1,508 research outputs found
Percolation for the stable marriage of Poisson and Lebesgue with random appetites
Let be a set of centers chosen according to a Poisson point process in
. Consider the allocation of to which is
stable in the sense of the Gale-Shapley marriage problem, with the additional
feature that every center has a random appetite , where
is a nonnegative scale constant and is a nonnegative random
variable. Generalizing previous results by Freire, Popov and Vachkovskaia
(\cite{FPV}), we show the absence of percolation when is small enough,
depending on certain characteristics of the moment of .Comment: 12 pages. Final versio
A generic model for spouse's pensions with a view towards the calculation of liabilities
We introduce a generic model for spouse's pensions. The generic model allows
for the modeling of various types of spouse's pensions with payments commencing
at the death of the insured. We derive abstract formulas for cashflows and
liabilities corresponding to common types of spouse's pensions. We show how the
standard formulas from the Danish G82 concession can be obtained as a special
case of our generic model. We also derive expressions for liabilities for
spouse's pensions in models more advanced than found in the G82 concession. The
generic nature of our model and results furthermore enable the calculation of
cashflows and liabilities using simple estimates of marital behaviour among a
population
Extra heads and invariant allocations
Let \Pi be an ergodic simple point process on R^d and let \Pi^* be its Palm
version. Thorisson [Ann. Probab. 24 (1996) 2057-2064] proved that there exists
a shift coupling of \Pi and \Pi^*; that is, one can select a (random) point Y
of \Pi such that translating \Pi by -Y yields a configuration whose law is that
of \Pi^*. We construct shift couplings in which Y and \Pi^* are functions of
\Pi, and prove that there is no shift coupling in which \Pi is a function of
\Pi^*. The key ingredient is a deterministic translation-invariant rule to
allocate sets of equal volume (forming a partition of R^d) to the points of
\Pi. The construction is based on the Gale-Shapley stable marriage algorithm
[Amer. Math. Monthly 69 (1962) 9-15]. Next, let \Gamma be an ergodic random
element of {0,1}^{Z^d} and let \Gamma^* be \Gamma conditioned on \Gamma(0)=1. A
shift coupling X of \Gamma and \Gamma^* is called an extra head scheme. We show
that there exists an extra head scheme which is a function of \Gamma if and
only if the marginal E[\Gamma(0)] is the reciprocal of an integer. When the law
of \Gamma is product measure and d\geq3, we prove that there exists an extra
head scheme X satisfying E\exp c\|X\|^d<\infty; this answers a question of
Holroyd and Liggett [Ann. Probab. 29 (2001) 1405-1425].Comment: Published at http://dx.doi.org/10.1214/009117904000000603 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Stable transports between stationary random measures
We give an algorithm to construct a translation-invariant transport kernel
between ergodic stationary random measures and on ,
given that they have equal intensities. As a result, this yields a construction
of a shift-coupling of an ergodic stationary random measure and its Palm
version. This algorithm constructs the transport kernel in a deterministic
manner given realizations and of the measures. The
(non-constructive) existence of such a transport kernel was proved in [8]. Our
algorithm is a generalization of the work of [3], in which a construction is
provided for the Lebesgue measure and an ergodic simple point process. In the
general case, we limit ourselves to what we call constrained densities and
transport kernels. We give a definition of stability of constrained densities
and introduce our construction algorithm inspired by the Gale-Shapley stable
marriage algorithm. For stable constrained densities, we study existence,
uniqueness, monotonicity w.r.t. the measures and boundedness.Comment: In the second version, we change the way of presentation of the main
results in Section 4. The main results and their proofs are not changed
significantly. We add Section 3 and Subsection 4.6. 25 pages and 2 figure
Insertion and deletion tolerance of point processes
We develop a theory of insertion and deletion tolerance for point processes. A process is insertion-tolerant if adding a suitably chosen random point results in a point process that is absolutely continuous in law with respect to the original process. This condition and the related notion of deletion-tolerance are extensions of the so-called finite energy condition for discrete random processes. We prove several equivalent formulations of each condition, including versions involving Palm processes. Certain other seemingly natural variants of the conditions turn out not to be equivalent. We illustrate the concepts in the context of a number of examples, including Gaussian zero processes and randomly perturbed lattices, and we provide applications to continuum percolation and stable matching
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