1,581 research outputs found
Gaussian limits for generalized spacings
Nearest neighbor cells in , are used to define
coefficients of divergence (-divergences) between continuous multivariate
samples. For large sample sizes, such distances are shown to be asymptotically
normal with a variance depending on the underlying point density. In ,
this extends classical central limit theory for sum functions of spacings. The
general results yield central limit theorems for logarithmic -spacings,
information gain, log-likelihood ratios and the number of pairs of sample
points within a fixed distance of each other.Comment: Published in at http://dx.doi.org/10.1214/08-AAP537 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Limit theorems for random point measures generated by cooperative sequential adsorption
We consider a finite sequence of random points in a finite domain of a
finite-dimensional Euclidean space. The points are sequentially allocated in
the domain according to a model of cooperative sequential adsorption. The main
peculiarity of the model is that the probability distribution of a point
depends on previously allocated points. We assume that the dependence vanishes
as the concentration of points tends to infinity. Under this assumption the law
of large numbers, the central limit theorem and Poisson approximation are
proved for the generated sequence of random point measures.Comment: 17 page
Three Problems of the Theory of Choice on Random Sets
This paper discusses three problems which are united not only by the common topic of
research stated in the title, but also by a somewhat surprising interlacing of the methods and
techniques used.
In the first problem, an attempt is made to resolve a very unpleasant metaproblem arising in
general choice theory: why the conditions of rationality are not really necessary or, in other words,
why in every-day life we are quite satisfied with choice methods which are far from being ideal. The
answer, substantiated by a number of results, is as follows: situations in which the choice function
"misbehaves" are very seldom met in large presentations.
In the second problem, an overview of our studies is given on the problem of statistical
properties of choice. One of the most astonishing phenomenon found when we deviate from scalar-extremal
choice functions is in stable multiplicity of choice. If our presentation is random, then a
random number of alternatives is chosen in it. But how many? The answer isn't trivial, and may be
sought in many different directions. As we shall see below, usually a bottleneck case was considered
in seeking the answer. It is interesting to note that statistical information effects the properties of the
problem very much.
The third problem is devoted to a model of a real life choice process. This process is
typically spread in time, and we gradually (up to the time of making a final decision) accumulate
experience, but once a decision is made we are not free to reject it. In the classical statement (i.e.
when "optimality" is measured by some number) this model is referred to as a "secretary problem",
and a great deal of literature is devoted to its study. We consider the case when the notions of
optimality are most general. As will be seen below, the best strategy is practically determined by
only the statistical properties of the corresponding choice function rather than its specific form
Gaussian limits for multidimensional random sequential packing at saturation (extended version)
Consider the random sequential packing model with infinite input and in any
dimension. When the input consists of non-zero volume convex solids we show
that the total number of solids accepted over cubes of volume is
asymptotically normal as . We provide a rate of
approximation to the normal and show that the finite dimensional distributions
of the packing measures converge to those of a mean zero generalized Gaussian
field. The method of proof involves showing that the collection of accepted
solids satisfies the weak spatial dependence condition known as stabilization.Comment: 31 page
- …