Dscrete Mathematcs an Theoretcal Computer Scence DMTCS vol. subm.), by the authors, Lmt Theorems for A Degenerate Fxe Pont Equaton Mchael Drmota an Markus Kuba Insttute of Dscrete Mathematcs an Geometry, TU Wen, Wener Hauptstrasse 8-0, A-040 Venna, Austra receve February 2007, revse..., accepte... We conser the stochastc fxe pont equaton X n = X n D +B that appears n several applcatons, for example n the Bolthausen-Szntman coalescent or n the analyss or recursve algorthms. In ths paper we prove a systematc stuy on these kns of stochastc fxe pont equatons. It turns out that epenng on the sngularty structure of the the moment generatng functon of D the lmtng strbuton s ether a Mttag-Leffler strbuton, a normal strbuton, or a spectrally negatve stable strbuton wth nex of stablty. Keywors: stochastc fxe pont equaton, generatng functons, stable lmt law Introucton The purpose of ths paper s to scuss the stochastc fxe pont equaton X n = Xn D + B n ), ) where we assume that X n = 0 for n 0 an that D s a screte ranom varable on the postve ntegers, B s ranom varable wth non-zero mean, an X n, D, an B are nepenent. Equatons of ths form naturally appear n the stochastc analyss of recursve algorthms For example, n see 7) fxe pont equatons of the form X n = XIn + B n, 2) are scusse, where I n has some strbuton on {0,,...,n } an also B n epens on n. However, the methos an results of 7) o not apply to ), where I n = n D s more or less) concentrate at n an at few values less. However, there s substantal lterature see ; 2; 5; 6; 8) an the references there) on the ranom varable Z n that counts the number of ranom cuts necessary to solate the root n ranom trees equvalently Both authors are supporte by the Austran Scence Founaton FWF) Project S9600. subm. to DMTCS c by the authors Dscrete Mathematcs an Theoretcal Computer Scence DMTCS), Nancy, France
2 Mchael Drmota an Markus Kuba one can conser the number of so-calle recors n trees). For example, for recursve trees, Z n satsfes the fxe pont equaton Z n = Zn Dn +, n > ), Z = 0, 3) where D n s a screte ranom varable, nepenent of Z n wth P{D n = k} = n, for k =,...,n. 4) n kk + ) Interestngly the same recursve relaton occurs n the Bolthausen-Szntman coalescent n bology. In ths context Z n counts the number of collson events that take place untl there s a sngle block an Z n also approxmates very well) the total branch length of ths coalescent. In 2) t was shown wth help of analytc tools that Z n has a stable lmtng strbuton whch also mples that the total branch length has ths lmt, see )). A probablstc proof can be foun n 5). Snce D n s close to ts lmt D gben by P{D = k} = /kk + ))) we expect that X n from above wll have the same lmtng behavour as Z n whch s n fact true compare wth the thr part of Theorem wth Lu) = log u). However, t s far from obvous to prove a tranfer between these two results. We wll comment on that n Secton 4. In orer to make the stuaton more transparent we have ece to work on the equaton ), where D an B o not epen on n. Theorem shows that uner natural assumptons on D an B) there are only three types of possble lmtng strbutons for X n, ether a Mttag-Leffler strbuton, a normal strbuton, or a spectrally negatve stable strbuton wth nex of stablty. Although s seems to be a non-trval problem to tranfer these results to relate equatons of the form X n = Xn Dn + B n one mght expect smlar results n the more general stuaton, see Secton 4. 2 Results In orer to formulate our man result we frst lst some facts on the appearng lmtng strbutons. Frst, let S = S α enote the Mttag-Leffler strbuton of nex α 0,)) that s efne by S = T α, where T has Laplapce transform Ee λt = e λα an correspons to the stable process T s,s 0) wth nex α: T s = s /α T. The moment generatng functon of S s gven by Ee λs = k 0 λk /Γ + αk), that s, S has moments ES k = k!/γ + αk). Secon, let Y enote a spectrally negatve stable strbuton wth nex of stablty, that s, characterstc functon of Y s gven by Ee λy = e λ log λ π 2 λ. We also nee a specal noton of a slowly varyng functon that we aopt from 3). A functon Lu) s sa to be of slow varaton at f there exsts a postve real number u 0 an an angle φ 0, π 2 ) such that Lu) s 0 an analytc n the oman Cu 0,φ) := {u C : π φ) argu u 0 ) π φ)} an f
Lmt Theorems for A Degenerate Fxe Pont Equaton 3 there exts a functon ǫx), efne for x 0 wth lm x ǫx) = 0, such that for all θ [ π φ),π φ] an u u 0 Lue θ ) Lu) < ǫu) an Lu log 2 u) Lu) < ǫu). Theorem Suppose that X n n Z) s a sequence of ranom varables that satsfes the strbutonal recurrence X n = Xn D + B for n, where X n = 0 for n 0, D s a screte ranom varable on the postve ntegers, B s ranom varable wth mean, an X n, D, an B are nepenent. Further assume that the probablty generatng functon hs) = Es B s analytc at s = wth h ) 0 an that gs) = Es D s analytc on the set Sε) = {s C : s < + ε, args ) 0} for some ε > 0 an has a local representaton of the form ) gs) = s) α As)L 5) s for s < ε an s Sε), where 0 < α, As) s analytc at s = wth A) 0, an Ls) s a functon of small varaton at n the above sense). Then we have the followng three kns of lmtng strbutons:. If 0 < α < then X n cn α S, /Ln) where c = h )/A) an S = S α enotes the Mttag-Leffler strbuton of nex α 0,). We also have convergence of all moments. 2. If α = an Lu) =, that s, gs) s analytc at s =, then X n satsfes a central lmt theorem of the form X n µn σ2 n where µ = h ) A) an σ2 = h )+h ) 2h ) 2 all moments. N0,), A) + h ) 2 A)+2A )) A) 3 3. If α = an f Lu) has the followng atonal propertes: a) Lu) an LuLu)) Lu/Lu)) Lu) as u,. We also have convergence of b) the equaton v = u/lu) has a unque soluton u Cu 0,φ) for v Cv 0,ψ) for some v 0 > 0 an ψ 0, π 2 )), c) Lu) s fferentable an L u) = Mu)/u, where Mu) s of slow varaton at wth the property that there exsts a functon ǫx), efne for x 0 wth lm x ǫx) = 0, such that Mαu) Mu) < ǫu) unformly for all α Cu 0,φ) wth /Lu) c Lu) as u,
4 Mchael Drmota an Markus Kuba then we have X n a n b n Y, 6) where a n = cn/lnmn)/ln))), b n = cnmn)/ln) 2, an Y enotes the spectrally negatve stable strbuton wth nex of stablty. It shoul be remarke that the restrctons on gs) are natural n the followng sense. Frst, f gs) s a probablty generatng functon an has a representaton of the form 5) then α has to be n the range 0 < α. Secon, t s of course also possble to work wth more general functons gs) that have some atonal error terms. However, n several usual cases these error terms can be put nto Lu). Therefore we have ece to work wth ths specfc kn of representaton. Fnally we want to menton the contons on Lu) are not that restrctve as they appear. For example, all functons of the form Lu) = log u) a log log u) b or of the form Lu) = e log u)β for some β < 2 ) satsfy all assumptons of the theorem. 3 Proof of Theorem All parts of the proof make use of the followng explct representaton for the generatng functon of the probablty generatng functons of X n. Lemma Let gs) = Es D an hs) = Es B enote the probablty generatng functon of D an B. Then we have fs,z) := Es Xn z n z hs) gz)) = z) hs)gz)). 7) n Proof: Set a n s) = Es Xn an p k = PD = k) for k ). Note that a n s) = for n 0. By ) we get a n s) = Es Xn D+B = hs) n p k a n k s) = hs) p k. k Ths mples an consequently 7). fs,z) = hs)gz)fs,z) + zhs) k = hs)gz)fs,z) + zhs) gz) z k= p k a n k s) + hs) k n p k z k z Wth help of 7) we can easly compute asymptotc expansons for moments that wll be also use to prove the frst part of Theorem ). Lemma 2 The moments of X n are asymptotcally gven by where c = h )/A). EX k n k! Γ + αk) ) cn α k n ), Ln)
Lmt Theorems for A Degenerate Fxe Pont Equaton 5 Proof: We use the relaton Frst we get whch mples n 0 fs,z) s EX n X n ) X n k + ))z n = k fs,z) s k = s. s= ) z hs) gz)) zh s) gz)) = z) hs)gz)) z) hs)gz)) 2 EX n z n = zh ) Az) z) α+ L n 0 By the transfer prncple of 3) ths rectly yels n α c EX n Γα + ) Ln). Smlarly we can procee further. In general the asymptotc leang term of k fs,z) s= equals s k z k!zh ) k gz) k z) gz)) k = k!zh ) k gz) k Az) k z) +αk L whch shows that ) EXn k = EX n X n ) X n k + )) + OEXn k k! cn α k ). Γ + αk) Ln) Of course ths mples the lemma. We are now able to prove the frst two parts of Theorem mmeately. The proof of the thr part s more nvolve. Proof of the frst part of Theorem : If we set Xn = Ln)X n /cn α ). Then Lemma 2 shows that E X n k k!/γ + αk). Hence, by the Frechet-Shohat theorem t follows that Xn S, where S enotes the Mttag-Leffler strbuton of nex α). Proof of the secon part of Theorem : If α = an Lu) = then gz) s analztc at z =. Consequently, for every fxe s C that s suffcently close to ) the functon z fs,z) = zhs)az) hs)gz) has a polar sngularty z 0 s) that s gven by the equaton hs)gz 0 s)) = 0. Hence, by stanar methos see 4)) ths proves a central lmt theorem for X n µn)/ σ 2 n wth ). z ) k µ = h ) A) an σ 2 = h ) + h ) 2h ) 2 A) + h ) 2 A) + 2A )) A) 3.
6 Mchael Drmota an Markus Kuba z s) 0 γ Fg. : Path of ntegraton Proof of the thr part of Theorem : The proof of the thr part of Theorem uses eas that are smlar to those of ; 2). We agan use the explct representaton 7) n orer to get asymptotc nformaton on the characterstc functon Ee λxn = [z n ]fe λ,z). In partcular, f Y n = X n a n )/b n enotes the normalze ranom varable for some sequences a n, b n ) then Ee λyn = e λan/bn [z n ]fe λ/bn,z) = e λtan/bn fe λ/bn,z)z n z, 2π where γ s close curve aroun the orgn that s contane n the analytcty regon of fe t/bn,z) see also Fgure ). We have to eal rectly wth the characterstc functon of Y n snce there s no moment convergence for moments of oer k 2. Ths s n complete contrast to the frst two cases, where we have convergence of all moments.) Note that the functon z fs,z) has usually) two sngulartes, namely the sngularty of gz) at z = that comes from the functon L z ) of slow varaton) an at z 0s), the soluton of the equaton hs)gz 0 s)) = 0 from the enomnator, that nuces a polar sngularty. It s easy to show that the functon s z 0 s) s contnuous, n partcular we have lm n z 0 e λ/bn ) = whch means that both sngulartes conce n the lmt. Nevertheless, t turn out that we can eal wth these two sngulartes more or less) separately. We make use of a path of ntegraton γ that s epcte n Fgure for a precse escrpton of γ see 2)) an assume that the asymptotc leang term of the ntegral comes from the part of the ntegraton that s close to the two sngulartes z = an z = z 0 s). For the sake of shortness we wll only ncate the man steps. A precse an complete asymptotc analyss can be easly complete by a varaton of the methos presente n 2). The sngular behavour of fs,z) aroun the sngularty z = s of the form zhs)az)l fs,z) = hs)gz) z γ ) hs)a)l hs) z ).
Lmt Theorems for A Degenerate Fxe Pont Equaton 7 Thus, by aaptng the methos of 3) we obtan a contrbuton from the corresponng part of the contour ntegraton that s of orer ) ) Ln) Mn) O = O n hs) Ln) f s = e λ/bn an b n = cnmn)/ln) 2. By the assumpton that L u) = Mu)/u, where Mu) s of slow varaton, t follows that Mu)/Lu) 0 as u. Hence, ths part of the ntegral s neglgble. The sngular behavour of fs,z) aroun the sngularty z = z 0 s) s gven by ) Snce g z 0 s)) Az 0 s))l Az 0 s))l fs,z) g z 0 s)) z 0s) z 0s) z z 0s) ). ) as s we get a contrbuton of the corresponng part of the ntegral that s asymptotcally of the form z 0 s) n + o)) s = e λ/bn. Summng up we obtan an asymptotc representaton for Ee λxn/bn = z 0 e λ/bn ) n + o)) = e n log z0eλ/bn) + o)). 8) As ncate above ths heurstc analyss can be rgorously proven by aaptng the methos of 2). Thus, we have to stuy the behavour of z 0 s). Lemma 3 Suppose that α = an that Lu) s of slow varaton an satsfes all assumptons of the thr part of Theorem. Let z 0 s) enote the soluton of the equaton hs)gz 0 s)) = 0 an set b n = cnmn)/ln) 2. Then ) w log z 0 e w/bn ) = b n L b nw L )) b nw + o 9) n for every fxe complex number w wth argw) π ψ for a properly chosen ψ 0, π 2 )). Proof: For smplcty we assume that az) =. The general case can be prove n a smlar way. By the assumptons on Mu) we have LuLu)) Lu) Mu)log Lu) an, thus, LuLu)) Lu) mples Mu)log Lu))/Lu) 0 as u. Next we wrte the unque soluton u of the equaton v = u/lu) as u = v Lv), where Lu) satsfes the equaton Lu) = Lu Lu)). It s now an easy exercse to show that Lu) Lu) as u an also Lαu) Lu) for every fxe α Cv 0,ψ)). We can be even more precsely. From Lu) = Lu Lu)) an the assumptons on Mu)) t follows that Lu) Lu) Mu)log Lu) Mu)log Lu) an consequently log Lu)/Lu)) Mu)log Lu)/Lu). Smlarly we obtan Lu) LuLu)) Mu)2 log Lu) Lu) an consequently for any fxe w C wth arg w π ψ) w b n Lbn /w) w b n Lb n /w Lb n /w)) wmn)2 log Ln) wmn)log Ln) b n Ln) 2 = = o cnln) ). n
8 Mchael Drmota an Markus Kuba ) Next note that the equaton hs)gz 0 s)) = s equvalent to v = u/lu) f we set v = / hs) an u = / z). Thus we can explctly compute z 0 s) as z 0 s) = ) / L / )). hs) hs) Hence, f we set s = e w/bn then we get log z 0 e w/bn ) = c n Lcn ) + O b 2 ) n, where c n = b n /w+ob 2 n ). It s now an easy exercse to replace c n asymptotcally by b n /w by estmatng the fference n the same style as above. Ths mmeatley leas to 9). We can now complete the proof of the thr part of Theorem. As n the proof of Lemma 3 we have u u LuLu)) L w w)) L Mu)log ulu) ) = Mu) log w + log Lu) ) Mu)log w. Lu/w) Of course, ths mples ululu)) ul u w L )) u w Smlarly, we can show that u w L u w u u ulu) 2 LuLu)) L w w))) L LnMn)/Ln)) Lb n Lb n )) = omn)). Mu) log w. ulu) 2 Consequently, f we set w = λ for some fxe real number λ we, thus, obtan from Lemma 3 nλ nlog z 0 e λ/bn ) = b n L b n λ L )) b n + o) λ nλ = b n LnMn)/Ln)) + nλ b n Lb n Lb n )) ) nλ + b n L b n λ L nλ )) b n + o) λ b n Lb n Lb n )) = λa ) n nmn) + o b n b n Ln) 2 Together wth 8) ths yels = λa n b n + λlogλ) + o). + nλmn) logλ) + o) b n Ln) 2 Ee λxn an)/bn = e λ logλ) + o) = e λ log λ π 2 λ + o) ) nλ b n LnMn)/Ln)) an completes the proof of the thr part of Theorem.
Lmt Theorems for A Degenerate Fxe Pont Equaton 9 4 Transfers of Lmtng Dstrbutons As alreay note above t seems to be a non-trval problem to tranfer a lmt theorem for X n to a corresponng lmt theorem for Z n recall ther efntons ) an 3)). For example, t s not true that Z n X n )/b n 0 n L 2. In fact, s seems that t s an nherent problem n ths context that one cannot apply moment methos snce the lmt theorem 6) has no counterpart on the level of moments. Nevertheless, t s possble to prove For ths purpose we note that X n can be alternatvely efne by Z n X n b n 0 n probablty. 0) X n = Xn Fn + n ), X = 0, where P{F n = k} s gven by the followng formula: P{F n = k} = kk + ) k n ), P{F n = n} = n. ) In ths short verson of the paper we wll only concentrate on ths specal choce of D n resp. F n. More general stuatons wll be consere n the full verson of the paper. We relate the ranom varables Z n an X n wth a quantty stue by Iksanov an Moehle 5). Let ξ ) N be a sequence of nepenent copes of a ranom varable ξ wth values n N := {,2...,}. For arbtrary but fxe n N, efne a two-mensonal couple) process R n) R n) 0,Sn) 0 ) := 0;0) an, for N, R n),s n) ) := R n),sn),s n) { ) + ξ,ξ ) f ξ < n R n) 0,ξ ) else., ) N0 recursvely va The process S n) ) N0 s a zeroelaye ranom walk S n) = ξ + ξ 2 + + ξ, N 0 ) an oes not ) N0 has non-ecreasng paths, starts n R n) 0 = 0 an satsfes R n) < n epen on n. The process R n) for all N 0. By nucton on t follows that R n) Let M n := { N R n) Rn) Furthermore let N n := mn{ N S n) S n), N 0. } enote the total number of jumps of the process R n) ) N0. n} enote the number of steps the ranom walk S n) ) N0 nees to reach a state larger than or equal to n. It was shown n 5) that the strbutons of M n an Z n are entcal f P{ξ = k} = /kk + )) an also that P{log n) 2 N n M n )/n > ǫ} 0. Hence we only have to relate X n wth N n. Set N 0 := 0 an observe that for n we have P{N n = } = P{mn{ N S n} = } = P{ξ n} = k n Further, for 2 m n we have P{N n = m} = n k=m P{ξ m = n k}p{n k = m } = n k=m kk + ) = n. P{N k = m } n k + )n k + 2).
0 Mchael Drmota an Markus Kuba Of course, ths s the same relaton as that for X n. Hence, N n = Xn an, thus, Z n X n )/b n 0 n probablty. Acknowlegements The authort want to thank Brgtte Chauvn an Alan Rouault from Versalles for pontng out reference 9) on the Mttag-Leffler strbuton. References [] M. Drmota, A. Iksanov, M. Moehle, an U. Roesler, Asymptotc results about the total branch length of the Bolthausen-Szntman coalescent, Stoch. Proc. Appl., to appear. [2] M. Drmota, A. Iksanov, M. Moehle, an U. Roesler, A lmtng strbuton for the number of cuts neee to solate the root of a ranom recursve tree, Ranom Struct. Algorthms, submtte. [3] Ph. Flajolet an A. M. Olyzko, Sngularty analyss of generatng functons, SIAM J. Dscrete Math., 3 990), 26 240. [4] Ph. Flajolet an R. Segewck, Analytc Combnatorcs, manuscrpt, http://algo.nra.fr/flajolet/publcatons/books.html. [5] A. Iksanov an M. Moehle, A probablstc proof of a weak lmt law for the number of cuts neee to solate the root of a ranom recursve tree, Electron. Comm. Probab. 2, 2007), to appear. [6] S. Janson, Ranom cuttng an recors n etermnstc an ranom trees, Ranom Struct. Algorthms 29 2006), 39 79. [7] R. Nennger an L. Rüschenorf, On the contracton metho wth egenerate lmt equaton, Ann. Prob. 32 2004), 2838 2856. [8] A. Panholzer, Destructon of recursve trees, In: Mathematcs an Computer Scence III, Brkhäuser, Basel, 2004, pp. 267 280. [9] J. Ptman, Combnatoral Stochastc Processes, Lecture Notes n Mathematcs 875, Sprnger, Berln, 2006.