Criticality for branching processes in random environment

By V. Afanasyev, Jochen Geiger, Götz Kersting, Vladimir Vatutin. 2005

223 citations recorded by the source corpus.

We study branching processes in an i.i.d. random environment, where the associated random walk is of the oscillating type. This class of processes generalizes the classical notion of criticality. The main properties of such branching processes are developed under a general assumption, known as Spitzer's condition in fluctuation theory of random walks, and some additional moment condition. We determine the exact asymptotic behavior of the survival probability and prove conditional functional limit theorems for the generation size process and the associated random walk. The results rely on a stimulating interplay between branching process theory and fluctuation theory of random walks.

Subfields: Management Information Systems, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/2c9a3e989eae6772bd8b55055d23dfe838c69047

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In Management Information Systems

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Asymptotic results for exponential functionals of Levy processes (2016, methodology): “)−)dr i ≤ Vˆ(x−)−1 Z ∞ 0 dV (y) Z x 0 e−α(y+x−z)Vˆ(y +x−z)dVˆ(z), where the last step follows by Theorem 20 in Bertoin (1996, p.176). By Corollary 5.3 in Kyprianou (2014, p.118) we have Vˆ(y) ∼ y/Pˆ[H(1)] as y → ∞. Then we can take γ ∈ (0,α) and C ≥ 0 so that e−(α−γ)yVˆ(y) ≤ C for y ≥ 0. It follows ”
Limit Theorems for Weakly Subcritical Branching Processes in Random Environment (2010, methodology): “Remark For examples where this assumption is fulfilled, see [4].”
Random walks in cones (2011, background): “• Asymptotic behavior of branching processes and random walks in random environment is closely connected to the behavior of random walks conditioned to stay positive, which are one-dimensional cases of a random walk conditioned to stay in a cone; see [1] and references therein.”
Conditional limit theorems for intermediately subcritical branching processes in random environment (2011, background): “6 in [4]), we obtain that for given ε > 0 and m ∈ N the estimate P(Sk ≥ Sm for some k > l) < ε is valid, if only l is chosen large enough.”
Moments, moderate and large deviations for a branching process in a random environment (2010, background): “The study of asymptotic properties of a branching process in a random environment has recently received attention, see for example Afanasyave, Geiger, Kersting & Vatutin (2005, [1] & [2]), Kozlov (2006, [16]), Bansaye & Berestycki (2009, [5]), Bansaye & Böinghoff (2010, [6]), Böinghoff & Kersting (2”
On the extinction of Continuous State Branching Processes with catastrophes (2013, background): “Actually, such asymptotic behaviors were known for branching processes in random environments in discrete time and space (see e.g. [GL01, GKV03, AGKV05]).”
Functional limit theorems for strongly subcritical branching processes in random environment (2005, background): “dence among the states of the environmental sequence (see [ 3 , 5]).”
Bernstein-gamma functions and exponential functionals of Levy Processes (2016, background): “These requirements are due to reduction to more general results on specific random sums/random walks, which can be found in [1, 39].”