71 research outputs found
A special family of Galton-Watson processes with explosions
The linear-fractional Galton-Watson processes is a well known case when many
characteristics of a branching process can be computed explicitly. In this
paper we extend the two-parameter linear-fractional family to a much richer
four-parameter family of reproduction laws. The corresponding Galton-Watson
processes also allow for explicit calculations, now with possibility for
infinite mean, or even infinite number of offspring. We study the properties of
this special family of branching processes, and show, in particular, that in
some explosive cases the time to explosion can be approximated by the Gumbel
distribution
Superprocesses as models for information dissemination in the Future Internet
Future Internet will be composed by a tremendous number of potentially
interconnected people and devices, offering a variety of services, applications
and communication opportunities. In particular, short-range wireless
communications, which are available on almost all portable devices, will enable
the formation of the largest cloud of interconnected, smart computing devices
mankind has ever dreamed about: the Proximate Internet. In this paper, we
consider superprocesses, more specifically super Brownian motion, as a suitable
mathematical model to analyse a basic problem of information dissemination
arising in the context of Proximate Internet. The proposed model provides a
promising analytical framework to both study theoretical properties related to
the information dissemination process and to devise efficient and reliable
simulation schemes for very large systems
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
Survival, extinction and approximation of discrete-time branching random walks
We consider a general discrete-time branching random walk on a countable set
X. We relate local, strong local and global survival with suitable inequalities
involving the first-moment matrix M of the process. In particular we prove
that, while the local behavior is characterized by M, the global behavior
cannot be completely described in terms of properties involving M alone.
Moreover we show that locally surviving branching random walks can be
approximated by sequences of spatially confined and stochastically dominated
branching random walks which eventually survive locally if the (possibly
finite) state space is large enough. An analogous result can be achieved by
approximating a branching random walk by a sequence of multitype contact
processes and allowing a sufficiently large number of particles per site. We
compare these results with the ones obtained in the continuous-time case and we
give some examples and counterexamples.Comment: 32 pages, a few misprints have been correcte
Mutation, selection, and ancestry in branching models: a variational approach
We consider the evolution of populations under the joint action of mutation
and differential reproduction, or selection. The population is modelled as a
finite-type Markov branching process in continuous time, and the associated
genealogical tree is viewed both in the forward and the backward direction of
time. The stationary type distribution of the reversed process, the so-called
ancestral distribution, turns out as a key for the study of mutation-selection
balance. This balance can be expressed in the form of a variational principle
that quantifies the respective roles of reproduction and mutation for any
possible type distribution. It shows that the mean growth rate of the
population results from a competition for a maximal long-term growth rate, as
given by the difference between the current mean reproduction rate, and an
asymptotic decay rate related to the mutation process; this tradeoff is won by
the ancestral distribution.
Our main application is the quasispecies model of sequence evolution with
mutation coupled to reproduction but independent across sites, and a fitness
function that is invariant under permutation of sites. Here, the variational
principle is worked out in detail and yields a simple, explicit result.Comment: 45 pages,8 figure
Dimension (in)equalities and H\"older continuous curves in fractal percolation
We relate various concepts of fractal dimension of the limiting set C in
fractal percolation to the dimensions of the set consisting of connected
components larger than one point and its complement in C (the "dust"). In two
dimensions, we also show that the set consisting of connected components larger
than one point is a.s. the union of non-trivial H\"older continuous curves, all
with the same exponent. Finally, we give a short proof of the fact that in two
dimensions, any curve in the limiting set must have Hausdorff dimension
strictly larger than 1.Comment: 22 pages, 3 figures, accepted for publication in Journal of
Theoretical Probabilit
Absolutely continuous spectrum for multi-type Galton Watson trees
We consider multi-type Galton Watson trees that are close to a tree of finite
cone type in distribution. Moreover, we impose that each vertex has at least
one forward neighbor. Then, we show that the spectrum of the Laplace operator
exhibits almost surely a purely absolutely continuous component which is
included in the absolutely continuous spectrum of the tree of finite cone type.Comment: to appear in Annales Henri Poincar\'
Experimental support of the scaling rule for demographic stochasticity
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/73613/1/j.1461-0248.2006.00903.x.pd
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