136,434 research outputs found
On martingale tail sums in affine two-color urn models with multiple drawings
In two recent works, Kuba and Mahmoud (arXiv:1503.090691 and
arXiv:1509.09053) introduced the family of two-color affine balanced Polya urn
schemes with multiple drawings. We show that, in large-index urns (urn index
between and ) and triangular urns, the martingale tail sum for the
number of balls of a given color admits both a Gaussian central limit theorem
as well as a law of the iterated logarithm. The laws of the iterated logarithm
are new even in the standard model when only one ball is drawn from the urn in
each step (except for the classical Polya urn model). Finally, we prove that
the martingale limits exhibit densities (bounded under suitable assumptions)
and exponentially decaying tails. Applications are given in the context of node
degrees in random linear recursive trees and random circuits.Comment: 17 page
Asymptotic results for a generalized PĂČlya urn with delay and an applications to clinical trials
In this paper a new PĂČlya urn model is introduced and studied; in particular, a strong law of large numbers and two central limit theorems are proven. This urn generalizes a model studied in Berti et al. (2004), May et al. (2005) and in Crimaldi (2007) and it has natural applications in clinical trials. Indeed, the model include both delayed and missing (or null) responses. Moreover, a connection with the conditional identity in distribution of Berti et al. (2004) is given.
Power-law behavior and condensation phenomena in disordered urn models
We investigate equilibrium statistical properties of urn models with
disorder. Two urn models are proposed; one belongs to the Ehrenfest class, and
the other corresponds to the Monkey class. These models are introduced from the
view point of the power-law behavior and randomness; it is clarified that
quenched random parameters play an important role in generating power-law
behavior. We evaluate the occupation probability with which an urn has
balls by using the concept of statistical physics of disordered systems. In
the disordered urn model belonging to the Monkey class, we find that above
critical density for a given temperature, condensation
phenomenon occurs and the occupation probability changes its scaling behavior
from an exponential-law to a heavy tailed power-law in large regime. We
also discuss an interpretation of our results for explaining of macro-economy,
in particular, emergence of wealth differentials.Comment: 16pages, 9figures, using iopart.cls, 2 new figures were adde
The CLT Analogue for Cyclic Urns
A cyclic urn is an urn model for balls of types where in each
draw the ball drawn, say of type , is returned to the urn together with a
new ball of type . The case is the well-known Friedman urn.
The composition vector, i.e., the vector of the numbers of balls of each type
after steps is, after normalization, known to be asymptotically normal for
. For the normalized composition vector does not
converge. However, there is an almost sure approximation by a periodic random
vector. In this paper the asymptotic fluctuations around this periodic random
vector are identified. We show that these fluctuations are asymptotically
normal for all . However, they are of maximal dimension only when
does not divide . For being a multiple of the fluctuations are
supported by a two-dimensional subspace.Comment: Extended abstract to be replaced later by a full versio
Analytic urns
This article describes a purely analytic approach to urn models of the
generalized or extended P\'olya-Eggenberger type, in the case of two types of
balls and constant ``balance,'' that is, constant row sum. The treatment starts
from a quasilinear first-order partial differential equation associated with a
combinatorial renormalization of the model and bases itself on elementary
conformal mapping arguments coupled with singularity analysis techniques.
Probabilistic consequences in the case of ``subtractive'' urns are new
representations for the probability distribution of the urn's composition at
any time n, structural information on the shape of moments of all orders,
estimates of the speed of convergence to the Gaussian limit and an explicit
determination of the associated large deviation function. In the general case,
analytic solutions involve Abelian integrals over the Fermat curve x^h+y^h=1.
Several urn models, including a classical one associated with balanced trees
(2-3 trees and fringe-balanced search trees) and related to a previous study of
Panholzer and Prodinger, as well as all urns of balance 1 or 2 and a sporadic
urn of balance 3, are shown to admit of explicit representations in terms of
Weierstra\ss elliptic functions: these elliptic models appear precisely to
correspond to regular tessellations of the Euclidean plane.Comment: Published at http://dx.doi.org/10.1214/009117905000000026 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Regenerative tree growth: Binary self-similar continuum random trees and Poisson--Dirichlet compositions
We use a natural ordered extension of the Chinese Restaurant Process to grow
a two-parameter family of binary self-similar continuum fragmentation trees. We
provide an explicit embedding of Ford's sequence of alpha model trees in the
continuum tree which we identified in a previous article as a distributional
scaling limit of Ford's trees. In general, the Markov branching trees induced
by the two-parameter growth rule are not sampling consistent, so the existence
of compact limiting trees cannot be deduced from previous work on the sampling
consistent case. We develop here a new approach to establish such limits, based
on regenerative interval partitions and the urn-model description of sampling
from Dirichlet random distributions.Comment: Published in at http://dx.doi.org/10.1214/08-AOP445 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
- âŠ