Gaps in samples of geometric random variables

Transcription

1 Discrt Mathmatics Not Gaps i sampls of gomtric radom variabls William M.Y. Goh a, Pawl Hitczko b,1 a Dpartmt of Mathmatics, Drxl Uivrsity, Philadlphia, PA 1914, USA b Dpartmts of Mathmatics ad Computr Scic, Drxl Uivrsity, Philadlphia, PA 1914, USA Rcivd 3 March 6; rcivd i rvisd form 8 Sptmbr 6; accptd Jauary 7 Availabl oli 15 Fbruary 7 Abstract I this ot w cotiu th study of gaps i sampls of gomtric radom variabls origiatd i Hitczko ad Kopfmachr [Gap-fr compositios ad gap-fr sampls of gomtric radom variabls. Discrt Math ] ad cotiud i Louchard ad Prodigr [Th umbr of gaps i squcs of gomtrically distributd radom variabls, Prprit availabl at umbr 81 o th list or at prodigr/pdffils/gapsapril7.pdf. ] I particular, sic th otio of a gap diffrs i ths two paprs, w driv som of th rsults obtaid i Louchard ad Prodigr [Th umbr of gaps i squcs of gomtrically distributd radom variabls, Prprit availabl at umbr 81 o th list or at prodigr/pdffils/gapsapril7.pdf. ] for gaps as dfid i Hitczko ad Kopfmachr [Gap-fr compositios ad gap-fr sampls of gomtric radom variabls. Discrt Math ]. 7 Elsvir B.V. All rights rsrvd. Kywords: Gaps; Gomtric radom variabls; Asymptotic aalysis; Mlli trasform 1. Itroductio A compositio of a atural umbr is said to b gap-fr if th part sizs occurrig i it form a itrval. I additio if th itrval starts at 1, th compositio is said to b complt. I[] gap-fr ad complt compositios wr studid ad it was show i particular, that th proportio of gap-fr or of complt compositios of tds to 1 as. I probabilistic laguag this ca b xprssd as follows; if by a radom compositio of w ma a compositio pickd accordig to th uiform discrt probability masur o th st of all compositios of, th th probability that a radomly chos compositio of is gap-fr or complt is asymptotically 1, as. Sic itgr compositios ar closly rlatd to sampls of i.i.d. gomtric radom variabls with paramtr 1 GEOM 1, aalogous rsults hold for such sampls. I fact, that was th approach tak i []; it was show that th 1 This author is supportd i part by th NSA grat H addrsss: wgoh@math.drxl.du W.M.Y. Goh, phitczko@math.drxl.du P. Hitczko. URL: phitcz P. Hitczko X/$ - s frot mattr 7 Elsvir B.V. All rights rsrvd. doi:1.116/j.disc

2 87 W.M.Y. Goh, P. Hitczko / Discrt Mathmatics probabilitis of big complt or gap-fr ar asymptotically th sam for compositios as for sampls of GEOM 1 radom variabls, ad it was show that th probability that such a sampl is complt is xactly 1. Of cours, oc o is itrstd i sampls of gomtric radom variabls, thr is o particular raso to rstrict cosidratios to th particular valu p = 1 of th paramtr p. This was do i may paprs ad, i fact, i [] as wll although it should b addd that th cas p = 1 diffrs from p = 1 i that that th probability of big gap-fr dos ot hav a limit as tds to ifiity, but xhibits a oscillatory bhavior. This study was cotiud by Louchard ad Prodigr [5] who cosidrd a mor gral qustio of th distributio of th umbr of gaps i a sampl of i.i.d. GEOMp radom variabls. Although thy usd th trm th umbr of gaps a closr ispctio of thir dfiitios ad argumts rvals that thir otio is diffrt tha th dfiitio usd i [] ad might b mor appropriatly calld th umbr of missd valus. For xampl a compositio 3, 4, 5, 3 has o gaps accordig to th dfiitio usd i [], but has two gaps amly 1 ad accordig to th otio usd i [5] as a mattr of fact th diffrc btw th umbr of gaps is ad th umbr of missd valus is is xactly th distictio btw gap-fr ad complt as dfid i []. Th sam otio of gap-fr i th cotxt of st partitios was usd as arly as 199 i [1]. To coform to th dfiitio usd i [1,] w say that th compositio or a sampl of gomtric radom variabls has r gaps if th rag of part sizs occurrig i it is a uio of r + 1 disjoit itrvals, ad r + 1 is th smallst umbr with that proprty. Accordig to that distictio, a compositio, 5, 5,, 8,, 6 of 3 has two gaps, amly th itrvals [3, 4] ad [7], whil it misss four valus: 1, 3, 4, ad 7; thus th two quatitis ar clarly diffrt. Louchard ad Prodigr carrid out th dtaild study of th umbr of valus missd i a sampl of gomtric radom variabls. Amog othr thigs thy foud th limitig distributio ad asymptotics of momts. I particular, thy provd. Thorm 1. If X is th umbr of valus missd i th sampl of i.i.d. GEOM 1 radom variabls, th, as, EX 1 ad varx. 1 Furthrmor PrX = r = 1, r =, 1,.... r+1 I this ot, asid from poitig out th diffrc i trmiology, w dscrib a fw rsults o th umbr of gaps dfid cosisttly with [] i a sampl of gomtric radom variabls. Bfor cotiuig w should mphasiz, that Louchard ad Prodigr cosidr th gral cas <p<1. To a larg xtt w will do th sam, but w would lik to mphasiz that th cotrast is most clar for th spcial cas p = 1. I particular, whil th rsults giv i 1 ad do ot xhibit oscillatory bhavior but as was show i [5] thydoifp = 1, our aalogs do hav oscillatios, v for p = 1.. Expctd valu ad th variac I this sctio w prov th followig coutrpart of 1: Propositio. If Y is th umbr of gaps i th sampl of i.i.d. GEOMp radom variabls, th, as, whr l1 + q EY = + η l q E l + o1, 3 η E x := l q kπi/ l q q R p1 + q kπi/ l q Γ kπi kπi/ l qx l q

3 W.M.Y. Goh, P. Hitczko / Discrt Mathmatics is a priodic fuctio with small amplitud. Furthrmor, vary = 1 + q1 + q1 + q + q l q l 1 + q 4 + η V l + o1, 4 whr η V x is a fuctio with faturs similar to thos of η E. Proof. Lt Γ 1, Γ,...b a squc of i.i.d. GEOMp radom variabls. Our proof rsts o a obsrvatio that thr is a gap bgiig at j + 1, j 1, if ad oly if k Γ k = j ad m Γ m = j + 1. This is actually tru for all j xcpt th o corrspodig to th largst obsrvd valu i th sampl. Thus, if w st { } G j := {Γ k = j} {Γ m = j + 1}, m=1 ad w follow th custom of idtifyig sts with thir idicator fuctios, w s that Y + 1 = j 1 G j, ad cosqutly Now, so that EY + 1 = G j = PrG j. 5 j=1 {Γ m = j + 1}\ m=1 {Γ m = j + 1,j}, m=1 PrG j = 1 pq j 1 pq j 1 pq j. 6 To procd furthr w will d som lmtary stimats which w isolat i th followig: Lmma 3. Lt x,y satisfy x + y 1. Th for vry 1 w hav i x 1 x =O1/x x, ii 1 x 1 y 1 x y =O1xy x y. Proof. For i writ x 1 x = x 1 x 1 = x 1 x kx 1 x 1 k k= 1 x 1 x kx 1 kx = Ox 1x k= = O 1 x x.

4 874 W.M.Y. Goh, P. Hitczko / Discrt Mathmatics For part ii w clarly hav 1 x 1 y = 1 x y + xy 1 x y. O th othr had 1 1 x y + xy 1 x y = xy 1 x y + xy k 1 x y 1 k k= xy1 x y + xy 1 = xy1 x 1 1 y 1 xy 1x 1y = O1xy x y, which provs ii. Usig Lmma 3 i ad 6 w s that PrG j = pq j 1 pq j 1 + O j 1 j 1 Sic th sum pq j pq j = j it rmais to valuat th sum pq j 1 pq j 1. j 1 To that d cosidr fx:= xpqj 1 xpqj 1 j=1 ad its Mlli trasform By ivrsio Mf s := = j=1 1 pq j 1 pq j 1. j 1 pq x pq x dx + O1 = O1, fxx s 1 dx = 1 pq j s = 1 p s 1 q s 1 Γs c+i j=1 x 1 x/q x s 1 dx /q s = xpqj 1 xpqj 1 x s 1 dx Γs q + 1 s q s p s 1 + q s q s. 1 fx= 1 Mf sx s ds, πi c i whr th vrtical li passig through c is i th fudamtal strip o which th Mlli trasform xists. I our cas, c may b ay umbr i 1,. Th lattr itgral may b valuatd by rsidu thorm; th itgrad Γs q + 1 s q s p s 1 + q s q s x s 1

5 W.M.Y. Goh, P. Hitczko / Discrt Mathmatics has simpl pols at χ k = kπi/ l q, k =, ±1, ±,...Th rsidu at χ is l1 + q l q 1 7 ad th rsidu at χ k, k = is Γχ k 1 + q χ k q χ kpx χ k l q. Thus, by cosidrig a larg rctagular box with vrtics c ± im ad R ± im, R,M >, M/ N ad omittig th stadard vrificatio that th itgrals alog th horizotal li sgmts ad th vrtical sgmt o th right vaish as M ad th R w obtai that f 1 l1 + q l q + l q q χ k R p χ k1 + q χ Γχ k k χ k l. Combiig this with 5 w obtai that l1 + q EY = + l q l q kπi/ l q q R p1 + q kπi/ l q Γ kπi kπi/ l ql, l q which provs 3. Th argumt for th variac is similar: vary = vary + 1 = E G k k 1 E G k k 1 = E j,k 1 = j,k 1 G j G k PrG k k 1 PrG j G k PrG k k 1 8 ad w d to fid PrG j G k. As w will ow show PrG j if j = k, if j k =1, PrG j G k = 1 pq j pq k 1 pq j 1 pq j pq k if j k. 1 pq j pq k pq j 1 pq j pq k 1 pq k

6 876 W.M.Y. Goh, P. Hitczko / Discrt Mathmatics Sic th first two assrtios ar clar, cosidr j k. Th PrG j G k = Pr {Γ l = j} {Γ m = k} {Γ r = j + 1,k+ 1} l=1 = Pr {Γ l = j} l=1 m=1 r=1 {Γ r = j + 1,k+ 1} r=1 Pr {Γ l = j} l=1 {Γ r = j + 1,k,k+ 1} r=1 = Pr {Γ r = j + 1,k+ 1} Pr {Γ r = j,j + 1,k+ 1} r=1 Pr {Γ r = j + 1,k,k+ 1} + Pr {Γ r = j,j + 1,k,k+ 1} r=1 r=1 r=1 = 1 pq j pq k 1 pq j 1 pq j pq k 1 pq j pq k 1 pq k + 1 pq j 1 pq j pq k 1 pq k. This, combid with 6 ad Lmma 3ii givs, for j k PrG j G k PrG j PrG k 1 pq j 1 pq k 1 pq j pq k which implis that PrG j G k Sic j k j k PrG j PrG k = j k + 1 pq j 1 pq k pq k 1 1 pq j pq k pq k pq k 1 pq j pq j 1 1 pq k pq j pq j pq j pq j 1 1 pq k pq k 1 1 pq j pq j 1 pq k pq k 1 1 = O pq j pq j pq k pq k, PrG j PrG k. PrG j PrG j PrG j+1 PrG j PrG k j,k 1 j 1 = j j 1PrG j j 1PrG PrG j PrG j+1, 8 implis that vary PrG j PrG j PrG j PrG j+1. 9 j 1 j 1 j 1

7 W.M.Y. Goh, P. Hitczko / Discrt Mathmatics Th first sum was alrady valuatd ad it is th mattr of th sam routi to aalyz th rmaiig two. W first appal to Lmma 3i to rplac th xact xprssios for PrG j by approximatios by xpotial fuctios. This givs, ad PrG j pq j 1 pq j 1 = pq j + p1+qqj 1 p1+qqj 1, PrG j PrG j+1 pq j pq j+1 1 pq j 1 1 pq j = p1+qqj p1+q1+qqj 1 p+qqj + p1+q q j 1. Mlli trasforms of th sums of ths asymptotic xprssios for PrG j ad PrG j PrG j+1 ar, rspctivly, Γs qs 1 p s s q s + 1 s 1 + q s q s q s 1 ad Γs qs 1 p s q s 1 + q s q1 + q s 1 q s + q s q s 1 q s 1. It follows that th itgrad i th Mlli ivrsio has simpl pols at xactly th sam χ k s as bfor ad th rst of th argumt ca b compltd by th sam rsidu calculatios as for th xpctd valu. I particular, th rsidua at χ = of th abov xprssios ar, rspctivly, l q l 1 + q + l1 + q l q ad l q1 + q + l1 + q1 + q + l q + q l1 + q. l q Combiig th abov two xprssios with 9 ad 7 w obtai that th mai o-oscillatory trm i th variac is l q l 1 + q1 + q1 + q + q 1 + q 4, as spcifid i 4. Rsidua at othr pols cotribut to oscillatory trms ad th xplicit xprssios for thm could b obtaid without difficulty, but w skip furthr dtails. Rmarks. i For q = 1 formula 3 bcoms EY l 3 l 1 + R l kπil 3/ l 1 Γ kπil 3/ l kπi kπi xp l l l. I particular, th oscillatory compot dos ot vaish. Its plot is giv i Fig. 1. This cotrasts with th situatio cosidrd by Louchard ad Prodigr i [5] whr thr wr oscillatios for q = 1 but ot for q = 1 s 1. Th sam applis to th variac. ii Th simpl midd approach prstd abov ca b usd to rcovr som of th rsults obtaid by Louchard ad Prodigr. For xampl, th radom variabl X rfrrd to i Thorm 1 satisfis X = j 1 V j,

8 878 W.M.Y. Goh, P. Hitczko / Discrt Mathmatics E-6 E-6 1E-6 E E-6 -E-6-3E-6 Fig. 1. Oscillatory cotributio to EY for p = 1/. whr Sic, V j = {Γ k = j} {Γ k = j}=v j w hav Hc, V j = EX = {Γ l >j}. l=1 {Γ k = j} {Γ k j} = V j {Γ k = j}\ {Γ k j 1}. j=1 {Γ k j 1}, {1 pq j 1 1 q j 1 }, 1 ad xactly th sam argumt as arlir could b usd to obtai th asymptotic xprssio for th xpctatio i 1. I particular, th o-oscillatory trm i th asymptotic xprssio for EX is l p/ l q, ad th xplicit xprssio for th oscillatios ca b obtaid by takig ito accout cotributios of th rsidua at χ k for k =. W omit furthr dtails, but w would lik to mtio that Louchard ad Prodigr usd diffrt approach which rquird a xtra argumts to show that th oscillatios vaish for p = q = 1. With our approach this is clar, sic for p = q 1 bcoms { EX = 1 1 j 1 1 } j 1 = 1, j=1 which givs th first part of 1.

9 W.M.Y. Goh, P. Hitczko / Discrt Mathmatics Th computatio for th scod part is ot much mor ivolvd. Bcaus of th last xprssio, th variac of X is qual to j<k PrV j V k, ad sic for j<k V j V k = {Γ l = j,k}\ {Γ l = j,γ l k 1}, l=1 this is furthr qual to j<k = = = l=1 { 1 1 j 1 k 1 1 j 1 } k 1 j=1 k=j+1 j=1 j=1 { 1 1 j 1 k 1 1 j 1 } k 1 { lim 1 1 M j 1 M 1 1 j 1 } j { 1 1 j 1 1 } j 1 =, as spcifid i th scod part of 1. It should b addd, howvr, that this mthod dos ot sm to b usful as far as computatio of highr momts of X or Y ar cocrd as o would hav to dal with icrasigly mor complicatd sums of th products of idicator fuctios. 3. A aalog of I this sctio w focus o a particular cas p = 1. Lt R,r b th vt that th sampl of i.i.d. GEOM 1 radom variabls has r gaps, ad lt p r b its probability. I othr words, p r = PrY = r whr Y is dfid i Propositio. Th Thorm 4. For r = w hav p = 1 1, 11 ad for r 1 p r = p r η P l + O β 1, 1 whr p r = 1 l m 1 p m r p m r 1 m4 m m 1, 13

10 88 W.M.Y. Goh, P. Hitczko / Discrt Mathmatics E E E E E Fig.. Th valus of p 1 for E- 3.44E- 3.44E E Fig. 3. Th valus of p for 1 5. ad η P x = l R k 1 m 1 p m r p m r 1 m!4 m m 1Γ m + kπi l kπi/ l x, ad < β < 1. Rmark. Whil statmt 1 sms circular, i ss that it rprsts p r i trms of a ifiit sris ivolvig that sam quatity, th poit to mak is that th sris covrgs xpotially fast, ad thus is much bttr amabl to various umrical valuatios icludig bouds o th approximatio rror tha th rcurrc 15 s.g. [4,] for similar istacs. Th fuctio η P has th usual faturs; i particular th magitud of its oscillatios is a fractio of 1 5 of th first trm i 1. Plots of th asymptotic xprssios for p r for 1 r 5 ar giv i Figs. 6. Proof of Thorm 4. Th proof is basd o what has bcom a wll stablishd, 4-stp program s [3] for a good xpositio: w will first driv a rcurrc rlatio for th p r, th covrt it ito a fuctioal quatio by poissoizatio. W will th aalyz th asymptotic bhavior of th solutio of this fuctioal quatio by mas of th Mlli trasform ad rsidu calculatio. Fially, w com back to th asymptotics of p r by aalytical dpoissoizatio. To bgi, for a st A, lt p r, A ad p r A dot PrR,r A ad PrR,r A, rspctivly. W lt m :=mi{γ k : 1 k },

11 W.M.Y. Goh, P. Hitczko / Discrt Mathmatics E E E E Fig. 4. Th valus of p 3 for Fig. 5. Th valus of p 4 for E E E E E Fig. 6. Th valus of p 5 for 1 5. b th miimum of a sampl of lgth. Th w hav p r = p r, m = 1 + p r, m > 1 = p r, m = 1 + p r m > 1Prm > 1 = p r, m = 1 + p r/,

12 88 W.M.Y. Goh, P. Hitczko / Discrt Mathmatics whr i th last stp w usd mmorylss proprty of gomtric radom variabls. I particular, p r, m = 1 = 1 1 p r, r, 14 ad sic it was show i [] that p,m = 1 = 1, 11 follows. For r 1 by coditioig o th umbr of Γ k s that ar qual to 1 w obtai p r, m = 1 = 1 k p r l I Γl =1 = k 1 1 = k p r l I Γl =1 = k + 1 p r l I Γl =1 = 1 1 = k {p kr, m k = 1 + p k r, m k } + 1 I r= 1 { 1 = k p k r, m k = } k p kr I r= = 1 k=r+1 1 k p kr, m k = k=r 1 k +k p kr I r=. Utilizig p,m = 1 = 1 ad 14 th abov ca b writt as: for r 1 ad r + 1, 1 1 p r = 1 1 k=r+1 p = k k p k r k W ow form a two-variabl fuctioal quatio by first lttig for 1 ad th B u := 1 p ru r = p ru r, r Pz, u := 1 r= z z B u!. k=r pk r 1 k, 15 W th hav 1 r u r z z 1 1! p r = Pz, u z/ Pz/,u,

13 W.M.Y. Goh, P. Hitczko / Discrt Mathmatics ad, o th othr had, 1 r = r 1 u r z z 1 1! p r = r 1 u r z z 1 1! p r + z z 1 1! p 1 u r z z! z z! + 1 = u r z r = r 1 k=r z/4 k k! z/ 1 1 k=r k=r+1 1 z/ k k! p k r 1 z/ k k! k r k k p k r pk r 1 k 1 1 k p k r z/ k k! } + 1 z z 1 u r z/ z/k 1 1 k! k p k r + 3z/4 u r z/4 z/4k p k r 1 k! k r = 1 z/ k r 1 u r z/ z/k k! =k+1 + u z/ 1 3z/4 u r z/4 z/4k k! k 1 r = 1 z/ k 1 r z/ z/k 1 1 k! k p k k 1 =k+1 k=r z/ k k! z/ k k! + 1 z 1 1 k p k r p k r + 1 z u r z/ z/k 1 1 k! k p k r k + u z/ 1 3z/4 Pz/4,u+ 1 z = 1 z/ {Pz/,u z/4 Pz/4,u 1 } z/ + u z/4 1 z/ Pz/4,u+ 1 z.

14 884 W.M.Y. Goh, P. Hitczko / Discrt Mathmatics This lads to th followig fuctioal quatio: Pz, u = Pz/,u Tz,u+ fz, 16 whr w hav st Tz,u:=1 u1 z/ z/4 Pz/4,u ad fz:= z/ 1 z/. For th rmaidr of our discussio u is a complx variabl that is rstrictd to a compact domai, say u 1. Th followig fact is slf-vidt, w omit th proof. Lmma 5. W hav a B u 1/1 u. b Pz, u is a tir fuctio of z. c For z, Pz, u 1/1 u is a boudd fuctio of z ad P,u=. It follows i particular, that th Mlli trasform MPs, u of Pz, u is wll-dfid ad is aalytic i th vrtical strip 1 < Rs <. Applyig th Mlli trasform to 16 w gt MPs, u = s MPs, u MT s, u + Mf s, u. 17 For Rs > 1 Mf s, u = z/ 1 z/ z s 1 dz = s 1Γs, so w tur our atttio to MT s, u. By Lmma 5 Pz/4,uis boudd for, hc its Mlli trasform MT s, u = 1 u 1 z/ z/4 Pz/4, uz s 1 dz is aalytic i th half pla Rs > 1. W first driv a xprssio valid for Rs > ad th xtd it by aalytic cotiuatio. Writ whr MT s, u = 1 ui 1 I, 18 I 1 = Now for Rs > I 1 = = 1 z/4 Pz/4, uz s 1 dz ad I = z/4 Pz/4, uz s 1 dz = B u!4 z/4 z/ z +s 1 dz = 1 1 B u +s 3z/4 Pz/4, uz s 1 dz. B u z/4!!4 Γ + s z/4 z s 1 dz itrchag of itgratio ad summatio is justifid sic th xprssio i qustio is absolutly covrgt. Likwis, w hav I = 1 B u Γ + s.!4

15 W.M.Y. Goh, P. Hitczko / Discrt Mathmatics Isrtig ths two quatios to 18 w obtai MT s, u = 1 u B u!4 Γ + s+s 1, 19 1 which is still valid for Rs > 1 by aalytic cotiuatio. It follows from 17 that MT s, u MPs, u = s Γs 1 ad by th Mlli ivrsio for 1 < σ < Pz, u = 1 πi σ+i σ i MT s, u s 1 Γs z s ds. W ow shift th itgratio path to th right sid of zro. Sic th fuctio MT s, u is implicit, a rigorous justificatio of that stp is rquird. Th followig lmma whos proof is postpod util th appdix will suffic for that purpos: Lmma 6. Lt s = σ + it, whr σ, t ar ral umbrs. Assum that σ is i a boudd st A cotaid i σ > 1. Th as t MT σ + ift,u =O A.34 t, whr th big O costat holds uiformly for σ A ad u 1. W ow cosidr th itgral aroud a rctagular cotour with vrtics, say, 1/ ± it,1 ± it, whr t vtually approachs. By Lmma 6 ad th assumptio z>, th itgrals alog th vrtical sids covrg absolutly ad th itgrals alog th horizotal sids go to zro as t. W gt from Pz, u = MT s, u Rs s Γsz s 1 k s=χ k + 1 πi 1+i 1 i MT s, u s 1 Th rsidu computatio is straightforward. W gt Pz, u = MT,u l l + 1 πi = 1 1 l 1+i 1 i m 1 1 l + 1 πi k = m 1 1+i 1 i Γs z s ds. k = MT s, u s Γs z s ds 1 1 ub m u m 1 m4 m kπi MT l,u kπi/ l z 1 ub m u m 1 m!4 m Γ m + kπi kπi/l z l MT s, u s 1 Γs z s ds.

16 886 W.M.Y. Goh, P. Hitczko / Discrt Mathmatics W rmark that bcaus of Lmma 6 th doubl sum abov ad th itgral alog σ = 1 ar absolutly covrgt. W will vrify i Appdix th applicability of th dpoissoizatio procss. This will imply i particular that it will suffic to focus o z through th positiv rals. But th th itgral abov is o1 so that as z through rals, Pz, u = 1 1 l m 1 whr th littl o costat is uiform for u 1 ad whr η P x := 1 l k = m 1 1 ub m u m 1 m4 m + 1 uη P l z + oz 1, 1 B m u m 1 m!4 m Γ m + kπi kπi/ l x. l Aftr dpoissoizatio w will gt for ay β,< β < 1, B u = P, u + O β 1, whr th big O costat holds uiformly for u 1. Makig xplicit what w hav from Eq. 1 B u = 1 1 l = 1 1 l m 1 m 1 1 ub m u m 1 m4 m + 1 uη P l + o 1 + O β 1 1 ub m u m 1 m4 m + 1 uη P l + O β 1, whr both th big O ad th littl o costats hold uiformly for u 1. Bcaus of uiformity w ca multiply th abov quatio by 1,r 1, ad itgrat th rsultig quatio with rspct to u alog th circl u = 1 u r+1. This provs th rsult. Appdix Proof of Lmma 6. Rcall 19 ad us th stimat of B u i Lmma 5a to gt MT σ + it,u 1 u B u!4 Γ + s+s !4 Γ + σ + it +σ + 1. Sic th situatio is symmtric with rspct to t it suffics to cosidr t. Not that 1 ad σ > 1 so that + σ >. This implis that +σ > 1 ad thus th last xprssio is boudd by 1!4 Γ + σ + it +σ = σ Γ + σ + it Γ + σ Γ + σ!.

17 W.M.Y. Goh, P. Hitczko / Discrt Mathmatics Now w focus o Γ + σ + it /Γ + σ. By Stirlig s formula w hav { xp + σ 1 l + σ + it t ta 1 t + σ + σ 1 } 1 l + σ 1 + O A + σ t = xp l σ tta 1 t + σ 1/ 1 + σ + σ + t 1/4 1 + O A { 1 xp + σ l t σ t } t 1 + σ ta O A + σ { } t 1 = xp + σh 1 + O A, + σ whr th fuctio hx is dfid by hx := 1 l1 + x x ta 1 x, x. Not that h = h = ad h x = ta 1 x. Hc hx is gativ for x. Lt { x gx := 3 if x, x + 3 if x. Th hx gx for x. Rplacig hx by gx w s that th lft-had sid of is boudd by O A { } t Γ + σ xp + σg + σ!. 1 W us th asymptotics Γ + σ Γ + 1 = σ O A, whr th big O costat holds uiformly ovr th boudd st A, ot oly for a idividual σ. Th sum is rducd to O A { } t 1 xp + σg + σ ε, 1 for a arbitrarily small ε>. W split th abov sum ito S 1 + S accordig to th dfiitio of gx: S 1 = +σ t/+σ+/3 l ε, Now +σ t/ S = +σ 1/3t /+σ l ε. S 1 = t +σ t/ +σ t/ +σ/3 l ε = σ/3 t +σ t/ /3 l ε. Sic 3 l.648 < th sum covrgs ad as t w hav S 1 = O A t. 3

18 888 W.M.Y. Goh, P. Hitczko / Discrt Mathmatics Nxt, S = +σ t/ +σ 1/3t /+σ l ε = +σ t/ 1/3t /+σ l ε. Not that + σ t/ implis t/ 1 + σ/. Sic σ is boudd, w hav t/ = O A 1 ad t + σ = t t + O A = t + O A1. Isrt this i S ad omit th possibly fiit may trms whos summatio idx is about t/ to obtai S = O A t l ε/ + O A 1/3t / l ε. 4 t/ Cosidr as a cotiuous variabl ad lt j := 1 t l ε. 3 It is asy to s that j attais th uiqu maximum at max = with th maximum valu t 3l ε.693t, j max = l ε 3 t. Dcompos th sum i 4 as 1/3t / l ε = t/.8t t/ 1/3t / l ε +.8t 1/3t / l ε. Sic th summads i th scod summatio ar dcrasig, rplacig th summatio by th corrspodig itgral ad stimatig th first sum trivially w gt.8t t jmax + 1/3t / l ε d.79t =.8t t l ε/ 3t + t =.3t l ε/ 3t + t /3t/ t l ε d t1/3+ l ε d. Th fuctio 1/3 + l ε is dcrasig, so th asymptotics for th itgral as t is straightforward. W hav t1/3+ l ε d = Ot 1 t1/ l ε = O.969t,.79 kowig that 1/ l W ow us th valu l / 3 = to ifr that 1/3t / l ε = O.961t. t/ Isrtig this i 4, w obtai S = O A.34t, which combid with 3 complts th proof.

19 W.M.Y. Goh, P. Hitczko / Discrt Mathmatics Justificatio of dpoissoizatio procss. Accordig to [3, Corollary 1, p.34] w d to costruct a liar co Lθ such that for som α, β,< α, β < 1, thr ar positiv costats A, B, R such that if z >R, z Lθ th Pz, u B z β, 5 ad if z R, z/ Lθ th Pz, u z A α z. 6 hold uiformly for u 1/. Vrificatio of 5: By Lmma 6 w choos θ =.3 actually ay positiv umbr lss that.34 will work ad w will us th Mlli ivrsio with σ = β. Sic Γ β it =O π/ ε t w gt Pz, u 1 MT β + it,u π β+it Γ β + itz β+it dt 1 = O β.34 t + π/ ε t z β+it dt = O β z β.34 t + π/ ε t t argz dt. Bcaus z L.3, argz.3. As a rsult, th abov itgral is absolutly covrgt ad 5 follows. Vrificatio of 6: W us th ida of icrasig domais. To this d, w choos λ = 3/ so that λ/ ad λ/4 ar lss tha 1. For som ξ > 1, dfi th domai D m by D m ={z : z/ L.3, ξ/4 z ξλ m+1 }, m=, 1,,.... W us iductio to vrify that 6 holds i all D m. Lt th α i qustio b ay umbr such that 1 > α > cos.3. W will s how to choos R,ξ i th squl. Now rwrit th fuctioal quatio 16 i th form Pz, u = Pz/,u 1 u1 z/ z/4 Pz/4,u+ z/ 1 z/. Multiply th abov by z ad furthr rduc th rsultig quatio to Pz, u z = Pz/,u z/ z/ 1 u z/ 1Pz/4,u z/4 + z/ 1. For z/ L.3, Rz z cos.3. Hc Pz, u z Pz/,u z/ 1/Rz + 1 u 1/Rz + 1 Pz/4,u z/4 + 1/Rz + 1 Pz/,u z/ 1/ z cos u 1/ z cos Pz/4,u z/ / z cos Choos ξ so larg ad fix it so that wh z ξ, w hav simultaously 1/ z cos.3 α 1 3, z 3/4α 1/ z cos u 1 3, 8 z α 1/ z cos

20 89 W.M.Y. Goh, P. Hitczko / Discrt Mathmatics Such choic for ξ is ralizabl sic cos.3 < α. With ξ chos, choos A>1 so that 6 is satisfid i D. This is fasibl du to th compactss of D. W will show by iductio that th 6 holds i all domais D m with rspct to th sam paramtrs. Assum that it holds i D m ad lt z D m+1.ifz D m, w ar do by th iductio hypothsis. If z D m+1 D m, th by th dfiitio of D m w hav 3 m+1 3 m+ ξ < z ξ. Multiplyig by 1 givs ξ 3ξ 4 4 ξ That is, 3 m+1 1 < z/ ξ 3 m ξ 3 m+1. ξ 3 m+1 4 z/ ξ. Hc z/ D m. Similarly, z/4 D m. By th iductio hypothsis ad 7, w hav Pz, u z B α z / 1/ z cos u 1/ z cos.3 + 1B α z /4 + 1/ z cos = 1/ z cos.3 α + 1 u z 3/4α 1/ z cos α z B 1/ z cos B α z. By 8 th xprssio i parthsis is lss tha 1 which givs 6 ad complts th iductio w may tak R = ξ. Rfrcs [1] W.M.Y. Goh, E. Schmutz, Gap-fr st partitios, Radom Struct. Algorithms [] P. Hitczko, A. Kopfmachr, Gap-fr compositios ad gap-fr sampls of gomtric radom variabl, Discrt Math [3] P. Jacqut, W. Szpakowski, Aalytical dpoissoizatio ad its applicatios, Thort. Comput. Sci [4] S. Jaso, W. Szpakowski, Aalysis of th asymmtric ladr lctio algorithm, Elctro. J. Combi [5] G. Lourchard, H. Prodigr, Th umbr of gaps i squcs of gomtrically distributd radom variabls, Prprit availabl at umbr 81 o th list or at prodigr/pdffils/gapsapril7.pdf.