311 research outputs found
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
On the survival of a class of subcritical branching processes in random environment
Let be the number of individuals in a subcritical BPRE evolving in
the environment generated by iid probability distributions. Let be the
logarithm of the expected offspring size per individual given the environment.
Assuming that the density of has the form for some a slowly varying function
and we find the asymptotic survival
probability and prove a Yaglom type conditional limit theorem for the process.
The survival probability decreases exponentially with an additional polynomial
term related to the tail of . The proof relies on a fine study of a random
walk (with negative drift and heavy tails) conditioned to stay positive until
time and to have a small positive value at time , with tending to
infinity
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
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 processes in random environment die slowly
Let be a branching process evolving in the random
environment generated by a sequence of iid generating functions and let be the
associated random walk with be
the left-most point of minimum of on the
interval and . Assuming that the
associated random walk satisfies the Doney condition we prove (under the quenched approach) conditional limit
theorems, as , for the distribution of and given . It is shown that
the form of the limit distributions essentially depends on the location of
with respect to the point $nt.
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