Branching processes in a Markov random environment

2014 ◽  
Vol 24 (6) ◽  
Author(s):  
Elena E. Dyakonova
Keyword(s):  
Limit Theorem ◽  
Invariance Principle ◽  
Random Environment ◽  
Branching Processes ◽  
Functional Limit Theorem ◽  
Functional Limit ◽  
Markov Random ◽  
Tail Behaviour ◽  
Conditional Invariance ◽  
Number Of Particles

AbstractThe paper is concerned with critical branching processes in a Markov random environment. A conditional functional limit theorem for the number of particles in a process and a conditional invariance principle are proved. The asymptotic tail behaviour for the distributions of the maximum and the total number of particles in a process is found.

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.

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 → ∞.

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 → ∞.

A limit theorem for the maxima of the para-critical simple branching process

1998 ◽  
Vol 30 (3) ◽  
pp. 740-756 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Limit Theorem ◽  
Branching Process ◽  
Branching Processes ◽  
Weak Limit ◽  
Hitting Time ◽  
Functional Limit Theorem ◽  
Diffusion Limit ◽  
Bessel Processes ◽  
Functional Limit ◽  
The Law

Let Mn denote the size of the largest amongst the first n generations of a simple branching process. It is shown for near critical processes with a finite offspring variance that the law of Mn/n, conditioned on no extinction at or before n, has a non-defective weak limit. The proof uses a conditioned functional limit theorem deriving from the Feller-Lindvall (CB) diffusion limit for branching processes descended from increasing numbers of ancestors. Subsidiary results are given about hitting time laws for CB diffusions and Bessel processes.

A limit theorem for the maxima of the para-critical simple branching process

1998 ◽  
Vol 30 (03) ◽  
pp. 740-756 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Limit Theorem ◽  
Branching Process ◽  
Branching Processes ◽  
Weak Limit ◽  
Hitting Time ◽  
Functional Limit Theorem ◽  
Diffusion Limit ◽  
Bessel Processes ◽  
Functional Limit ◽  
The Law

Let M n denote the size of the largest amongst the first n generations of a simple branching process. It is shown for near critical processes with a finite offspring variance that the law of M n /n, conditioned on no extinction at or before n, has a non-defective weak limit. The proof uses a conditioned functional limit theorem deriving from the Feller-Lindvall (CB) diffusion limit for branching processes descended from increasing numbers of ancestors. Subsidiary results are given about hitting time laws for CB diffusions and Bessel processes.

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