be the set of dces k,..., k { E[N ] l = k + + k E[N ] + } l N E[N ] 4.9 5

26 whee we set l := k + + k. Sce ude D, C, { k E[N ] + l } ad sce Poposto 3.4, specalzed to the ufo case, gves E[N ] = k, t follows that C C #, ad 4.8 s thus futhe uppe-bouded by l +N k + +k P ax Z ε + P D c k C #, =l + Now, vew of 4.0, t s eough to show the covegece to zeo of the fst te o the ght-had sde of To do so, set E = Ω ad, fo, } Ek,..., k = { N k,..., k E[N k,..., k ] x, wth ad let E = x = l, 4.3 k,...,k C E k,..., k. 4.3 Ou ext goal s to show that asyptotcally, E has full pobablty. Poposto 4. Let, the l + P E c = 0. I ode to pove Poposto 4., we fst eed the followg techcal esult, poved Appedx A.: Lea 4.4 Fo x [, +, let K x = x + x+ x + x+. The, fo soe costats c, C 0, +, x K x C exp c, x We poceed ow to the poof of Poposto 4.: Poof. Pop. 4. Clealy, P E c P Ek,..., k c k,...,k C ax k,...,k C Theefoe, to pove the lea, t s eough to show that: l + ax k,...,k C P E k,..., k c. P E k,..., k c =

27 Now, fo each, Popostos 3.3 ad 3.4 asset that, N = N k,..., k = N,, whee the N, ae depedet ad wth pobablty geeatg fucto Next, E [ ] k x N, =. x P E k,..., k c = P N E[N ] > x = P N, k > x + P The fst te 4.35 s bouded by Θ k + +k x, whee k N, > x Θ kx := exp tx + k + k l e t t> = x + k x+k x + k x+k k, k 4.37 sce the zato 4.36 occus at t = l x + k/x + k. The secod te 4.35 s bouded by Θ l k + +k x, whee Θ l kx := exp tx k + k l e t t> = k x k x k x k x k, k 4.39 obsevg that, fo x k, the zato 4.38 occus at t = l k x/k x. Fo the pevous bouds ad 4.35, t s clea that 4.34 wll follow fo l + ax k,...,k C Θ k + +k x = 0, 4.40 fo {l, }. To obta such as lt, we ake use of Lea 4.4, wth x = x = l, otg also that C { k,..., k : k + + k E[N ] + l } { k,..., k : k + + k ax,..., p + l }. Fst, fo =, whe k + + k x, 4.33 wtes as Θ x k + +k x C exp ck + + k, k + + k 7 x k + + k

28 = C exp cx, so that ax Θ k k + +k x + +k x C e c l 0, +, whee above, ad below, C s a fte postve costat whose value ght chage fo a le to aothe. Fo x k + + k ax,..., p + l = / + l, 4.33 wtes as Θ k + +k x C exp ck + + k so that = C exp c C exp c x k + + k x / + l x k + + k,, x k + + k ax Θ k + +k x l exp c / + 0, l x k + +k /+ l +, guaateeg 4.40 wth =. Next, let = l ad cosde the followg thee cases: k + + k x /, x / k + + k x ad x k + + k ax,..., p + l = / + l. Whe k + + k x /, 4.38 esues that fo all t > 0: Θ l k + +k x exp tk + + k x k + + k l e t exp t x. 4.4 Whe x / k + + k x, 4.38 esues that fo all t > 0: Θ l k + +k x exp tk + + k x k + + k l e t exp x l e t. 4.4 Whe x k + +k /+ l, 4.39 ad 4.33 Lea 4.4 esue that: Θ l x x k + +k x C exp ck + + k, k + + k k + + k x = C exp c k + + k 8

29 C exp l c / + l Gatheg togethe the bouds 4.4, 4.4 ad 4.43 poves 4.40, fo = l. Cobg ths last fact wth the coespodg esult fo =, ad va 4.40 ad 4.34, poves Poposto 4.. Now, thaks to Poposto 4., to pove the covegece to zeo, as +, of the fst te o the ght-had sde of 4.30, t s eough to pove the sae esult fo l P ax +N k + +k Z ε E, 4.44 k C #, =l + whee the Z ae gve 4.5,.e., Z = N T,T /, =,...,,. Ou ext eleetay poposto, the ultate befoe closg ths secto, povdes tal estates o the patal sus of the Z ottg the dces fo a whle. Poposto 4. Let Z be d ado vaables as 4.5. The, fo sutable postve ad fte costats c ad C, all x > 0, ad all postve tege k, k P Z x k P Z x k P Z x k P Z x exp tx + k + k l e t =: Θ kx,4.45 t>0 x x C exp c k,, 4.46 k exp tx k + k l e t =: Θ l kx,4.47 t>0 x x C exp c k,, fo x k k Poof. Recall fo 4.5 that Z = N T,T /,, ad fo 3., Hece, usg the otato 4.36, E [x N T k [ P Z x e tx +k E exp tn T t>0,t ] = x. 4.49,T ] k = Θ kx, 9

30 ad 4.46 follows fo 4.36 ad 4.37 the poof of Poposto 4. wth ts otato ad fo 4.33 Lea 4.4. Slaly, usg the otato 4.38 k k P Z x = P N T,T x t>0 e tx k E [ exp tn T,T ] k = Θ l kx. whch s As pevously obseved va 4.39, whe x k, the zato fo Θ l k x occus at t = l k x/k x, ad, oce aga, 4.33 o Lea 4.4 esue We ae ow eady to ove towads copletg ths secto. Fo ts vey defto 4.9, C #, = C { E[N ] l = k + + k E[N ] + l N E[N ] } { k + + k E[N ] + } l { E[N ] l = k + + k E[N ] + l N E[N ] } { k + + k ax p + } l,..., { E[N ] l = k + + k E[N ] + l N E[N ] }. Theefoe, ecallg also fo 3.5 that E[N ] = k + + k, 4.44 s uppe bouded by: l +N k + +k P ax k + +k /+ Z l k + +k l E[N ]+ l N k + +k =l + ε E l + P ax k + +k /+ ax Z ε ecall 4.3 l x P 3x 3x k + +k l E[N ]+ l +x ax ax l + l E[N ]+ l +x x =l + ax P x l E[N ]+ l +x ax l E[N ]+ l +x 0 x l + =l + =l + Z ε Z ε Θ l ε + Θ ε,

31 whee, the ext to last equalty, we used the usual shap the d case boudg of the axu va the ube of tes tes the axal pobablty; whle the last oe, x was chaged to 0 x = l. Ou fal task s to show that l + x ax Θ 0 x ε = 0, 4.5 fo {l, }. Ths eles aga o Lea 4.4 ad Poposto 4.. Fo =, whe k < ε, 4.45 ad 4.46 etal that, whle, fo ε k x, they etal that, Θ kε C exp cε ; 4.5 Θ kε C exp cε /k C exp cε /x = C exp cε / l Theefoe, fo =, 4.5 follows fo 4.5 ad Let us ow tu ou atteto to = l. Whe ε k x, 4.48 etals that, Θ l kε C exp cε /k C exp cε /x = C exp cε / l Fo k ε /, 4.47 etals that, fo ay t > 0, Θ l kε exp tk ε k l e t exp εt / Fo ε / k ε, 4.47 etals that, fo ay t > 0, Θ l kε exp tk ε / k l e t exp εt / l e t Theefoe, fo = l, 4.5 follows fo 4.54, 4.55, ad Gatheg all the teedate esults, fo ay =,...,, l l P + ax +N k + +k Z ε E = 0, ad theefoe, k C #, l P ax + k C =l + N +k =k + +k + Z ε = 0. The goal of ths secto has thus bee acheved: the quattes 4.8 ad 4.9 have the sae weak lt. 3

32 4.3 The Costats To deal wth the thd heustc lt, we ow eed to obta the covegece of the ado set of costats towads a detestc set of costats. Ths fact wll follow fo the vaous eductos obtaed to date as well as ew aguets developed fo ow o. To stat wth, let us ecall two eleetay facts about covegece dstbuto. The fst fact assets that f f s a sequece of Boel fuctos such that x x ples that f x f x, ad f X s a sequece of ado vaables such that X X, the f X f X. Ideed, va the Skoohod epesetato theoe fo C 0 [0, ]-valued ado vaables, thee exsts a pobablty space ad C 0 [0, ]-valued ado vaables Y,, such that Y L = X,, ad Y Y wth pobablty oe. But, by hypothess, f Y f Y, wth pobablty oe. Theefoe f X f X. The secod eleetay fact s as follows: Let X be a sequece of ado vaables such that X ± Y, the X Y, whee x + = axx, 0 ad x = x, 0. Ideed, PX + x PX x PX x, fo all x R. Usg these two eleetay facts, let us etu to ou deadozato poble. Recallg 4.9, ad usg the polygoal stuctue of the pocesses B X ad B Y, we have M := ax F X B X, k F Y B Y, k, k C whee F X u, t = F Y u, t = u p X u p Y u u t u t, 4.57 t u t, 4.58 fo u = u,..., u C 0 [0, ] ad t = t,..., t [0, ]. Now, let C ± = { k = k : =,...,, 0 k ad wth x = l as 4.3, ad let Sce M ± = ax k C ± N X N X = p k + F X B X, k k p ± x }, 4.59 F Y B Y, k N X E[N X] N X E[N X], 3

33 wth a sla stateet eplacg X by Y, the codto k N X N X N Y N Y, the defto.. of C, wtes as k / p + RX, Y / whee N RX, Y = X E[N X] N X E[N X] N Y E[N Y ] N Y E[N Y ]. 4.6 Now let F := { N E[N ] x } E, wth E defed 4.3. Fo 4.0 ad Poposto 4., we have l + P F c = 0 ad, o F, R X, Y x, fo all. Theefoe, whe F s ealzed, C. s ecapsulated as follows: C C C +, ad Clealy, whee ow { C ± = M ± M M M = ax FX B X, t F Y B Y, t, 4.63 t C ± t = t [0, ] : =,...,, t p ± x } Next, theefoe Slaly, P M x P {M x} F + P F c P {M x} F + P F c l sup + P M x + P F c, PM x l sup PM x P M x = P {M x} F + P {M x} F c P {M + x} F 33

34 ad theefoe P M + x P F c, l f PM x l f P M + x Cobg 4.65 ad 4.66 wth the secod eleetay fact descbed above, ou goal s ow to show that the covegece dstbuto of both M + ad M towads M = ax FX B X, t F Y B Y, t, 4.67 t V holds tue, whee V := Vp,..., p = { t = t [0, ] : =,...,, } t p. To do so, fst ote that by Doske s theoe B X, B Y B X, B Y ad we ow wsh to apply the fst eleetay fact, ecalled above, to the fuctos f ± u, v = ax FX u, t F Y v, t, 4.68 t C ± ad f u, v = ax FX u, t F Y v, t t V Wth these otatos, M ± = f ± B X, B Y ad M = f B X, B Y. I othe wods, we wsh to show that u, v u, v C 0 [0, ] ples that f u, v f u, v. To stat wth, f ± u, v f u, v f ± u, v f ± u, v + f ± u, v f u, v, 4.70 ad we cotue by estatg f ± u, v f ± u, v. But, f ± u, v f ± u, v ax F X u, t F Y v, t F X u, t F Y v, t t C ± ax ax F X u, t F X u, t, F Y v, t F Y v, t t C ± c ax ax t C ± u t ut, v t vt 4.7, 4.7 akg use of Lea , ad by the leaty of both F X ad F Y, wth espect to the fst aguet, 4.7 ad whee, futhe, c s a fte postve costat depedg explctly o. Theefoe, f ± u, v f ± u, v c ax u u, v v, 34

35 ad so f u, v u, v, t follows that f ± u, v f ± u, v 0. I ode to coplete the poof of M ± M ad thus that of M M, let us ow estate the ght-ost expesso At fst, ote that C V C +, hece f u, v f u, v f + u, v Next, va 4.68 ad 4.69, set f + u, v = ax t C + θ u,v t, ad f u, v = ax t V θ u,v t, whee θ u,v t = F X u, t F Y v, t. Sce C C +, fo, t follows as show ext that f u, v ax t C θ u,vt. Ideed, l + f u, v ax t C θ u,vt ad f the pevous equalty wee stct, thee would ow be K 0, + such that ax θ u,v t K < ax t C t θ u,v t. C The left-had sde equalty ples that fo all, ad t C, θ u,v t K, cotadctg the ght-had sde equalty. Sce C + C + +, fo, t also follows that f + u, v ax t C+ θ u,vt. Ideed, we have l + f + u, v ax t C θ u,vt ad f the pevous equalty wee stct, thee would be K 0, + such that ax θ u,v t K > ax t C + t θ u,v t. C+ The left-had sde equalty ples that fo ay, thee exsts t C + wth θ u,v t K. Up to a subsequece t t C+ ad by the cotuty of θ u,v, θ u,v t K, whch s cosstet wth the pevous ght-had sde equalty. Fally, sce C = V, the teo of V, ad sce C+ = V = V, the closue of V, we have l f u, v = ax θ u,vt f u, v = ax θ u,vt = l f + t V + u, v t V + It eas to show that the axu of θ u,v o V s attaed o V fo P B X,B Y -alost all u, v,.e., that P ax θ t V/,...,/ B X,B Y t = ax θ B X,B Y t = t V/,...,/ Wth 4.75, 4.74 etals l + f ± u, v = f u, v fo P B X,B Y -alost all u, v,.e., the ght-ost expesso 4.70 coveges to 0 ad, as pevously explaed, ths gves M ± M ad M M. I ode to coplete 4.75 we atcpate, the secod equalty below, o the esults of Secto 4.4 whch paaetes ae chaged va: s = u, s +s = u,..., s + +s = u ad whee we pove that θb X,B Y t t V/,...,/ L = θb X,B Y s = θb s V,...,,B u, u W 35

36 whee W = {0 = u 0 u u u = }, θ B,B u = B + B u B u 4.76 B + B u B u, ad wth B ad B two depedet, stadad, -desoal Bowa o [0, ]. The popety 4.75 s thus equvalet to P ax θb u W,B u = ax θ B,B u = u W The advatage of 4.77 ove 4.75 s that the foe volves two stadad Bowa otos each oe havg depedet coodates. Roughly speakg, the popety 4.77 should be deved fo the followg obsevato: whe u W, the u k = u k+, fo soe dex k, ad fo such a u, the su tes. Lettg u ε be gve by u ε, = u, k +, ad u ε,k+ = u k + ε, B u B u cotas oly we have = B u ε, B u ε, B u B u + B k+ u k + ε B k+ u k + B k+ u k B k+ u k + ε. The tes B k+ u k + ε B k+ u k ad B k+ u k B k+ u k + ε ae depedet of B u B u ad fo stadad popetes of Bowa oto, alost suely, the su B k+ u k + ε B k+ u k + B k+ u k B k+ u k + ε takes postve value fo abtaly sall ε > 0. Sce the sae s tue fo the secod te 4.76 elatve to B, t follows that the vcty of each u W, thee s u ε W wth θ B,B u ε > θ B,B u. Theefoe, ax θ u W B,B u s attaed W, ad so both 4.77 ad 4.75 hold tue, leadg to M M. 4.4 Fal Step: A Lea Tasfoato By cobg the esults of the pevous thee subsectos, we poved that LCI / ax B,X B,X t B,X t, V/,...,/ 36

37 B,Y B,Y t B,Y t 4.78, whee the axu s take ove t = t,..., t V/,..., /. Now, va the lea tasfoatos of the paaetes gve by s = t, =,...,, s 0 = t 0 = 0, ad Bowa scalg, the ght-had sde of 4.78 becoes equal, law, to: ax B,X B,X s B,X s, 0=s 0 s s B,Y B,Y s B,Y s Next, fo all t [0, ] ad =,...,, let us toduce the followg two potwse lea tasfoatos: B,X t = B t B t, B,Y t = B t B t, whee B ad B ae two, stadad, -desoal Bowa oto o [0, ]. Clealy B,X t,..., B,X t 0 t has the coect covaace atx 4.3, ad slaly fo B, eplacg X by Y. Moeove, B,X = B B,X s B,X s + B B s B s + B s B s = B + B B s + B s B s.4.80 Fally, wth the help of 4.80 ad the coespodg detty fo Y, 4.79 becoes: ax B + B s B s, 0=s 0 s s s = B + B s B s, 4.8 ad the poof of Theoe. s ove. 37

38 5 Cocludg Reaks Let us dscuss below soe potetal extesos to Theoe. ad soe questos we beleve ae of teest. Fo the poof peseted above, the passage fo two to thee o oe sequeces s clea: the u ove two Bowa fuctoals becoes a u ove thee o oe Bowa fuctoals, ad such a passage apples to the cases touched upo below. It s also clea fo the poof developed above, that a theoe fo two sequeces of d o-ufo ado vaables s also vald. Hee s what t should look lke: Let X = X ad Y = Y be two sequeces of d ado vaables wth values A = {α < α < < α }, a totally odeed fte alphabet of cadalty ad wth a coo law,.e., X L = Y. Let p ax = of p ax. The, LCI p ax pax = ax 0=t 0 t t k t k = kpax k kpax k ax PX = α ad let k be the ultplcty,,..., k k B + B + k k B t B t, B t B t,5. whee B ad B ae two k-desoal stadad Bowa otos defed o [0, ]. So, fo stace, f p ax s uquely attaed the the ltg law 5. s the u of two ceteed Gaussa ado vaables. Usg the sadwchg techques developed [HL], a fte coutable alphabet esult ca also be obtaed wth 5.. The loss of depedece sde the sequeces, ad the loss of detcal dstbutos, both wth ad betwee the sequeces s oe challegg. Results fo these stuatos wll be peseted elsewhee. The legth of the logest ceasg subsequece of a ado wod s well kow to have a equvalet tepetato pecolato theoy: Ideed, cosde the followg dected last-passage pecolato odel Z +: let Π, be the set of dected paths Z + fo 0, 0 to, wth ut steps gog ethe Noth o East. Gve ado vaables ω,, 0,, ad tepetg each ω, as the legth of te spet by a path at the vetex,, the last-passage te to, s gve by T, = ax π Π,, π ω,. 5. See Bodeau ad Mat [BM], ad the efeeces thee, fo detals. I ou ado wod cotext, whe X = X s a sequece of d ado vaables takg the values 38

39 a totally odeed fte alphabet { } α < α < < α of sze, takg ω, = {X =α } ad ω 0, = 0,, whch fo each ae depedet ado vaables, the legth of the logest ceasg subsequece of the ado wod s equal to the last passage-te T,, see [BH]. Now LCI, the legth of the logest coo ad ceasg subsequeces, eoys a sla pecolato theoy tepetato, but Z 3 +. Let Π 3,, be the set of paths Z 3 + fo 0, 0, 0 to,, takg ethe ut steps towads the top o steps, of ay legth, the hozotal plae but ethe paallel to the x-axs o to the y-axs,.e., { Π 3,, := u, u,..., u + Z+ 3 + : u = 0, 0,, u + =,,, u + u { 0, 0,, a, b, 0 wth a, b N \ {0} } }, =,, +. Gve weghts ω,,k, 0, 0, k, o the lattce, we ca cosde a quatty aalogous to T, 5., aely, T 3,, := ω,,k. ax π Π 3,,,,k π I the ado wod cotext, takg ω,,k = {X =Y =α k } ad ω 0,0,k = 0, k, as weghts, gves LCI = T 3,,. Note that whe X = Y, T 3,, ecoves T, sce T, s uchaged f, 5., Π, s eplaced by { Π, := u, u,..., u + Z+ + : u = 0,, u + =,, u + u { 0,, a, b wth a, b N \ {0} } }, =,..., +. Moe geeally, fo p 3 sequeces of lettes X l = X l, l p, we ca slaly cosde { Π p+,..., := u, u,..., u + Z p+ +: + u = 0,..., 0,, u + =,...,, u + u { 0,,..., 0,, a,..., a p, 0 wth a N\{0} } }, =,..., +, ad T p,...,, := ax π Π p+,...,,,..., p,k π ω,..., p,k The, obseve that LCI, fo the p sequeces, s equal to T p,...,,, whee ow ω,..., p,k = {X = =X p =α k } ad ω 0,...,0,k = 0, k =,...,, ae depedet ado vaables. I vew of Theoe. ad of [BM], oe would expect that fo fxed ad fo expoetal ea oe d weghts ω.,.,., T 3,, coveges, whe popely ceteed, by, ad scaled, by, towads ax 0=t 0 t t t = B t B t, 39. B t B t,

40 wth also the tval odfcato fo T p. Statg wth Bayshkov [Ba] ad Gave, Tacy ad Wdo [GTW] see, also [BGH], fo a futhe descpto ad up to date efeeces a stog teacto has bee show to exst betwee Bowa fuctoals, ogatg queug theoy wth Gly ad Whtt [GW] see also Seppäläe [Sep], ad axal egevalues of Gaussa ado atces. Lkewse, we hypothesze that the ax/ fuctoals obtaed hee do eoy a sla stog coecto whch ght exted to specta ad Youg dagas. Could t be that the ght-had sde of. wth o wthout the lea tes has the sae law as the axal egevalue of a ado atx odel? Eve the bay case, t would be teestg to fd the law of the pocesses ax 0 t B t B /, B t B / t 0 ad ax 0 t B t, B t t 0 whee, say, B ad B ae two depedet stadad lea Bowa otos. Vey pelay wok o these pobles was stated wth Mac Yo, befoe hs utely death, ad ths text s dedcated to hs eoy. To fsh, ote that the LCIS poble fo two o oe ufo ado peutatos of {,,..., } has ot bee studed ethe, although t cetaly deseves to be. I pot of fact, t s show [HI] that, fo ay two depedet ufo ado peutatos σ ad σ of {,,..., }, ad fo ay x R, PLC σ, σ x = PLI σ x, whee LI σ s the legth of the logest ceasg subsequeces of σ. Theefoe, ths equalty law shows the eegece of the Tacy-Wdo dstbuto, whch had soetes bee speculated, as the coespodg ltg law. Ideed, oce we ae gve the esult of Bak, Deft ad Johasso [BDJ] o the ltg law of LI σ, a coespodg esult actually equvalet to t fo LC σ, σ s edate. I fact, ay of the esults o LI σ peseted Rok [Ro], such as the law of lage ubes of Veshk ad Keov [VK] ae stataeously tasfeable to equvalet vesos fo LC σ, σ. Moeove, fo p 3 depedet ad ufo ado peutatos σ, σ,..., σ p, the ethodology developed [HI] easly shows that LC σ, σ,..., σ p d = LCI σ,..., σ p, whee d = deotes equalty dstbuto. Theefoe, the study of logest coo ad ceasg subsequeces ado wods o ado peutatos whch ght appea, at fst, qute atfcal s actually tately elated to the study of logest coo subsequeces. A Appedx A. Poofs of techcal leas Poof of Lea 4. Fst, ax k=,...,k ak b k axk=,...,k ak + c k b k + d k ax k=,...,k ak b k ak + c k b k + d k. 40

41 Next, the esult wll follows fo the eleetay equalty a b a + c b + d c d, A. whch s vald fo all a, b, c, d R. Ideed, set D = a b a + c b + d ad assue wthout loss of geealty that a b. If a + c b + d, the D = a a + c = c c. If b + d a + c, the D = a b d ad so wheeve a b + d, A. s edate, whle f a b + d, the D = a b d d = d sce a b 0 ad d b a 0. Poof of Lea 4. Let D = { N E[N ] } < x, ad fo ε > 0, let { Z A ε = [N,E[N ]] ε }. Sce P A ε P A ε D + PD c, ad sce l PD c = 0. ad t s eough to show l + P A ε D = 0. But, by Kologoov s axal equalty, P A ε D P Z ax ε k E[N ] <x x VaZ ε 0, [k,e[n ]] +. s ufoly tegable. Poposto 3. ad Re- Poof of Lea 4.3 Fst, we show that B k Nk / ak 3. gve [ B k E B k Nk Nk p] = N N k + o P / [ E p N N k p] + o p/ ] = / p/ E [ N N k p + o p/. But N N k = ɛ,k whee ɛ,k ae d wth ɛ,k = whe X = α, ɛ,k = whe X = α k ad ɛ,k = 0 othewse. Hece, by the classcal Mackewcz- Zygud equalty, so soe costat C p, ] E [ N N k p [ p] = E ɛ 4

42 Theefoe, fo ay p > ad B k ad [ B k sup E [ p/ ] C p E ɛ,k C p p/. Nk p] < + Nk / s ufoly tegable. Next, fo B k [ B k p] E B k = [/] Z k + [ [/] p p/ E / [] Z k [/]+. Z k : p] [ + p p/ Z k E [/]+ p ] [ [/] ] p/ [ C p p/ E Z k + p p/ Z k E p ], usg aga the Mackewcz-Zygud equalty. Cotug, usg covexty, [ [/] C p p/ E Hece, fo ay p >, ad B k ] [/] p/ Z k C p p/ E [[/] p/ [ B k p] sup E < +, ] Z k p C [ p E Z k p/ ]. p / s ufoly tegable ad theefoe, fo above, so s B k N k / B k /. Fally, ode to show 4.7, t s eough to pove B k Nk B k P 0, +. A. Settg A = { N k / l }, Hoeffdg s equalty esues that l + PA c = 0. Theefoe, sce B k Nk B k = N k Z k, we have B k P Nk B k ε { N k P 4 =[/]+ Z k =[/]+ ε } A + PA c,

43 ad Kologoov s axal equalty etals that P ax l [/ l,/+ l ] l Z k =[/]+ ε [ /+ l ε E ] Z k =[/]+ l E[ Z k ], fshg the poof of A. ad thus of 4.7. Poof of Lea 4.4 Cosde thee cases: x, x hee u v eas l + u /v = 0 ad x,.e., c x c, fo two fte costats c ad c, ad expad K x accodgly. Fst, let x : the, x+ + x x+ x+ + x x+ K x = = exp x + l + x x + l + x x = exp x + x x 8 + o x + = exp x 4 + 3x3 x 3 x 8 + o + o = exp x x 4 + o, whch yelds 4.33 case x. Next, let x : the, x x + o x + x+ K x = x + x+ = x + x x+ 4 x + x x+ x = 4 exp x + l + x + l + x x x x = 4 exp x + x x x + o x + x x = 4 exp + 3 x x 73 x + o x = exp + 3 x x + l x l + o 4 x x x x + o x. A.3 Sce x, the lage ode the expoetal A.3 s x l ad, ths ecove a boud of the fo 4.33 ths case. Fally, cosde the case x, say x = α wth α >. 43

44 The, K x = α + α+ α + α+ = exp cα, whch s aga of the fo 4.33, sce cα = l α + α+ /α + α+ s postve fo all α > ad s also bouded. A. O [HLM] The pupose of ths Appedx s to povde soe ssg steps the poof of the a theoe [HLM] devoted to the bay case as well as to coect the eos peset thee. The otatos ad ubeg ae as [HLM]. I patcula, ecall that N esp. N s the ube of zeos X,..., X esp. Y,..., Y. Poof of 3. Recall aga fo [HLM] that V = ax 0 k N N, Clealy, X = ax 0 t X,, B + B B B + + = B B t., k, B, A.4 ad deote by the dex fo whch the u A.4 s attaed. Next, f N N /, the V X ; ad slaly f the axu defg V s attaed at soe k /, the V X. Othewse, N N / wth, oeove, the axu defg V attaed at k [/, N N ] ad so: V =, Now, va A.4, V X B +, B + k = B B B B + B k B k 44, k. B B + + B B