Branching Process In Random Environment Research Articles

Limit Theorems for Weakly Subcritical Branching Processes in Random Environment

For a branching process in random environment, it is assumed that the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. Interestingly there is the possibility that the process may at the same time be subcritical and, conditioned on nonextinction, “supercritical.” This so-called weakly subcritical case is considered in this paper. We study the asymptotic survival probability and the size of the population conditioned on nonextinction. Also a functional limit theorem is proved, which makes the conditional supercriticality manifest. A main tool is a new type of functional limit theorems for conditional random walks.

Harmonic moments of branching processes in random environments

We consider harmonic moments of branching processes in general random environments. For a sequence of square integrable random variables, we give some conditions such that there is a positive constant c that every variable in this sequence belong to Open image in new window or Open image in new window uniformly.

Surviving particles for subcritical branching processes in random environment

The asymptotic behavior of a subcritical Branching Process in Random Environment (BPRE) starting with several particles depends on whether the BPRE is strongly subcritical (SS), intermediate subcritical (IS) or weakly subcritical (WS). In the (SS+IS) case, the asymptotic probability of survival is proportional to the initial number of particles, and conditionally on the survival of the population, only one initial particle survives a . s . These two properties do not hold in the (WS) case and different asymptotics are established, which require new results on random walks with negative drift. We provide an interpretation of these results by characterizing the sequence of environments selected when we condition on the survival of particles. This also raises the problem of the dependence of the Yaglom quasistationary distributions on the initial number of particles and the asymptotic behavior of the Q-process associated with a subcritical BPRE.

Age-dependent branching processes in random environments

We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ 0, ξ 1,…) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξ n ) on ℝ+, and reproduce independently new particles according to a probability law p(ξ n ) on ℕ. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean E ξ Z(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments.

Proliferating parasites in dividing cells: Kimmel’s branching model revisited

We consider a branching model introduced by Kimmel for cell division with parasite infection. Cells contain proliferating parasites which are shared randomly between the two daughter cells when they divide. We determine the probability that the organism recovers, meaning that the asymptotic proportion of contaminated cells vanishes. We study the tree of contaminated cells, give the asymptotic number of contaminated cells and the asymptotic proportions of contaminated cells with a given number of parasites. This depends on domains inherited from the behavior of branching processes in random environment (BPRE) and given by the bivariate value of the means of parasite offsprings. In one of these domains, the convergence of proportions holds in probability, the limit is deterministic and given by the Yaglom quasistationary distribution. Moreover, we get an interpretation of the limit of the Q-process as the size-biased quasistationary distribution.

On subcritical multi-type branching process in random environment

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 $n \to \infty$.

THE CONSTRUCTION OF MULTITYPE CANONICAL MARKOV BRANCHING CHAINS IN RANDOM ENVIRONMENTS

The investigation for branching processes has a long history by their strong physics background, but only a few authors have investigated the branching processes in random environments. First of all, the author introduces the concepts of the multitype canonical Markov branching chain in random environment (CMBCRE) and multitype Markov branching chain in random environment (MBCRE) and proved that CMBCRE must be MBCRE, and any MBCRE must be equivalent to another CMBCRE in distribution. The main results of this article are the construction of CMBCRE and some of its probability properties.

Bisexual Galton-Watson Branching Processes in Random Environments

In this paper, we consider a bisexual Galton-Watson branching process whose offspring probability distribution is controlled by a random environment process. Some results for the probability generating functions associated with the process are obtained and sufficient conditions for certain extinction and for non-certain extinction are established.

A note on asymptotic behavior of Galton-Watson branching processes in random environments

In this paper, we investigate Galton-Watson branching processes in random environments. In the case where the environmental process is a Markov chain which is positive recurrent or has a transition matrix Q (θ,α) such that sup,, Q (θ,α) > 0 for some a, we prove that the model has the asymptotic behavior being similar to that of Galton-Watson branching processes. In other case where the environments are non-stationary independent, the sufficient conditions are obtained for certain extinction and uncertain extinction for the model.

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

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 $n \to \infty$.

Functional limit theorems for strongly subcritical branching processes in random environment

For a strongly subcritical branching process ( Z n ) n ⩾ 0 in random environment the non-extinction probability at generation n decays at the same exponential rate as the expected generation size and given non-extinction at n the conditional distribution of Z n has a weak limit. Here we prove conditional functional limit theorems for the generation size process ( Z k ) 0 ⩽ k ⩽ n as well as for the random environment. We show that given the population survives up to generation n the environmental sequence still evolves in an i.i.d. fashion and that the conditioned generation size process converges in distribution to a positive recurrent Markov chain.

On the Recurrence and Transience of State-Dependent Branching Processes in Random Environment

The state-dependent branching process in random environment $Z_n$---the generalization of the Smith--Wilkinson model---is considered. The Lamperti--Kersting criteria for the recurrence or transience of growth models are applied to the process $X_n:=\log Z_n$, $Z_n\ge 3$.

On weighted branching processes in random environment

In this paper we investigate the nonnegative martingale W n=Z n/μ n( U ), n⩾0 and its a.s. limit W, when ( Z n ) n⩾0 is a weighted branching process in random environment with stationary ergodic environmental sequence U=( U n ) n⩾0 and μ n ( U) denotes the conditional expectation of Z n given U for n⩾0. We find necessary and sufficient conditions for W to be nondegenerate, generalizing earlier results in the literature on ordinary branching processes in random environment and also weighted branching processes. In the important special case of i.i.d. random environment, a Z log Z -condition turns out to be crucial. Deterministic and nonvarying environments are treated as special cases. Our arguments adapt the probabilistic proof of Biggins’ theorem for branching random walks given by Lyons (1997) to our situation.

A Note on Diffusion-Type Approximation to Branching Processes in Random Environments

We extend the known results on diffusion-type approximation to branching processes in random environments. In particular, the range of the initial values of the processes can be much wider, moment conditions are more general, and the approximant can be a discontinuous process. The proof is based on the author's estimates for diffusion approximation to branching processes in varying environments.

Reduced Branching Processes in Random Environment: The Critical Case

Let $Z_n$ be the number of particles at time $n=0,1,2,\ldots$ in a branching process in random environment, $Z_0=1$, and let~$Z_{m,n}$ be the number of such particles in the process at time $m\in [0,n]$, each of which has a nonempty offspring at time~n. It is shown that if the offspring generating functions $f_k(s)$ of the particles of the kth generation are independent and identically distributed for all $k=0,1,2,\ldots$ with ${\bf E} \log f^{\prime}_k(1)=0$ and $\sigma ^2={\bf E} (\log f^{\prime}_k(1))^2\in (0, \infty)$, then, under certain additional restrictions, the sequence of conditional processes \biggl\{\frac{1}{\sigma \sqrt{n}} \,\log Z_{[nt],n}, t\in [0,1]\,\big|\,Z_n>0\biggr\} converges, as $n\to\infty$, in distribution in Skorokhod topology to the process $\{\inf_{t\le u\le 1}W^+(u),\, t\in [0,1]\}$, where $\{W_+(t), t\in [0,1]\}$ is the Brownian meander.

Limit theorems for subcritical branching processes in random environment

Let ( Z n ) n⩾0 be a branching process in random environment represented by a sequence of i.i.d. generating functions ( f n ) n⩾0 . In the subcritical case, Elog f′(1)<0, the non-extinction probability at generation n decays exponentially fast, the rate depending on whether E[ f′(1)log f′(1)] is less, equal or greater than 0. We determine the exact asymptotic of the non-extinction probability P( Z n >0) in all three cases under suitable integrability assumptions. Moreover, we show that Z n conditioned on Z n >0 has a non-degenerate limit law.

Branching processes with deteriorating random environments

We introduce Galton-Watson-style branching processes in random environments which are deteriorating rather than stationary or independent. Some primary results on process growth and extinction probability are shown, and two simple examples are given.

Branching processes with deteriorating random environments

We introduce Galton-Watson-style branching processes in random environments which are deteriorating rather than stationary or independent. Some primary results on process growth and extinction probability are shown, and two simple examples are given.

Many visits to a single site by a transient random walk in random environment

We consider a transient random walk on Z in random environment, and study the almost sure asymptotics of the supremum of its local time. Our main result states that if the random walk has zero speed, there is a (random) sequence of sites and a (random) sequence of times such that the walk spends a positive fraction of the times at these sites. This was known for a recurrent random walk in random environment (Random Walk in Random and Non-Random Environments, World Scientific, Singapore, 1990; Stochastic Process. Appl. 76 (1998) 231). Our method of proof is different and relies on the connection of random walk in random environment with branching processes in random environment used in Kesten et al. (Compositio Math. 30 (1975) 145).

A note on diffusion-type approximation to branching processes in random environments

A note on diffusion-type approximation to branching processes in random environments