On the derivative martingale in a branching random walk

Buraczewski, Dariusz; Iksanov, Alexander; Mallein, Bastien

doi:10.48550/arxiv.2002.05215

Cited by 2 publications

(2 citation statements)

References 31 publications

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“…The result holds under weaker assumptions on the offspring distribution; see [41]. Also, an analogous result for branching random walk has been proven recently in [16].…”

Section: Descendants Of a Single Particlementioning

confidence: 59%

Yaglom-type limit theorems for branching Brownian motion with absorption

Maillard,

Schweinsberg

2020

Preprint

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We consider one-dimensional branching Brownian motion in which particles are absorbed at the origin. We assume that when a particle branches, the offspring distribution is supercritical, but the particles are given a critical drift towards the origin so that the process eventually goes extinct with probability one. We establish precise asymptotics for the probability that the process survives for a large time t, building on previous results by Kesten (1978) and Berestycki, Berestycki, and Schweinsberg (2014). We also prove a Yaglom-type limit theorem for the behavior of the process conditioned to survive for an unusually long time, providing an essentially complete answer to a question first addressed by Kesten (1978). An important tool in the proofs of these results is the convergence of a certain observable to a continuous state branching process. Our proofs incorporate new ideas which might be of use in other branching models.

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“…The result holds under weaker assumptions on the offspring distribution; see [41]. Also, an analogous result for branching random walk has been proven recently in [16].…”

Section: Descendants Of a Single Particlementioning

confidence: 59%

Yaglom-type limit theorems for branching Brownian motion with absorption

Maillard,

Schweinsberg

2020

Preprint

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“…In[17], the +1 in the definition of µZ was mistakenly missing, as pointed out to us by the authors of[7] 2. In fact, the statement is slightly weaker in[17], but the proof can be readily adapted so as to yield the current statement, by following the proof idea of Proposition 4.1.…”

mentioning

confidence: 96%

1-stable fluctuations in branching Brownian motion at critical temperature I: The derivative martingale

Maillard¹,

Pain²

2019

Ann. Probab.

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Let (Z t ) t≥0 denote the derivative martingale of branching Brownian motion, i.e. the derivative with respect to the inverse temperature of the normalized partition function at critical temperature. A well-known result by Lalley and Sellke [Ann. Probab., 15(3):1052-1061, 1987 says that this martingale converges almost surely to a limit Z ∞ , positive on the event of survival. In this paper, our concern is the fluctuations of the derivative martingale around its limit. A corollary of our results is the following convergence, confirming and strengthening a conjecture by Mueller and Munier [Phys. Rev. E, 90:042143, 2014]:where S is a spectrally positive 1-stable Lévy process independent of Z ∞ . In a first part of the paper, a relatively short proof of (a slightly stronger form of) this convergence is given based on the functional equation satisfied by the characteristic function of Z ∞ together with tail asymptotics of this random variable. We then set up more elaborate arguments which yield a more thorough understanding of the trajectories of the particles contributing to the fluctuations. In this way, we can upgrade our convergence result to functional convergence. This approach also sets the ground for a follow-up paper, where we study the fluctuations of more general functionals including the renormalized critical additive martingale.All proofs in this paper are given under the hypothesis E L(log L) 3 < ∞, where the random variable L follows the offspring distribution of the branching Brownian motion. We believe this hypothesis to be optimal.

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On the derivative martingale in a branching random walk

Cited by 2 publications

References 31 publications

Yaglom-type limit theorems for branching Brownian motion with absorption

Yaglom-type limit theorems for branching Brownian motion with absorption

1-stable fluctuations in branching Brownian motion at critical temperature I: The derivative martingale

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