392 research outputs found
Second class particles and cube root asymptotics for Hammersley's process
We show that, for a stationary version of Hammersley's process, with Poisson
sources on the positive x-axis and Poisson sinks on the positive y-axis, the
variance of the length of a longest weakly North--East path from
to is equal to , where is the
location of a second class particle at time . This implies that both
and the variance of are of order .
Proofs are based on the relation between the flux and the path of a second
class particle, continuing the approach of Cator and Groeneboom [Ann. Probab.
33 (2005) 879--903].Comment: Published at http://dx.doi.org/10.1214/009117906000000089 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A new look at distances and velocities of neutron stars
We take a fresh look at the determination of distances and velocities of
neutron stars. The conversion of a parallax measurement into a distance, or
distance probability distribution, has led to a debate quite similar to the one
involving Cepheids, centering on the question whether priors can be used when
discussing a single system. With the example of PSRJ0218+4232 we show that a
prior is necessary to determine the probability distribution for the distance.
The distance of this pulsar implies a gamma-ray luminosity larger than 10% of
its spindown luminosity. For velocities the debate is whether a single
Maxwellian describes the distribution for young pulsars. By limiting our
discussion to accurate (VLBI) measurements we argue that a description with two
Maxwellians, with distribution parameters sigma1=77 and sigma2=320 km/s, is
significantly better. Corrections for galactic rotation, to derive velocities
with respect to the local standards of rest, are insignificant.Comment: Has appeared in Journal of Astrophysics and Astronomy special issue
on 'Physics of Neutron Stars and Related Objects', celebrating the 75th
birth-year of G. Srinivasan. Ten pages, nine figure
Hammersley's process with sources and sinks
We show that, for a stationary version of Hammersley's process, with Poisson
``sources'' on the positive x-axis, and Poisson ``sinks'' on the positive
y-axis, an isolated second-class particle, located at the origin at time zero,
moves asymptotically, with probability 1, along the characteristic of a
conservation equation for Hammersley's process. This allows us to show that
Hammersley's process without sinks or sources, as defined by Aldous and
Diaconis [Probab. Theory Related Fields 10 (1995) 199-213] converges locally in
distribution to a Poisson process, a result first proved in Aldous and Diaconis
(1995) by using the ergodic decomposition theorem and a construction of
Hammersley's process as a one-dimensional point process, developing as a
function of (continuous) time on the whole real line. As a corollary we get the
result that EL(t,t)/t converges to 2, as t\to\infty, where L(t,t) is the length
of a longest North-East path from (0,0) to (t,t). The proofs of these facts
need neither the ergodic decomposition theorem nor the subadditive ergodic
theorem. We also prove a version of Burke's theorem for the stationary process
with sources and sinks and briefly discuss the relation of these results with
the theory of longest increasing subsequences of random permutations.Comment: Published at http://dx.doi.org/10.1214/009117905000000053 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Space-time stationary solutions for the Burgers equation
We construct space-time stationary solutions of the 1D Burgers equation with
random forcing in the absence of periodicity or any other compactness
assumptions. More precisely, for the forcing given by a homogeneous Poissonian
point field in space-time we prove that there is a unique global solution with
any prescribed average velocity. These global solutions serve as one-point
random attractors for the infinite-dimensional dynamical system associated to
solutions to the Cauchy problem. The probability distribution of the global
solutions defines a stationary distribution for the corresponding Markov
process. We describe a broad class of initial Cauchy data for which the
distribution of the Markov process converges to the above stationary
distribution.
Our construction of the global solutions is based on a study of the field of
action-minimizing curves. We prove that for an arbitrary value of the average
velocity, with probability 1 there exists a unique field of action-minimizing
curves initiated at all of the Poissonian points. Moreover action-minimizing
curves corresponding to different starting points merge with each other in
finite time.Comment: 50 pages. In this version: small technical corrections in Lemmas 6.1
and 6.
Asymptotic expansion of the minimum covariance determinant estimators
In arXiv:0907.0079 by Cator and Lopuhaa, an asymptotic expansion for the MCD
estimators is established in a very general framework. This expansion requires
the existence and non-singularity of the derivative in a first-order Taylor
expansion. In this paper, we prove the existence of this derivative for
multivariate distributions that have a density and provide an explicit
expression. Moreover, under suitable symmetry conditions on the density, we
show that this derivative is non-singular. These symmetry conditions include
the elliptically contoured multivariate location-scatter model, in which case
we show that the minimum covariance determinant (MCD) estimators of
multivariate location and covariance are asymptotically equivalent to a sum of
independent identically distributed vector and matrix valued random elements,
respectively. This provides a proof of asymptotic normality and a precise
description of the limiting covariance structure for the MCD estimators.Comment: 21 page
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