Reduced Branching Processes

Yakymiv, A. L.

doi:10.1137/1125069

Cited by 16 publications

(21 citation statements)

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“…One of the important characteristics of a genealogical tree is the so-called reduced branching process Z m,n , 0 m n, in which Z m,n equals the number of particles in the process under consideration at time m n, each of which has a nonempty offspring at time n. Properties of the reduced processes for the Galton-Watson processes, continuous-time Markov branching processes, and Bellman-Harris branching processes have been analyzed in [5], [6], [9], [10], [11], [12], [13], [14], [15], [22], and [23].…”

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confidence: 99%

Reduced Branching Processes in Random Environment: The Critical Case

Vatutin¹

2003

Theory Probab. Appl.

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Let Zn be the number of particles at time n = 0, 1, 2, . . . in a branching process in random environment, Z 0 = 1, and let Zm,n be the number of such particles in the process at time m ∈ [0, n], each of which has a nonempty offspring at time n. It is shown that if the offspring generating functions f k (s) of the particles of the kth generation are independent and identically distributed for all k = 0, 1, 2, . . . with E log f k (1) = 0 and σ 2 = E(log f k (1)) 2 ∈ (0, ∞), then, under certain additional restrictions, the sequence of conditional processesKey words. critical branching process in random environment, reduced process, functional limit theorem, random walk PII. S0040585X97979421Branching processes in random environment are a rather popular object of investigation: see [1], [2], [3], [7], [8], [16], [18], [19], [20], [21], [22], [23], [24], [25], and [26].An informal description of these processes in the setting of interest for us appears as follows. A process is initiated at time n = 0 by a single particle of the zero generation which lives up to moment n = 1, and at the end of its life, produces offspring in accordance with a generating function f 0 (s) which is random, that is, selected from the set of all probability generating functions of integer-valued nonnegative variables in accordance with a probability distribution. The particles of the first generation live up to moment n = 2 and at the ends of their lives produce offspring according to a common generating function f 1 (s) which is a probabilistic copy of f 0 (s), and so on. Let Z n be the number of particles in the process at time n = 0, 1, . . . . The evolution law of Z n can be described in the following way:A branching process in random environment is said to be subcritical if ρ = E log f 0 (1) < 0, critical if ρ = 0, and supercritical if ρ > 0.In the present paper we study the critical processes in random environment. It is known that such processes die out with probability 1. The problem of finding the asymptotic behavior of the survival probability P{Z n > 0} as n → ∞ for the critical branching processes in random environment with generating functions of the general form proved to be difficult. Thus, Kozlov [7] proved as early as 1976 that if the * Received by the editors August 27, 2001. This work was supported in part by Russian Foundation for Basic Research grants 99-01-00012, 00-15-96136, and INTAS 99-01317.

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confidence: 99%

Reduced Branching Processes in Random Environment: The Critical Case

Vatutin¹

2003

Theory Probab. Appl.

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“…When the critical offspring distribution ρ has infinite variance, the study of scaling limits of T * n goes back to Vatutin [17] and Yakymiv [18]. If we scale the graph distance by the factor n −1 , as n → ∞ the discrete reduced tree n −1 T * n converges to a random continuous tree ∆ (α) that we now describe.…”

Section: Introductionmentioning

confidence: 93%

Typical Behavior of the Harmonic Measure in Critical Galton–Watson Trees with Infinite Variance Offspring Distribution

Lin

2017

J Theor Probab

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We study the typical behavior of the harmonic measure in large critical Galton-Watson trees whose offspring distribution is in the domain of attraction of a stable distribution with index α ∈ (1, 2]. Let µ n denote the hitting distribution of height n by simple random walk on the critical Galton-Watson tree conditioned on non-extinction at generation n. We extend the results of [11] to prove that, with high probability, the mass of the harmonic measure µ n carried by a random vertex uniformly chosen from height n is approximately equal to n −λα , where the constant λ α > 1 α−1 depends only on the index α. In the analogous continuous model, this constant λ α turns out to be the typical local dimension of the continuous harmonic measure. Using an explicit formula for λ α , we are able to show that λ α decreases with respect to α ∈ (1, 2], and it goes to infinity at the same speed as (α − 1) −2 when α approaches 1.

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“…All other things equal, we would expect a decrease in the population size. This effect thus counteracts the increase in size due to heavier tails and, for low enough values of β, the size of In Section 2 we present the so-called reduced branching process describing the genealogy of the particles alive at time t. In this section we recall the known limit processes for the reduced branching processes in the cases in which 0 < α ≤ 1 obtained in [3] and [9]. Then we state the main result of this paper, Theorem 2, giving a new limit structure as t tends to ∞ for the reduced branching process in the case in which α = 0 and (2) holds.…”

Section: Introductionmentioning

confidence: 99%

Reduced Branching Processes with Very Heavy Tails

Lagerås

Sagitov²

2008

J. Appl. Probab.

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The reduced Markov branching process is a stochastic model for the genealogy of an unstructured biological population. Its limit behavior in the critical case is well studied for the Zolotarev-Slack regularity parameter α ∈ (0, 1]. We turn to the case of very heavytailed reproduction distribution α = 0 assuming that Zubkov's regularity condition holds with parameter β ∈ (0, ∞). Our main result gives a new asymptotic pattern for the reduced branching process conditioned on nonextinction during a long time interval.

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Reduced Branching Processes

Cited by 16 publications

References 4 publications

Reduced Branching Processes in Random Environment: The Critical Case

Reduced Branching Processes in Random Environment: The Critical Case

Typical Behavior of the Harmonic Measure in Critical Galton–Watson Trees with Infinite Variance Offspring Distribution

Reduced Branching Processes with Very Heavy Tails

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