Properties of the nonextinction probability of general critical branching processes under weak constrainis

Топчий, В. А.

doi:10.1007/bf00969331

Cited by 2 publications

(5 citation statements)

References 5 publications

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“…As was shown in [11,Corollary B] (see also [7,Lemma 3.2] for an adaptation to the discretetime setting), the relation (4) holds even for the GWO processes satisfying the conditions (1), (3), and a < ∞.…”

Section: Rulementioning

confidence: 80%

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Critical branching as a pure death process coming down from infinity

Sagitov

2023

J. Appl. Probab.

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We consider the critical Galton–Watson process with overlapping generations stemming from a single founder. Assuming that both the variance of the offspring number and the average generation length are finite, we establish the convergence of the finite-dimensional distributions, conditioned on non-extinction at a remote time of observation. The limiting process is identified as a pure death process coming down from infinity. This result brings a new perspective on Vatutin’s dichotomy, claiming that in the critical regime of age-dependent reproduction, an extant population either contains a large number of short-living individuals or consists of few long-living individuals.

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Section: Rulementioning

confidence: 80%

“…Borrowing an idea from [11], suppose to the contrary that t n := min{t:tQ(t) ≥ n} is finite for any natural n. It follows that…”

Section: Proof Of Lemmas and Propositionsmentioning

confidence: 99%

Critical branching as a pure death process coming down from infinity

Sagitov

2023

J. Appl. Probab.

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“…Paper [7] states more general results in a setting with Var(N ) ≤ ∞. In particular, the statement of Theorem 2 can be extracted from [7,Corollary B]. Nonetheless, here we give a direct autonomous proof of Theorem 2, which paves the way to the proof of the main result of this paper, Theorem 1.…”

Section: Asymptotic Formula For the Survival Probabilitymentioning

confidence: 83%

“…In its turn, Corollary 2 is a straightforward version of Corollary 1, recovering the classical formula for the critical GW-process, see [1]. Paper [7] states more general results in a setting with Var(N ) ≤ ∞. In particular, the statement of Theorem 2 can be extracted from [7,Corollary B].…”

Section: Asymptotic Formula For the Survival Probabilitymentioning

confidence: 99%

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Limit theorems for pure death processes coming down from infinity

Sagitov

2017

J. Appl. Probab.

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We consider a pure death process (Z(t), t ≥ 0) with death rates λn satisfying the condition ∞ n=2 λ −1 n < ∞ of coming from infinity, Z(0) = ∞, down to an absorbing state n = 1. We establish limit theorems for Z(t) as t → 0, which strengthen the results that can be extracted from [1]. We also prove a large deviation theorem assuming that λn regularly vary as n → ∞ with an index β > 1. It generalises a similar statement with β = 2 obtained in [4] for λn = n 2 .

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Properties of the nonextinction probability of general critical branching processes under weak constrainis

Cited by 2 publications

References 5 publications

Critical branching as a pure death process coming down from infinity

Critical branching as a pure death process coming down from infinity

Limit theorems for pure death processes coming down from infinity

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