Search

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.

We consider a branching random walk. Biggins and Kyprianou (2004) proved that, in the boundary case, the associated derivative martingale converges almost surely to a finite nonnegative limit, whose law serves as a fixed point of a smoothing transformation (Mandelbrot's cascade). In this paper, we give a necessary and sufficient condition for the nontriviality of the limit in this boundary case.

Search Results

Refine search

Refine search

Actions for selected content:

2 results

Path to survival for the critical branching processes in a random environment

A necessary and sufficient condition for the nontrivial limit of the derivative martingale in a branching random walk

Search Results

Refine search

Refine search

Actions for selected content:

Save Search

2 results

Path to survival for the critical branching processes in a random environment

A necessary and sufficient condition for the nontrivial limit of the derivative martingale in a branching random walk