Limit theorems for intermediately subcritical and strongly subcritical branching processes in a random environment

2001 ◽  
Vol 11 (2) ◽  
Author(s):  
V. I. AFANASYEV
Keyword(s):  
Random Environment ◽  
Limit Theorems ◽  
Branching Processes

Functional limit theorems for high-level subcritical branching processes in random environment

2014 ◽  
Vol 24 (5) ◽  
Author(s):  
Valeriy I. Afanasyev
Keyword(s):  
Random Environment ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Moment Generating Function ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
The Moment ◽  
High Level ◽  
Passage Times

AbstractThe paper is concerned with subcritical branching process in random environment. It is assumed that the moment-generating function of steps of the associated random walk is equal to 1 for some positive value of the argument. Functional limit theorems for sizes of various generations and passage times to various levels are put forward.

scholarly journals Branching processes in random environment die slowly

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Vladimir Vatutin ◽  
Andreas Kyprianou
Keyword(s):  
Random Walk ◽  
Random Environment ◽  
Generating Functions ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Limit Distributions ◽  
Conditional Limit Theorems ◽  
International Audience ◽  
Conditional Limit

International audience Let $Z_n,n=0,1,\ldots,$ be a branching process evolving in the random environment generated by a sequence of iid generating functions $f_0(s),f_1(s),\ldots,$ and let $S_0=0$, $S_k=X_1+ \ldots +X_k,k \geq 1$, be the associated random walk with $X_i=\log f_{i-1}^{\prime}(1), \tau (m,n)$ be the left-most point of minimum of $\{S_k,k \geq 0 \}$ on the interval $[m,n]$, and $T=\min \{ k:Z_k=0\}$. Assuming that the associated random walk satisfies the Doney condition $P(S_n > 0) \to \rho \in (0,1), n \to \infty$, we prove (under the quenched approach) conditional limit theorems, as $n \to \infty$, for the distribution of $Z_{nt}, Z_{\tau (0,nt)}$, and $Z_{\tau (nt,n)}, t \in (0,1)$, given $T=n$. It is shown that the form of the limit distributions essentially depends on the location of $\tau (0,n)$ with respect to the point $nt$.

scholarly journals Limit Theorems for Weakly Subcritical Branching Processes in Random Environment

2010 ◽  
Vol 25 (3) ◽  
pp. 703-732 ◽  
Author(s):  
V. I. Afanasyev ◽  
C. Böinghoff ◽  
G. Kersting ◽  
V. A. Vatutin
Keyword(s):  
Random Environment ◽  
Limit Theorems ◽  
Branching Processes

scholarly journals Functional limit theorems for strongly subcritical branching processes in random environment

2005 ◽  
Vol 115 (10) ◽  
pp. 1658-1676 ◽  
Author(s):  
V.I. Afanasyev ◽  
J. Geiger ◽  
G. Kersting ◽  
V.A. Vatutin
Keyword(s):  
Random Environment ◽  
Limit Theorems ◽  
Branching Processes ◽  
Functional Limit ◽  
Functional Limit Theorems

scholarly journals Limit theorems for decomposable branching processes in a random environment

2015 ◽  
Vol 52 (3) ◽  
pp. 877-893 ◽  
Author(s):  
Vladimir Vatutin ◽  
Quansheng Liu
Keyword(s):  
Random Environment ◽  
Survival Probability ◽  
Limit Theorems ◽  
Branching Processes ◽  
Constant Environment

We study the asymptotics of the survival probability for the critical and decomposable branching processes in a random environment and prove Yaglom-type limit theorems for these processes. It is shown that such processes possess some properties having no analogues for the decomposable branching processes in a constant environment.

Sign in / Sign up

Export Citation Format

Share Document