514 research outputs found
The Capacity of Some P\'olya String Models
We study random string-duplication systems, which we call P\'olya string
models. These are motivated by DNA storage in living organisms, and certain
random mutation processes that affect their genome. Unlike previous works that
study the combinatorial capacity of string-duplication systems, or various
string statistics, this work provides exact capacity or bounds on it, for
several probabilistic models. In particular, we study the capacity of noisy
string-duplication systems, including the tandem-duplication, end-duplication,
and interspersed-duplication systems. Interesting connections are drawn between
some systems and the signature of random permutations, as well as to the beta
distribution common in population genetics
Computable de Finetti measures
We prove a computable version of de Finetti's theorem on exchangeable
sequences of real random variables. As a consequence, exchangeable stochastic
processes expressed in probabilistic functional programming languages can be
automatically rewritten as procedures that do not modify non-local state. Along
the way, we prove that a distribution on the unit interval is computable if and
only if its moments are uniformly computable.Comment: 32 pages. Final journal version; expanded somewhat, with minor
corrections. To appear in Annals of Pure and Applied Logic. Extended abstract
appeared in Proceedings of CiE '09, LNCS 5635, pp. 218-23
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
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)
Cutting edges at random in large recursive trees
We comment on old and new results related to the destruction of a random
recursive tree (RRT), in which its edges are cut one after the other in a
uniform random order. In particular, we study the number of steps needed to
isolate or disconnect certain distinguished vertices when the size of the tree
tends to infinity. New probabilistic explanations are given in terms of the
so-called cut-tree and the tree of component sizes, which both encode different
aspects of the destruction process. Finally, we establish the connection to
Bernoulli bond percolation on large RRT's and present recent results on the
cluster sizes in the supercritical regime.Comment: 29 pages, 3 figure
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