Minima in branching random walks

Addario‐Berry, Louigi; Reed, Bruce

doi:10.1214/08-aop428

Cited by 126 publications

(270 citation statements)

References 30 publications

(69 reference statements)

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“…(iii) Under (3.12) and suitable integrability assumptions, Addario-Berry and Reed [1] obtain a very precise asymptotic estimate of E[inf |x|=n V (x)], as well as an exponential upper bound for the deviation probability for inf |x|=n V (x) − E[inf |x|=n V (x)], which, in particular, implies (3.16).…”

Section: Central Limit Theoremmentioning

confidence: 74%

“…The new particles form the first generation. Each of the new particles produces offspring according to the same probability distribution, independently of each 1 Sometimes also referred to as a Bienaymé-Galton-Watson tree. other and of everything else in the generation.…”

Section: Galton-watson Trees and Extinction Probabilitiesmentioning

confidence: 99%

“…Since E(X 1 ) = 0 (which is a consequence of the assumption ψ ′ (1) = 0), the random walk (S k ) is centered; so the probability above is n −(3/2)+o (1) . Combining this with (3.25), (3.26) and (3.27) yields…”

Section: Small Moments Of Partition Function: Upper Bound In Theorem 35mentioning

confidence: 99%

“…If β = 1, we write simply Z n instead of Z n,1 . Z n,β = n −3β/2+o (1) , in probability. An important step in the proof of Theorems 3.2 and 3.3 is to estimate all small moments of Z n and Z n,β , respectively.…”

mentioning

confidence: 99%

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Random Walks and Trees

Shi

2011

ESAIM: Proc.

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Abstract. These notes provide an elementary and self-contained introduction to branching random walks.Section 1 gives a brief overview of Galton-Watson trees, whereas Section 2 presents the classical law of large numbers for branching random walks. These two short sections are not exactly indispensable, but they introduce the idea of using size-biased trees, thus giving motivations and an avant-goût to the main part, Section 3, where branching random walks are studied from a deeper point of view, and are connected to the model of directed polymers on a tree.Tree-related random processes form a rich and exciting research subject. These notes cover only special topics. For a general account, we refer to the St-Flour lecture notes of Peres [47] References 38 Galton-Watson treesWe start by studying a few basic properties of supercritical Galton-Watson trees. The main aim of this section is to introduce the notion of size-biased trees. In particular, we see in Subsection 1.3 how this allows us to prove the well-known Kesten-Stigum theorem. This notion of size-biased trees will be developed in forthcoming sections to study more complicated models. Galton-Watson trees and extinction probabilitiesWe are interested in processes involving (rooted) trees. The simplest rooted tree is the regular rooted tree, where each vertex has a fixed number (say m, with m > 1) of offspring. For example, here is a rooted binary tree:Let Z n denote the number of vertices (also called particles or individuals) in the n-th generation, thenIn probability theory, we often encounter trees where the number of offspring of a vertex is random. The easiest case is when these random numbers are i.i.d., which leads to a Galton-Watson treeA Galton-Watson tree starts with one initial ancestor (sometimes, it is possible to have several or even a random number of initial ancestors, in which case it will be explicitly stated). It produces a certain number of offspring according to a given probability distribution. The new particles form the first generation. Each of the new particles produces offspring according to the same probability distribution, independently of each 1 Sometimes also referred to as a Bienaymé-Galton-Watson tree. other and of everything else in the generation. And the system regenerates. We write p i for the probability that a given particle has i children, i ≥ 0; thus To avoid trivial discussions, we assume throughout that p 0 + p 1 < 1.As before, we write Z n for the number of particles in the n-th generation. It is clear that if Z n = 0 for a certain n, then Z j = 0 for all j ≥ n. One of the first questions we ask is about the extinction probability q := P{Z n = 0 eventually}.(1.1) 4 ESAIM: PROCEEDINGSIt turns out that the expected number of offspring plays an important role. Let(1.2) Theorem 1.1. Let q be the extinction probability defined in (1.1).(i) The extinction probability q is the smallest root of the equation f (s) = s for s ∈ [0, 1], whereis the generating function of the reproduction law.(ii) In particular, q = 1 if m ≤...

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Section: Central Limit Theoremmentioning

confidence: 74%

Section: Galton-watson Trees and Extinction Probabilitiesmentioning

confidence: 99%

Section: Small Moments Of Partition Function: Upper Bound In Theorem 35mentioning

confidence: 99%

mentioning

confidence: 99%

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Random Walks and Trees

Shi

2011

ESAIM: Proc.

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“…. , A N ) and L(R) denotes the law of the random variable R. A fixed point of the smoothing transform is given by any µ ∈ P (R) such that, if R has distribution µ, the equation The homogeneous equation (1.1) is used for example, to study interacting particle systems [9] or the branching random walk [1,12]. In recent years, from practical reasons, the inhomogeneous equation has gained importance.…”

Section: Introductionmentioning

confidence: 99%

Precise tail asymptotics of fixed points of the smoothing transform with general weights

Buraczewski¹,

Damek²,

Zienkiewicz³

2015

Bernoulli

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We consider solutions of the stochastic equationAiRi + B, where N > 1 is a fixed constant, Ai are independent, identically distributed random variables and Ri are independent copies of R, which are independent both from Ai's and B. The hypotheses ensuring existence of solutions are well known. Moreover under a number of assumptions the main being E|A1| α = 1/N and E|A1| α log |A1| > 0, the limit limt→∞ t α P[|R| > t] = K exists. In the present paper, we prove positivity of K.

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Advanced Branching Processes

Simatos

2011

Wiley Encyclopedia of Operations Research and Management Science

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In this article, we discuss three important classes of advanced models of branching processes, namely branching processes in random environment, branching random walks, and continuous state branching processes. These models have recently been the object of intense research activity, and we highlight some of their most basic properties, linked for instance to extinction problems or long‐time behavior.

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Minima in branching random walks

Cited by 126 publications

References 30 publications

Random Walks and Trees

Random Walks and Trees

Precise tail asymptotics of fixed points of the smoothing transform with general weights

Advanced Branching Processes

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