## Connecting Yule process, bisection and binary search tree via martingales (2004)

Venue: | J. Iranian Statistical Society |

Citations: | 10 - 4 self |

### BibTeX

@ARTICLE{Chauvin04connectingyule,

author = {B. Chauvin and A. Rouault},

title = {Connecting Yule process, bisection and binary search tree via martingales},

journal = {J. Iranian Statistical Society},

year = {2004},

volume = {3},

pages = {89--116}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present new links between some remarkable martingales found in the study of the Binary Search Tree or of the bisection problem, looking at them on the probability space of a continuous time binary branching process. Key words. Binary search tree, branching random walk, Yule process, convergence of martingales, functional equations. A.M.S. Classification. 60J25, 60J80, 60J85, 68W40, 60G42, 60G44. 1

### Citations

463 | Branching processes
- Athreya, Ney
- 1972
(Show Context)
Citation Context ... and phrases: Binary search tree, branching random walk, Yule process, convergence of martingales, functional equations.2 Chauvin and Rouault disjoint subtrees. The Yule process (see Athreya and Ney =-=[2]-=-) is a continuous time binary branching process in which an ancestor has an exponential 1 distributed lifetime and at his death, gives rise to two children with independent exponential 1 lifetimes and... |

219 |
Evolution of Random Search Trees
- Mahmoud
- 1992
(Show Context)
Citation Context ...ctional equations. A.M.S. Classification. 60J25, 60J80, 60J85, 68W40, 60G42, 60G44. 1 Introduction This paper is a kind of game with martingales around the Binary Search Tree (BST) model (see Mahmoud =-=[25]-=-). The BST process, under the random permutation model is an increasing sequence of binary trees (Tn)n≥0 storing data, in such a way that for every integer n, Tn has n + 1 leaves; the growing from tim... |

101 | On some exponential functionals of Brownian motion
- Yor
- 1992
(Show Context)
Citation Context ...straint (41) leads to κ = 4. Taking into account the unicity, we see that E exp −xM(∞, 1/4) = (1 + √ 2x)e −√ 2x , which allows to identify the law of M(∞, 1/4). As it is noticed in another context in =-=[30]-=- p.110-111, this is the Laplace transform of the density G(t) = 1 √ 2πt 5 e−1/2t , t ≥ 0 of � 2γ 3/2) −1 , where we use the notation γα for a variable with distribution gamma with parameter α, i.e. P(... |

58 |
Martingale convergence in the branching random walk
- Biggins
- 1978
(Show Context)
Citation Context ... (positive) martingale associated to this process is � Mn(θ) = e θx−nΛ(θ) Zn(dx) . IR Let M∞(θ) the a.s. limit. Under a “k log k” type condition, we have (Biggins convergence theorem, for instance in =-=[5, 6]-=-) - if θΛ ′ (θ) − Λ(θ) < 0, the convergence is also L 1 and EM∞(θ) = 1, - if θΛ ′ (θ) − Λ(θ) ≥ 0, then M∞(θ) = 0 a.s.. By analogy with the Galton-Watson process, we call “supercritical” the values of ... |

54 |
Fixed points of the smoothing transformation
- Durrett, Liggett
- 1983
(Show Context)
Citation Context ...mits of martingales. The solutions of these equations have distributions which are fixed points of so-called smoothing transformations, as defined in Holley and Liggett [18] or in Durrett and Liggett =-=[17]-=-. Owing to known results on uniqueness of their Laplace transforms (Liu [24], [25], Kyprianou [21], Biggins and Kyprianou [8]) we get equalities in law (Proposition 4.1). Section 5 is devoted to proof... |

43 | Branching processes and their applications in the analysis of tree structures and tree algorithms - Devroye - 1998 |

42 |
On growing random binary trees
- Pittel
- 1984
(Show Context)
Citation Context ... LAMA, Bâtiment Fermat, Université de Versailles F-78035 Versailles. e-mail: Name@math.uvsq.fr 1sBy embedding, the BST is a Yule-tree process stopped at τn, the first time when n+1 individuals exist (=-=[28, 7, 12]-=-). A continuous time family of martingales (M(t, z), t ≥ 0) is attached to this model; it is defined in section 2.2. In a recent paper [11] where several models are embedded in the probability space o... |

38 | The profile of binary search trees
- Chauvin, Drmota, et al.
- 2001
(Show Context)
Citation Context ... precisely, we call BST martingale where C0(z) = 1 and Cn(z) = M BST n (z) = � n−1 � k=0 k + 2z k + 1 u∈∂Tn z |u| � Uk(n) = Cn(z) Cn(z) k zk = (−1)n � � −2z , n ≥ 1 . n It is proved (in Jabbour & al. =-=[10, 18, 11]-=-) that in the supercritical range z ∈ (z− c , z + c ), this martingale converges in L1 to a nondegenerate limit M BST ∞ (z) and converges a.s. to 0 elsewhere, in particular for the critical values z− ... |

36 |
Fixed points of a generalized smoothing transformation and applications to the branching random walk
- LIU
- 1998
(Show Context)
Citation Context ... of smoothing transforms that are needed for our study. There is a broad literature on this topic. One of the more recent contribution is [8]. We choose to give these results under assumptions of Liu =-=[23]-=-, [24] (see also Kyprianou [20]), which are fulfilled in our examples. Let us consider a branching random walk as in Section 2.1 with Z = �N δxi i=1 and P (N = 0) = 0. We defined the martingale � Mn(θ... |

33 |
Uniform convergence of martingales in the branching random walk
- Biggins
- 1992
(Show Context)
Citation Context ... (positive) martingale associated to this process is � Mn(θ) = e θx−nΛ(θ) Zn(dx) . IR Let M∞(θ) the a.s. limit. Under a “k log k” type condition, we have (Biggins convergence theorem, for instance in =-=[5, 6]-=-) - if θΛ ′ (θ) − Λ(θ) < 0, the convergence is also L 1 and EM∞(θ) = 1, - if θΛ ′ (θ) − Λ(θ) ≥ 0, then M∞(θ) = 0 a.s.. By analogy with the Galton-Watson process, we call “supercritical” the values of ... |

33 | Measure change in multitype branching
- Biggins, Kyprianou
(Show Context)
Citation Context ...rmations, as defined in Holley and Liggett [18] or in Durrett and Liggett [17]. Owing to known results on uniqueness of their Laplace transforms (Liu [24], [25], Kyprianou [21], Biggins and Kyprianou =-=[8]-=-) we get equalities in law (Proposition 4.1). Section 5 is devoted to proofs of theorems of Section 3. In particular, we show that equalities in law are (a.s.) equalities between random variables. In ... |

32 | An analytic approach to the height of binary search trees. Algorithmica
- Drmota
- 2001
(Show Context)
Citation Context ...seen as fixed point of smoothing transforms, as called by Holley-Liggett [17]. Taking Laplace transforms we get functional equations which are related to those investigated in Drmota’s recent papers (=-=[14, 15]-=-). We stress that unicity of solutions of functional equations provides here a.s. identities of random variables. Section 5 is devoted to proofs of theorems. Let us now precise some notations. In the ... |

30 |
Multiplicative martingales for spatial branching processes, in
- Neveu
- 1987
(Show Context)
Citation Context ...less directly imported from theorem 3 in [21]. It is written here to make the proof self-contained. 13sMultiplicative martingales appeared many times in the literature, for instance in Neveu, Chauvin =-=[26, 9]-=- in branching brownian motion frame, in Biggins and Kyprianou [8] for discrete branching random walks and in Kyprianou [21] for branching Lévy processes. They are studied for themselves in relation wi... |

24 |
General branching processes as Markov fields
- Jagers
- 1989
(Show Context)
Citation Context ...of this sequence. This theorem holds without any serious difficulty as soon as the notion of “line” is precisely defined, which is necessary since several notions exist 2 ; “Optional lines” in Jagers =-=[19]-=- require measurability of the stopping rule with respect to the process until the line. More restrictively, “stopping lines” in Chauvin [9] and Kyprianou [21, 22] or frosts (in the fragmentation frame... |

23 | A di®usion limit for a class of randomly-growing binary trees. Probab. Theory Related Fields
- Aldous, Shields
- 1988
(Show Context)
Citation Context ...les 1) The Bisection martingale Passing to logarithms in the bisection problem, we get a discrete time branching random walk whose reproduction measure is Z BIS = δ− log U + δ − log(1−U), where U ∼ U(=-=[0, 1]-=-) 1 . We have Λ(θ) = log E[U −θ + (1 − U) −θ ] = log 2 1 − θ . Let us operate a change of parameter, setting z = 1−θ 2 corresponding martingale is so that Λ(θ) = − log z. The M BIS n (z) := � |u|=n e ... |

23 |
On generalized multiplicative cascades. Stochastic Process
- Liu
- 2000
(Show Context)
Citation Context ...o (positive) solutions of z log z − z + 2 = 0 , 2 i.e. (c ′ and c are notations of Drmota [15]) z − c = c′ 2 = 0.186..., z+ c = c = 2.155... 2 For z = z− c (resp. z + c ), applying Theorem 2.5 of Liu =-=[24]-=-, we see that the derivative (M BIS n ) ′ (z) converges a.s. to a limit denoted M ′BIS ∞ (z) which is positive (resp. negative) and has infinite expectation. 2) The Yule-time martingale 1 U([0, 1]) is... |

21 | Probability on Trees: An Introductory Climb
- Peres
- 1999
(Show Context)
Citation Context ... clear that (for z �= 1/2) M(∞, z) is different from M BST ∞ (z). This is consistent with the fact that τn does not define a stopping line. 2 let us also mention the close notion of “cutset” in Peres =-=[27]-=- 9s4 Smoothing transforms and limit distributions The random limits mentioned in Section 2 satisfy “duplication” relations, which come from the binary branching structure of the underlying processes. ... |

21 | Rouault Discretization methods for homogeneous fragmentations
- Bertoin, A
- 2005
(Show Context)
Citation Context ... of expectation 0, which converge a.s. to a random variable of constant sign, of infinite expectation, under appropriate conditions (Kyprianou [21], Liu [25], BigginsKyprianou [8] and Bertoin-Rouault =-=[4]-=-). The details are given below. 2.2 Examples Bisection martingale Passing to logarithms in the bisection problem, we get a discrete time branching random walk whose reproduction measure is Z BIS = δ− ... |

19 |
Product martingales and stopping lines for branching Brownian motion
- Chauvin
- 1991
(Show Context)
Citation Context ... since several notions exist 2 ; “Optional lines” in Jagers [19] require measurability of the stopping rule with respect to the process until the line. More restrictively, “stopping lines” in Chauvin =-=[9]-=- and Kyprianou [21, 22] or frosts (in the fragmentation frame, cf Bertoin [3]) require measurability of the stopping rule with respect to the branch from the root to some node of the line. The above m... |

15 | How fast does a general branching random walk spread? In Classical and modern branching processes
- Biggins
- 1994
(Show Context)
Citation Context ... LAMA, Bâtiment Fermat, Université de Versailles F-78035 Versailles. e-mail: Name@math.uvsq.fr 1sBy embedding, the BST is a Yule-tree process stopped at τn, the first time when n+1 individuals exist (=-=[28, 7, 12]-=-). A continuous time family of martingales (M(t, z), t ≥ 0) is attached to this model; it is defined in section 2.2. In a recent paper [11] where several models are embedded in the probability space o... |

14 |
Generalized potlatch and smoothing processes
- Holley, Liggett
- 1981
(Show Context)
Citation Context ...Section 4: we write the (stochastic) equations satisfied by the limits of martingales. The solutions of these equations can be seen as fixed point of smoothing transforms, as called by Holley-Liggett =-=[17]-=-. Taking Laplace transforms we get functional equations which are related to those investigated in Drmota’s recent papers ([14, 15]). We stress that unicity of solutions of functional equations provid... |

13 | Martingales and large deviations for binary search trees. Random Structure and Algorithms
- Jabbour-Hattab
- 2001
(Show Context)
Citation Context ...mber of leaves in each generation. A polynomial that codes the profile (called the level polynomial) allows to define a family of martingales (M BST n (z), n ≥ 0) where z is a positive real parameter =-=[18]-=-. It is defined in section 2.3. There are (at least) two ways of connecting the BST model to branching random walks, and take advantage of related probabilistic methods and results. • The first one co... |

12 |
Martingale convergence and the stopped branching random walk. Probab. Theory Related Fields 116
- KYPRIANOU
- 2000
(Show Context)
Citation Context ...ions exist 2 ; “Optional lines” in Jagers [19] require measurability of the stopping rule with respect to the process until the line. More restrictively, “stopping lines” in Chauvin [9] and Kyprianou =-=[21, 22]-=- or frosts (in the fragmentation frame, cf Bertoin [3]) require measurability of the stopping rule with respect to the branch from the root to some node of the line. The above mentioned general theore... |

10 | Stochastic analysis of tree-like data structures
- Drmota
- 2004
(Show Context)
Citation Context ...ndependence between subtrees, it gives a discrete time branching random walk; call it the Yule-generation process. • The second way consists in “approaching” the BST by the so-called bisection model (=-=[12, 13]-=-). It is also known as the Kolmogorov’s rock model. An object (rock) is initially of mass one. At time 1 it is broken into two rocks with uniform size. At time n each rock (there are 2 n ) is broken i... |

10 |
Slow variation and uniqueness of solutions to the functional equation in the branching random walk
- KYPRIANOU
- 1998
(Show Context)
Citation Context ...re needed for our study. There is a broad literature on this topic. One of the more recent contribution is [8]. We choose to give these results under assumptions of Liu [23], [24] (see also Kyprianou =-=[20]-=-), which are fulfilled in our examples. Let us consider a branching random walk as in Section 2.1 with Z = �N δxi i=1 and P (N = 0) = 0. We defined the martingale � Mn(θ) = e θx−Λ(θ) Z(dx) and the der... |

9 |
Spatial growth of a branching process of particles living in Rd
- Uchiyama
- 1982
(Show Context)
Citation Context ...tion of individuals alive at time t, the (positive) martingale associated to this process is � M(t, θ) = e θx−tL(θ) Zt(dx) . IR Its behavior as t → ∞ is similar to the above, with L instead of Λ ([6],=-=[29]-=-). We denote by M(∞, θ) its limit. For critical values of θ, the derivative ∂ ∂θ Mn(θ) (or ∂ ∂θ M(t, θ)) are martingales of expectation 0, which converge a.s. to a r.v. of constant sign, of infinite e... |

6 |
A note on branching L¶evy processes. Stochastic Process
- Kyprianou
- 1999
(Show Context)
Citation Context ...ions exist 2 ; “Optional lines” in Jagers [19] require measurability of the stopping rule with respect to the process until the line. More restrictively, “stopping lines” in Chauvin [9] and Kyprianou =-=[21, 22]-=- or frosts (in the fragmentation frame, cf Bertoin [3]) require measurability of the stopping rule with respect to the branch from the root to some node of the line. The above mentioned general theore... |

5 |
Exercises in probability
- Chaumont, Yor
- 2003
(Show Context)
Citation Context ...s. E[M BST ∞ (z)|FBIS ∞ Set for a while, X := MBST ∞ Summarizing (33) and (26), we have ] = MBIS (z) . (33) ∞ (z), Y := M BIS ∞ (z) and A := F BIS ∞ . E[X|A] = Y and X law = Y . From Exercise 1.11 in =-=[9]-=- this implies X = Y a.s. 5.2 Proof of Theorem 3.3 It is exactly the same line of argument as in the above subsection, using (25) instead of (26) and (14) instead of (17). 5.3 Proof of Theorem 3.4 Sinc... |

4 |
Additive martingales and probability tilting for homogeneous fragmentations
- Bertoin, Rouault
(Show Context)
Citation Context ... Yule martingale (e −t Nt, t ≥ 0) which is known to converge a.s. to a random variable ξ ∼ E(1). The behavior of this family follows the same rule as above. Moreover, it was proved in Bertoin-Rouault =-=[4]-=- (see also [11]) that for z = z ± c , the derivative M ′ (t, z) converges a.s. to a limit denoted by M ′ (∞, z), of constant sign and infinite expectation. 3) The Yule-generation martingale The Yule-g... |

1 |
Saccà D.: Multiple total stable models are definitely needed to solve unique solution problems
- unknown authors
- 1996
(Show Context)
Citation Context ... − c , z + c ) . Let us summarize some results on fixed points of smoothing transforms that are needed for our study. There is a broad literature on this topic. One of the more recent contribution is =-=[8]-=-. We choose to give these results under assumptions of Liu [23], [24] (see also Kyprianou [20]), which are fulfilled in our examples. Let us consider a branching random walk as in Section 2.1 with Z =... |

1 |
Martingales, Embedding and Tilting of Binary Trees. Preprint available at http://fermat.math.uvsq.fr/ marckert/papers.html
- Chauvin, Klein, et al.
- 2003
(Show Context)
Citation Context ...ped at τn, the first time when n+1 individuals exist ([28, 7, 12]). A continuous time family of martingales (M(t, z), t ≥ 0) is attached to this model; it is defined in section 2.2. In a recent paper =-=[11]-=- where several models are embedded in the probability space of the Yule-tree process, the martingale (M BST n (z), n ≥ 0) appears as a projection of the martingale (M(t, z), t ≥ 0). Besides, consider ... |

1 |
An analytic approach to some height of binary search trees II
- Drmota
- 2003
(Show Context)
Citation Context ...seen as fixed point of smoothing transforms, as called by Holley-Liggett [17]. Taking Laplace transforms we get functional equations which are related to those investigated in Drmota’s recent papers (=-=[14, 15]-=-). We stress that unicity of solutions of functional equations provides here a.s. identities of random variables. Section 5 is devoted to proofs of theorems. Let us now precise some notations. In the ... |