Limit theorems for decomposable branching processes in a random environment

We study the asymptotics of the survival probability for the critical and decomposable branching processes in 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 constant environmentComment: 21 page

Random walk with heavy tail and negative drift conditioned by its minimum and final values

We consider random walks with finite second moment which drifts to $-\infty$ and have heavy tail. We focus on the events when the minimum and the final value of this walk belong to some compact set. We first specify the associated probability. Then, conditionally on such an event, we finely describe the trajectory of the random walk. It yields a decomposition theorem with respect to a random time giving a big jump whose distribution can be described explicitly.Comment: arXiv admin note: substantial text overlap with arXiv:1307.396

Branching random walk in Z^4 with branching at the origin only

For the critical branching random walk in $\mathbb{Z}^{4}$ with branching at the origin only we find the asymptotic behavior of the probability of the event that there are particles at the origin at moment $t\rightarrow \infty$ and prove a Yaglom type conditional limit theorem for the number of individuals at the origin given that there are particles at the origin