Large deviations of bisexual branching process in random environment

For a bisexual branching process in a random environment, the conditional mean growth rate per mating unit is introduced, and its related properties are studied, then the upper and lower bounds of conditional mean of the process are obtained. The limiting behaviors of the number of mating units in each generation normalized by these two bounds are respectively investigated. Particularly, the limiting behaviors of the number of females and males in each generation normalized by slightly modi ed bounds are discussed too, and equivalence theorems between the normalized number of mating units and that of females or males in each generation are obtained.

On weighted branching processes in random environment

We introduce a branching process in a sparse random environment as an intermediate model between a Galton–Watson process and a branching process in a random environment. In the critical case we investigate the survival probability and prove Yaglom-type limit theorems, that is, limit theorems for the size of population conditioned on the survival event.

Let (Z n) n$\ge$0 with Z n = (Z n (i, j)) 1$\le$i,j$\le$p be a p multi-type critical branching process in random environment, and let M n be the expectation of Z n given a fixed environment. We prove theorems on convergence in distribution of sequences of branching processes Zn |Mn| /|Z n | > 0 and ln Zn $\sqrt$ n /|Z n | > 0. These theorems extend similar results for single-type critical branching process in random environment.

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.

A generalization of the well-known result concerning the survival probability of a critical branching process in random environment $Z_k$ is considered. The triangular array scheme of branching processes in random environment $Z_{k,n}$ that are close to $Z_k$ for large $n$ is studied. The equivalence of the survival probabilities for the processes $Z_{n,n}$ and $Z_n$ is obtained under rather natural assumptions on the closeness of $Z_{k,n}$ and $Z_k$. Bibliography: 7 titles.

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.

Branching Processes in Random Environment (BPREs) $(Z_n:n\geq0)$ are the generalization of Galton-Watson processes where 'in each generation' the reproduction law is picked randomly in an i.i.d. manner. The associated random walk of the environment has increments distributed like the logarithmic mean of the offspring distributions. This random walk plays a key role in the asymptotic behavior. In this paper, we study the upper large deviations of the BPRE $Z$ when the reproduction law may have heavy tails. More precisely, we obtain an expression for the limit of $-\log \mathbb{P}(Z_n\geq \exp(\theta n))/n$ when $n\rightarrow \infty$. It depends on the rate function of the associated random walk of the environment, the logarithmic cost of survival $\gamma:=-\lim_{n\rightarrow\infty} \log \mathbb{P}(Z_n \gt 0)/n$ and the polynomial rate of decay $\beta$ of the tail distribution of $Z_1$. This rate function can be interpreted as the optimal way to reach a given "large" value. We then compute the rate function when the reproduction law does not have heavy tails. Our results generalize the results of Böinghoff & Kersting (2009) and Bansaye & Berestycki (2008) for upper large deviations. Finally, we derive the upper large deviations for the Galton-Watson processes with heavy tails.

On the survival of branching processes in random environments

Branching processes in random environment (Z n : n ≥ 0) are the generalization of Galton-Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical regime, the process survives with a positive probability and grows exponentially on the non-extinction event. We focus on rare events when the process takes positive values but lower than expected. More precisely, we are interested in the lower large deviations of Z, which means the asymptotic behavior of the probability {1 ≤ Z n ≤ exp(nθ)} as n → ∞. We provide an expression for the rate of decrease of this probability under some moment assumptions, which yields the rate function. With this result we generalize the lower large deviation theorem of Bansaye and Berestycki (2009) by considering processes where ℙ(Z 1 = 0 | Z 0 = 1) > 0 and also much weaker moment assumptions.

Lower deviations of branching processes in random environment with geometrical offspring distributions

Branching processes exhibit a particularly rich longtime behaviour when evolving in a random environment. Then the transition from subcriticality to supercriticality proceeds in several steps, and there occurs a second ‘transition’ in the subcritical phase (besides the phase-transition from (sub)criticality to supercriticality). Here we present and discuss limit laws for branching processes in critical and subcritical i.i.d. environment. The results rely on a stimulating interplay between branching process theory and random walk theory. We also consider a spatial version of branching processes in random environment for which we derive extinction and ultimate survival criteria.

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

A conditional log-Laplace functional (CLLF) for a class of branching processes in random environments is derived. The basic idea is the decomposition of a dependent branching dynamic into a no-interacting branching and an interacting dynamic generated by the random environments. CLLF will play an important role in the investigation of branching processes and superprocesses with interaction.

In this paper, growth of branching processes in random environment is considered. In particular it is shown that this process either "explodes" at an exponential rate or else becomes extinct w.p.1. A classification theorem outlining the cases of "explosion or extinction" is given. To prove these theorems, the associated branching process (the process conditioned on each particle having infinite descent) and the reduced branching process (the particles of the process having infinite descent) are introduced. The method of proof used, in general, is direct probabilistic computation, in contrast with the classical functional iteration method.