51 research outputs found
An algebraic approach to Polya processes
P\'olya processes are natural generalization of P\'olya-Eggenberger urn
models. This article presents a new approach of their asymptotic behaviour {\it
via} moments, based on the spectral decomposition of a suitable finite
difference operator on polynomial functions. Especially, it provides new
results for {\it large} processes (a P\'olya process is called {\it small} when
1 is simple eigenvalue of its replacement matrix and when any other eigenvalue
has a real part ; otherwise, it is called large)
Limit distributions for large P\'{o}lya urns
We consider a two-color P\'{o}lya urn in the case when a fixed number of
balls is added at each step. Assume it is a large urn that is, the second
eigenvalue of the replacement matrix satisfies . After
drawings, the composition vector has asymptotically a first deterministic term
of order and a second random term of order . The object of
interest is the limit distribution of this random term. The method consists in
embedding the discrete-time urn in continuous time, getting a two-type
branching process. The dislocation equations associated with this process lead
to a system of two differential equations satisfied by the Fourier transforms
of the limit distributions. The resolution is carried out and it turns out that
the Fourier transforms are explicitly related to Abelian integrals over the
Fermat curve of degree . The limit laws appear to constitute a new family of
probability densities supported by the whole real line.Comment: Published in at http://dx.doi.org/10.1214/10-AAP696 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Smoothing equations for large P\'olya urns
Consider a balanced non triangular two-color P\'olya-Eggenberger urn process,
assumed to be large which means that the ratio sigma of the replacement matrix
eigenvalues satisfies 1/2<sigma <1. The composition vector of both discrete
time and continuous time models admits a drift which is carried by the
principal direction of the replacement matrix. In the second principal
direction, this random vector admits also an almost sure asymptotics and a
real-valued limit random variable arises, named WDT in discrete time and WCT in
continous time. The paper deals with the distributions of both W. Appearing as
martingale limits, known to be nonnormal, these laws remain up to now rather
mysterious.
Exploiting the underlying tree structure of the urn process, we show that WDT
and WCT are the unique solutions of two distributional systems in some suitable
spaces of integrable probability measures. These systems are natural extensions
of distributional equations that already appeared in famous algorithmical
problems like Quicksort analysis. Existence and unicity of the solutions of the
systems are obtained by means of contracting smoothing transforms. Via the
equation systems, we find upperbounds for the moments of WDT and WCT and we
show that the laws of WDT and WCT are moment-determined. We also prove that WDT
is supported by the whole real line and admits a continuous density (WCT was
already known to have a density, infinitely differentiable on R\{0} and not
bounded at the origin)
Support and density of the limit -ary search trees distribution
The space requirements of an -ary search tree satisfies a well-known phase
transition: when , the second order asymptotics is Gaussian. When
, it is not Gaussian any longer and a limit of a complex-valued
martingale arises. We show that the distribution of has a square integrable
density on the complex plane, that its support is the whole complex plane, and
that it has finite exponential moments. The proofs are based on the study of
the distributional equation W\egalLoi\sum_{k=1}^mV_k^{\lambda}W_k, where
are the spacings of independent random variables
uniformly distributed on , are independent copies of W
which are also independent of and is a complex
number
Digital search trees and chaos game representation
In this paper, we consider a possible representation of a DNA sequence in a
quaternary tree, in which on can visualize repetitions of subwords. The
CGR-tree turns a sequence of letters into a digital search tree (DST), obtained
from the suffixes of the reversed sequence. Several results are known
concerning the height and the insertion depth for DST built from i.i.d.
successive sequences. Here, the successive inserted wors are strongly
dependent. We give the asymptotic behaviour of the insertion depth and of the
length of branches for the CGR-tree obtained from the suffixes of reversed
i.i.d. or Markovian sequence. This behaviour turns out to be at first order the
same one as in the case of independent words. As a by-product, asymptotic
results on the length of longest runs in a Markovian sequence are obtained
Variable length Markov chains and dynamical sources
Infinite random sequences of letters can be viewed as stochastic chains or as
strings produced by a source, in the sense of information theory. The
relationship between Variable Length Markov Chains (VLMC) and probabilistic
dynamical sources is studied. We establish a probabilistic frame for context
trees and VLMC and we prove that any VLMC is a dynamical source for which we
explicitly build the mapping. On two examples, the ``comb'' and the ``bamboo
blossom'', we find a necessary and sufficient condition for the existence and
the unicity of a stationary probability measure for the VLMC. These two
examples are detailed in order to provide the associated Dirichlet series as
well as the generating functions of word occurrences.Comment: 45 pages, 15 figure
B-urns
The fringe of a B-tree with parameter is considered as a particular
P\'olya urn with colors. More precisely, the asymptotic behaviour of this
fringe, when the number of stored keys tends to infinity, is studied through
the composition vector of the fringe nodes. We establish its typical behaviour
together with the fluctuations around it. The well known phase transition in
P\'olya urns has the following effect on B-trees: for , the
fluctuations are asymptotically Gaussian, though for , the
composition vector is oscillating; after scaling, the fluctuations of such an
urn strongly converge to a random variable . This limit is -valued and it does not seem to follow any classical law. Several properties
of are shown: existence of exponential moments, characterization of its
distribution as the solution of a smoothing equation, existence of a density
relatively to the Lebesgue measure on , support of . Moreover, a
few representations of the composition vector for various values of
illustrate the different kinds of convergence
A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics
A ``hybrid method'', dedicated to asymptotic coefficient extraction in
combinatorial generating functions, is presented, which combines Darboux's
method and singularity analysis theory. This hybrid method applies to functions
that remain of moderate growth near the unit circle and satisfy suitable
smoothness assumptions--this, even in the case when the unit circle is a
natural boundary. A prime application is to coefficients of several types of
infinite product generating functions, for which full asymptotic expansions
(involving periodic fluctuations at higher orders) can be derived. Examples
relative to permutations, trees, and polynomials over finite fields are treated
in this way.Comment: 31 page
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