995 research outputs found
Critical scaling of stochastic epidemic models
In the simple mean-field SIS and SIR epidemic models, infection is
transmitted from infectious to susceptible members of a finite population by
independent coin tosses. Spatial variants of these models are proposed, in
which finite populations of size are situated at the sites of a lattice and
infectious contacts are limited to individuals at neighboring sites. Scaling
laws for both the mean-field and spatial models are given when the infection
parameter is such that the epidemics are critical. It is shown that in all
cases there is a critical threshold for the numbers initially infected: below
the threshold, the epidemic evolves in essentially the same manner as its
branching envelope, but at the threshold evolves like a branching process with
a size-dependent drift.Comment: Published at http://dx.doi.org/10.1214/074921707000000346 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
Statistical regularities of self-intersection counts for geodesics on negatively curved surfaces
Let be a compact, negatively curved surface. From the (finite)
set of all closed geodesics on of length , choose one, say
, at random and let be the number of its
self-intersections. It is known that there is a positive constant
depending on the metric such that in
probability as . The main results of this paper concern
the size of typical fluctuations of about . It
is proved that if the metric has constant curvature -1 then typical
fluctuations are of order , in particular,
converges weakly to a nondegenerate probability distribution. In contrast, it
is also proved that if the metric has variable negative curvature then
fluctuations of are of order , in particular, converges weakly to a Gaussian
distribution. Similar results are proved for generic geodesics, that is,
geodesics whose initial tangent vectors are chosen randomly according to
normalized Liouville measure
Spatial Epidemics: Critical Behavior in One Dimension
In the simple mean-field SIS and SIR epidemic models, infection is
transmitted from infectious to susceptible members of a finite population by
independent p-coin tosses. Spatial variants of these models are proposed, in
which finite populations of size N are situated at the sites of a lattice and
infectious contacts are limited to individuals at neighboring sites. Scaling
laws for these models are given when the infection parameter p is such that the
epidemics are critical. It is shown that in all cases there is a critical
threshold for the numbers initially infected: below the threshold, the epidemic
evolves in essentially the same manner as its branching envelope, but at the
threshold evolves like a branching process with a size-dependent drift. The
corresponding scaling limits are super-Brownian motions and Dawson-Watanabe
processes with killing, respectively
Consistency of Bayes estimators of a binary regression function
When do nonparametric Bayesian procedures ``overfit''? To shed light on this
question, we consider a binary regression problem in detail and establish
frequentist consistency for a certain class of Bayes procedures based on
hierarchical priors, called uniform mixture priors. These are defined as
follows: let be any probability distribution on the nonnegative integers.
To sample a function from the prior , first sample from
and then sample uniformly from the set of step functions from
into that have exactly jumps (i.e., sample all jump locations
and function values independently and uniformly). The main result states
that if a data-stream is generated according to any fixed, measurable
binary-regression function , then frequentist consistency
obtains: that is, for any with infinite support, the posterior of
concentrates on any neighborhood of . Solution of an
associated large-deviations problem is central to the consistency proof.Comment: Published at http://dx.doi.org/10.1214/009053606000000236 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Occupation Statistics of Critical Branching Random Walks in Two or Higher Dimensions
Consider a critical nearest neighbor branching random walk on the
-dimensional integer lattice initiated by a single particle at the origin.
Let be the event that the branching random walk survives to generation
. We obtain limit theorems conditional on the event for a variety of
occupation statistics: (1) Let be the maximal number of particles at a
single site at time . If the offspring distribution has finite th
moment for some integer , then in dimensions 3 and higher,
; and if the offspring distribution has an exponentially
decaying tail, then in dimensions 3 and higher, and
in dimension 2. Furthermore, if the offspring
distribution is non-degenerate then for
some . (2) Let be the number of multiplicity- sites
in the th generation, that is, sites occupied by exactly particles. In
dimensions 3 and higher, the random variables converge jointly to
multiples of an exponential random variable. (3) In dimension 2, the number of
particles at a "typical" site (that is, at the location of a randomly chosen
particle of the th generation) is of order , and the number of
occupied sites is
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