124 research outputs found
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.
Criticality for branching processes in random environment
We study branching processes in an i.i.d. random environment, where the
associated random walk is of the oscillating type. This class of processes
generalizes the classical notion of criticality. The main properties of such
branching processes are developed under a general assumption, known as
Spitzer's condition in fluctuation theory of random walks, and some additional
moment condition. We determine the exact asymptotic behavior of the survival
probability and prove conditional functional limit theorems for the generation
size process and the associated random walk. The results rely on a stimulating
interplay between branching process theory and fluctuation theory of random
walks.Comment: Published at http://dx.doi.org/10.1214/009117904000000928 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Conditional limit theorems for intermediately subcritical branching processes in random environment
The Hitting Times with Taboo for a Random Walk on an Integer Lattice
For a symmetric, homogeneous and irreducible random walk on d-dimensional
integer lattice Z^d, having zero mean and a finite variance of jumps, we study
the passage times (with possible infinite values) determined by the starting
point x, the hitting state y and the taboo state z. We find the probability
that these passages times are finite and analyze the tails of their cumulative
distribution functions. In particular, it turns out that for the random walk on
Z^d, except for a simple (nearest neighbor) random walk on Z, the order of the
tail decrease is specified by dimension d only. In contrast, for a simple
random walk on Z, the asymptotic properties of hitting times with taboo
essentially depend on the mutual location of the points x, y and z. These
problems originated in our recent study of branching random walk on Z^d with a
single source of branching
Limit theorems for weakly subcritical branching processes in random environment
For a branching process in random environment it is assumed that the
offspring distribution of the individuals varies in a random fashion,
independently from one generation to the other. Interestingly there is the
possibility that the process may at the same time be subcritical and,
conditioned on nonextinction, 'supercritical'. This so-called weakly
subcritical case is considered in this paper. We study the asymptotic survival
probability and the size of the population conditioned on non-extinction. Also
a functional limit theorem is proven, which makes the conditional
supercriticality manifest. A main tool is a new type of functional limit
theorems for conditional random walks.Comment: 35 page
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