Recurrence and Transience of Critical Branching Processes in Random Environment with Immigration and an Application to Excited Random Walks

Abstract

We establish recurrence and transience criteria for critical branching processes in random environments with immigration. These results are then applied to the recurrence and transience of a recurrent random walk in a random environment on ℤ disturbed by cookies inducing a drift to the right of strength 1.

Exact convergence rate in the central limit theorem for a branching process with immigration in a random environment

“…This means that the population size at every generation is influenced by both the environment and immigration populations. In recent years, many results were obtained for BPIRE, including moments, limit theorems, convergence rates and so on; see for example [4,5,7,17,18]. In this paper, we are interested in the central limit theorem associated to the process.…”

On the speed of a one-dimensional random walk in a random environment perturbed by cookies of strength one

“…In [5] and [6] the author studied a left-transient (respectively recurrent) one-dimensional random walk in a random environment that is perturbed by cookies of maximal strength and established criteria for transience and recurrence. In the current article, we study the speed of this random walk.…”

“…We recall the model from [5,6] and explain it in a few words. Choose a sequence (p x ) x∈Z , with p x ∈ (0, 1) for all x ∈ Z, at random and put on every integer x ∈ Z a random number M x of cookies (M x ∈ N 0 ).…”

“…If there is no cookie, he goes to x + 1 with probability p x and to x − 1 with probability 1 − p x . For illustrations of the model see [5,Fig. 1] or [6, Figure 1].…”

“…Throughout the paper, all or parts of the following assumptions on P will be needed, compare also to [5,6…”