We consider the classical Yaglom limit theorem for a branching Markov process X = (X t, t ≥ 0), with nonlocal branching mechanism in the setting that the mean semigroup is critical, that is, its leading eigenvalue is zero. In particular, we show that there exists a constant c(f ) such that (Formula Presented) where e c(f ) is an exponential random variable with rate c(f ) and the convergence is in distribution. As part of the proof, we also show that the probability of survival decays inversely proportionally to time. Although Yaglom limit theorems have recently been handled in the setting of branching Brownian motion in a bounded domain and superprocesses, (Probab. Theory Related Fields 173 (2019) 999–1062; Electron. Commun. Probab. 23 (2018) 42), these results do not allow for nonlocal branching which complicates the analysis. Our approach and the main novelty of this work is based around a precise result for the scaled asymptotics for the kth martingale moments of X (rather than the Yaglom limit itself).

Original languageEnglish
Pages (from-to)2373–2408
Number of pages36
JournalAnnals of Probability
Issue number6
Early online date23 Oct 2022
Publication statusPublished - 30 Nov 2022

Bibliographical note

The SCH, AEK and MW were supported by EPSRC grant EP/P009220/1.


  • Branching markov process
  • Neutron transport equation
  • Perron– frobenius decomposition
  • Quasi-stationary limit
  • Semigroup theory
  • Yaglom limit

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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