DRMOTA, M. and GITTENBERGER, B. (1997) On the profile of random trees. Rand. Str. Alg. 10, 421--451.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....time of the normalized Brownian excursion e( atlevelt,i.e. #(t) lim ##0 1 # # 1 0 I [t,t #] e(u) du. Aldous [1] conjectured that t # # G (n) #t # n# # n would converge weakly, as a stochastic process, to t # # #(t) 2. Aldous s conjecture was settled by Drmota and Gittenberger [9]. As noted by these last authors, their result entails the weak convergence of W n # n to the maximum m of the Brownian excursion, as #(t) is itself a Brownian excursion changed of time [5] Previously, the weak convergence of W n # n to m was proven directly by Takacs (1993) However weak ....

M. Drmota & B. Gittenberger, (1997) On the profile of random trees. Random Structures Algorithms 10, No. 4, 421--451.

.... [19] compare Biane [6] Th eor eme 3) This version of Theorem 1 has been observed by Aldous in special cases (as the geometric offspring distributions) and conjectured in the general finite variance case, compare [3] A first proof of the conjecture was given 3 by Drmota and Gittenberger [11]. They mastered the formidable task to obtain convergence of the finite dimensional distributions as well as tightness, using generating functions and thereby generalizing work of Kennedy [25] on the one dimensional distributions. Pitman [34] surrounded the difficulties by imbedding the problem ....

....the limit n 1, as was shown by Aldous, such that the asymptotic height profile can still be described by Brownian local time. For ff 2 however, the dependence structure survives in the limit. The combinatorics of CGW (n) trees have been widely studied by means of generating functions (see f.e. [11, 17, 25, 31]) Probabilistic methods have been introduced in Kolchin [28] and in particular by Aldous [2, 3] Our proof of theorem 1 is based on two probabilistic constructions, which are valid for the infinite variance case, too. The first one will be described in section 2, it establishes a connection ....

Drmota, M. and Gittenberger, B.: On the profile of random trees. Random Struct. Alg. 10, 421-451 (1997) 22

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DRMOTA, M. and GITTENBERGER, B. (1997) On the profile of random trees. Rand. Str. Alg. 10, 421--451.

....Another correspondence which was pointed out by Aldous [1] is considering simply generated trees as representations of Galton Watson branching processes conditioned on the total progeny. Under this point of view the offspring distribution induces the weights (1. 1) for more details see [1] or also [4]) Date: March 7, 2000. This research was supported by the Austrian Science Foundation FWF, grant P10187 MAT, and by the Stiftung Aktion Osterreich Ungarn, grant 34oeu24. Department of Geometry, TU Wien, Wiedner Hauptstrasse 8 10 113, A 1040 Wien, Austria, email: ....

M. Drmota and B. Gittenberger, On the profile of random trees, Random Struct. Alg. 10 (1997), 421--451.

.... scaled excursion X at a, denoted by (a) have been studied by several authors (note that for an excursion of length we have: a) d j p (a= p ) See for instance Getoor and Sharpe [8] Knight [16] Cohen and Hooghiemstra [3] Hooghiemstra [11] Drmota and Gittenberger [5]. Several representations of the one dimensional Brownian Excursion local time density are known. Results for the two dimensional local time density can be found in [3] and [5] In both papers indirect methods have been used: Approaching via queuing theory and random trees, respectively. In this ....

....for instance Getoor and Sharpe [8] Knight [16] Cohen and Hooghiemstra [3] Hooghiemstra [11] Drmota and Gittenberger [5] Several representations of the one dimensional Brownian Excursion local time density are known. Results for the two dimensional local time density can be found in [3] and [5]. In both papers indirect methods have been used: Approaching via queuing theory and random trees, respectively. In this paper we aim at deriving expressions for all dimensions and offer two different methods to do this: a direct and an indirect one. The first one is a direct computation by means ....

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DRMOTA, M. and GITTENBERGER, B. (1997) On the profile of random trees. Rand. Str. Alg. 10, 421--451. 23

....converges to reflecting Brownian bridge (rBB) i.e. the process identical in law to (jW (t) Gamma tW (1)j; 0 t 1) where W (t) is a one dimensional Brownian motion (BM) or roughly speaking rBB is a BM of length 1 reflected at 0 and conditioned to have zeros at 0 and 1. In view of the results in [9, 11] this suggests that the process l n (t) n Gamma1=2 Ln (t p n) t 0, where Ln (t) btc 1 Gamma t)L n (btc) t Gamma btc)L n (btc 1) for non integral t 0; converges weakly to the local time process for rBB. In fact, we will prove Theorem 1.1. Let B(t) denote reflecting Brownian ....

.... nm;k u m z n n = 1 1 Gamma a k (z; u) with a k (z; u) y k (z; ua(z) where y 0 (z; u) u y i 1 (z; u) ze y i (z;u) i 0; and a(z) is the well known tree function given by its functional equation a(z) z exp(a(z) This follows immediately from the combinatorial setup (details see [9]) Hence the characteristic function of n Gamma1=2 Ln (k) is OE kn (t) n n n [z n ] i 1 Gamma y k i z; e it= p n a(z) jj Gamma1 and that of Gamma n Gamma1=2 Ln (k 1 ) n Gamma1=2 Ln (k p ) Delta is given by OE k1 Delta Delta Deltak pn (t 1 ; t p ....

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M. Drmota and B. Gittenberger, On the profile of random trees, Random Struct. Alg. 10 (1997), 421-- 451.

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