scholarly journals Survival probability of a critical multi-type branching process in random environment

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Elena Dyakonova
Keyword(s):  
Random Walk ◽  
Random Environment ◽  
Survival Probability ◽  
Branching Process ◽  
International Audience

International audience We study a multi-type branching process in i.i.d. random environment. Assuming that the associated random walk satisfies the Doney-Spitzer condition, we find the asymptotics of the survival probability at time $n$ as $n \to \infty$.

scholarly journals On subcritical multi-type branching process in random environment

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Elena Dyakonova
Keyword(s):  
Random Walk ◽  
Random Environment ◽  
Survival Probability ◽  
Branching Process ◽  
Random Variables ◽  
International Audience ◽  
Watson Process ◽  
Independent Identically Distributed

International audience We investigate a multi-type Galton-Watson process in a random environment generated by a sequence of independent identically distributed random variables. Suppose that the associated random walk constructed by the logarithms of the Perron roots of the reproduction mean matrices has negative mean and assuming some additional conditions, we find the asymptotics of the survival probability at time $n$ as $n \to \infty$.

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 $nt$.

Multitype weakly subcritical branching processes in random environment

2021 ◽  
Vol 31 (3) ◽  
pp. 207-222
Author(s):  
Vladimir A. Vatutin ◽  
Elena E. Dyakonova
Keyword(s):  
Random Walk ◽  
Random Environment ◽  
Survival Probability ◽  
Branching Process ◽  
Branching Processes ◽  
Left Eigenvector ◽  
Mean Values ◽  
Long Time ◽  
The Mean ◽  
Independent Identically Distributed

Abstract A multi-type branching process evolving in a random environment generated by a sequence of independent identically distributed random variables is considered. The asymptotics of the survival probability of the process for a long time is found under the assumption that the matrices of the mean values of direct descendants have a common left eigenvector and the increment X of the associated random walk generated by the logarithms of the Perron roots of these matrices satisfies conditions E X < 0 and E XeX > 0.

On the Survival Probability of a Branching Process in a Random Environment

1997 ◽  
Vol 29 (01) ◽  
pp. 38-55 ◽  
Author(s):  
J. C. D'souza ◽  
B. M. Hambly
Keyword(s):  
State Space ◽  
Random Environment ◽  
Survival Probability ◽  
Branching Process ◽  
Large Deviation ◽  
Convergence Results ◽  
Finite State ◽  
Large Deviation Result ◽  
Finite State Space

We derive a variety of estimates for the survival probability of a branching process in a random environment. There are three cases of interest, the critical, weakly and strongly subcritical. The large deviation result, first obtained by Dekking for the class of finite state space i.i.d. environments, is shown to hold in more general environments. We also obtain some finer convergence results.

On the Survival Probability of a Branching Process in a Random Environment

1997 ◽  
Vol 29 (1) ◽  
pp. 38-55 ◽  
Author(s):  
J. C. D'souza ◽  
B. M. Hambly
Keyword(s):  
State Space ◽  
Random Environment ◽  
Survival Probability ◽  
Branching Process ◽  
Large Deviation ◽  
Convergence Results ◽  
Finite State ◽  
Large Deviation Result ◽  
Finite State Space

We derive a variety of estimates for the survival probability of a branching process in a random environment. There are three cases of interest, the critical, weakly and strongly subcritical. The large deviation result, first obtained by Dekking for the class of finite state space i.i.d. environments, is shown to hold in more general environments. We also obtain some finer convergence results.

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.

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