A Conditional Functional Limit Theorem for a Decomposable Branching Process

2021 ◽  
pp. 1-17
Author(s):  
V. I. Afanasyev
Keyword(s):  
Limit Theorem ◽  
Branching Process ◽  
Functional Limit Theorem ◽  
Functional Limit

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.

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.

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 → ∞.

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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