A 2-spine decomposition of the critical Galton-Watson tree and a probabilistic proof of Yaglom’s theorem

Ren, Yanfei; Song, Renming; Sun, Zhenyao

doi:10.1214/18-ecp143

Cited by 14 publications

(26 citation statements)

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“…As mentioned earlier in Subsection 1.1, this 2-spine decomposition theorem for superprocesses is an analog of the 2-spine decomposition theorem for Galton-Watson trees in [37], and is closely related to the multi-spine theory appeared in [20], [21], [24] and [1]. Of course, depend on the choice of F , there are many versions of Theorem 1.2.…”

Section: Resultsmentioning

confidence: 80%

“…It was first introduced by Harris and Roberts [20] in the context of branching processes. Our 2-spine methods for Galton-Watson trees [37] and for superprocesses in this paper are both inspired by [20]. An analogous k-spine decomposition theorem also appeared in [21] and [24] in the context of continuous time Galton-Watson processes.…”

mentioning

confidence: 95%

“…One of those two spines is longer than the other. The spirit of our proof in [37] is to show that the immigration along the longer spine at generation n is distributed approximately likeŻ n , while the immigration along the shorter spine at generation n is distributed approximately likeŻ ′ [U ·n] . HereŻ n andŻ ′ n are independent Z n -transforms of Z n .…”

mentioning

confidence: 99%

“…Roughly speaking, we haveZ (n) n law ≈Ż n +Ż ′ [U ·n] , and therefore, if X is the weak limit of Zn n conditioned on {Z n > 0}, then X is a positive random variable satisfying (1.3). In this paper, we adapt the method of [37] to develop a 2-spine decomposition for critical superprocesses and then use this 2-spine decomposition to give probabilistic proofs of Kolmogorov type and Yaglom type results for superprocesses. The spirit of this paper is similar to that of [37], but the arguments are more complicated.…”

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confidence: 99%

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Spine Decompositions and Limit Theorems for a Class of Critical Superprocesses

2019

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In this paper we first establish a decomposition theorem for size-biased Poisson random measures. As consequences of this decomposition theorem, we get a spine decomposition theorem and a 2-spine decomposition theorem for some critical superprocesses. Then we use these spine decomposition theorems to give probabilistic proofs of the asymptotic behavior of the survival probability and Yaglom's exponential limit law for critical superprocesses.2010 Mathematics Subject Classification. 60J80, 60F05. Key words and phrases. Critical superprocess, size-biased Poisson random measure, spine decomposition, 2-spine decomposition, asymptotic behavior of the survival probability, Yaglom's exponential limit law, martingale change of measure.

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Section: Resultsmentioning

confidence: 80%

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confidence: 95%

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confidence: 99%

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confidence: 99%

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Spine Decompositions and Limit Theorems for a Class of Critical Superprocesses

2019

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“…A similar idea of establishing weak convergence through a comparison of the equations satisfied by the distributions has already been used by us in [22,23]. We characterized the exponential distribution using its double size-biased transform; and to help us make the comparison, we investigated the double size-biased transform of the corresponding processes.…”

Section: 3mentioning

confidence: 99%

Limit theorems for a class of critical superprocesses with stable branching

Ren

Song

Sun

2020

Stochastic Processes and their Applications

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We consider a critical superprocess {X; P µ } with general spatial motion and spatially dependent stable branching mechanism with lowest stable index γ 0 > 1. We first show that, under some conditions, P µ ( X t = 0) converges to 0 as t → ∞ and is regularly varying with index (γ 0 − 1) −1 . Then we show that, for a large class of non-negative testing functions f , the distribution of {X t (f ); P µ (·| X t = 0)}, after appropriate rescaling, converges weakly to a positive random variable z (γ0−1) with Laplace transform E[e −uz (γ 0 −1) ] = 1 − (1 + u −(γ0−1) ) −1/(γ0−1) .

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Convergence of genealogies through spinal decomposition with an application to population genetics

Foutel-Rodier,

Schertzer

2023

Probab. Theory Relat. Fields

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Consider a branching Markov process with values in some general type space. Conditional on survival up to generation N, the genealogy of the extant population defines a random marked metric measure space, where individuals are marked by their type and pairwise distances are measured by the time to the most recent common ancestor. In the present manuscript, we devise a general method of moments to prove convergence of such genealogies in the Gromov-weak topology when $$N \rightarrow \infty $$ N → ∞ . Informally, the moment of order k of the population is obtained by observing the genealogy of k individuals chosen uniformly at random after size-biasing the population at time N by its kth factorial moment. We show that the sampled genealogy can be expressed in terms of a k-spine decomposition of the original branching process, and that convergence reduces to the convergence of the underlying k-spines. As an illustration of our framework, we analyse the large-time behavior of a branching approximation of the biparental Wright–Fisher model with recombination. The model exhibits some interesting mathematical features. It starts in a supercritical state but is naturally driven to criticality. We show that the limiting behavior exhibits both critical and supercritical characteristics.

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A 2-spine decomposition of the critical Galton-Watson tree and a probabilistic proof of Yaglom’s theorem

Abstract: In this note we propose a two-spine decomposition of the critical Galton-Watson tree and use this decomposition to give a probabilistic proof of Yaglom's theorem.2010 Mathematics Subject Classification. 60J80, 60F05.

Cited by 14 publications

References 10 publications

Spine Decompositions and Limit Theorems for a Class of Critical Superprocesses

Spine Decompositions and Limit Theorems for a Class of Critical Superprocesses

Limit theorems for a class of critical superprocesses with stable branching

Convergence of genealogies through spinal decomposition with an application to population genetics

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