A limit theorem for a critical branching process of general type

1983 ◽  
Vol 34 (3) ◽  
pp. 713-717 ◽  
Author(s):  
S. M. Sagitov
Keyword(s):  
Limit Theorem ◽  
General Type ◽  
Branching Process ◽  
Critical Branching Process

Reduced multitype critical branching processes in random environment

2018 ◽  
Vol 28 (1) ◽  
pp. 7-22 ◽  
Author(s):  
Elena E. Dyakonova
Keyword(s):  
Limit Theorem ◽  
Random Environment ◽  
Branching Process ◽  
Branching Processes ◽  
Critical Branching Process ◽  
Number Of Particles

Abstract We consider a multitype critical branching process Zn, n = 0, 1,…, in an i.i.d. random environment. Let Zm,n be the number of particles in this process at time m having descendants at time n. A limit theorem is proved for the logarithm of Znt,n at moments nt,0 ≤ t ≤ 1, conditioned on the survival of the process Zn up to moment n when n → ∞.

scholarly journals Path to survival for the critical branching processes in a random environment

2017 ◽  
Vol 54 (2) ◽  
pp. 588-602 ◽  
Author(s):  
Vladimir Vatutin ◽  
Elena Dyakonova
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Random Environment ◽  
Branching Process ◽  
Branching Processes ◽  
Functional Limit Theorem ◽  
Levy Process ◽  
Initial Part ◽  
Functional Limit ◽  
Critical Branching Process

Abstract A critical branching process {Zk, k = 0, 1, 2, ...} in a random environment is considered. A conditional functional limit theorem for the properly scaled process {log Zpu, 0 ≤ u < ∞} is established under the assumptions that Zn > 0 and p ≪ n. It is shown that the limiting process is a Lévy process conditioned to stay nonnegative. The proof of this result is based on a limit theorem describing the distribution of the initial part of the trajectories of a driftless random walk conditioned to stay nonnegative.

scholarly journals Berry-Esseen bound for nearly critical branching processes with immigration

2019 ◽  
pp. 42-49
Author(s):  
Ya. Khusanbaev ◽  
S. Sharipov ◽  
V. Golomoziy
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Rate Of Convergence ◽  
Central Limit ◽  
Branching Process ◽  
Branching Processes ◽  
Branching Process With Immigration ◽  
Critical Branching Process

In this paper, we consider a nearly critical branching process with immigration. We obtain the rate of convergence in central limit theorem for nearly critical branching processes with immigration.

On Largest Offspring in a Critical Branching Process with Finite Variance

2013 ◽  
Vol 50 (03) ◽  
pp. 791-800 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Limit Theorem ◽  
Branching Process ◽  
Shape Parameter ◽  
Weak Limit ◽  
Finite Variance ◽  
Poisson Measure ◽  
Doubly Stochastic ◽  
Weak Limit Theorem ◽  
Watson Process ◽  
Critical Branching Process

Continuing the work in Bertoin (2011) we study the distribution of the maximal number X * k of offspring amongst all individuals in a critical Galton‒Watson process started with k ancestors, treating the case when the reproduction law has a regularly varying tail F̅ with index −α for α &gt; 2 (and, hence, finite variance). We show that X * k suitably normalized converges in distribution to a Fréchet law with shape parameter α/2; this contrasts sharply with the case 1&lt; α&lt;2 when the variance is infinite. More generally, we obtain a weak limit theorem for the offspring sequence ranked in decreasing order, in terms of atoms of a certain doubly stochastic Poisson measure.

Limit theorem for multitype critical branching process evolving in random environment

2015 ◽  
Vol 25 (3) ◽  
Author(s):  
Elena E. Dyakonova
Keyword(s):  
Limit Theorem ◽  
Random Environment ◽  
Branching Process ◽  
Functional Limit Theorem ◽  
Functional Limit ◽  
Critical Branching Process ◽  
Number Of Particles

AbstractWe investigate a multitype critical branching process in an i.i.d. random environment. A functional limit theorem is proved for the logarithm of the number of particles in the process at moments nt, 0 ≤ t ≤ 1, conditioned on its survival up to moment n → ∞.

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