1,148 research outputs found

### Confidence intervals for nonhomogeneous branching processes and polymerase chain reactions

We extend in two directions our previous results about the sampling and the
empirical measures of immortal branching Markov processes. Direct applications
to molecular biology are rigorous estimates of the mutation rates of polymerase
chain reactions from uniform samples of the population after the reaction.
First, we consider nonhomogeneous processes, which are more adapted to real
reactions. Second, recalling that the first moment estimator is analytically
known only in the infinite population limit, we provide rigorous confidence
intervals for this estimator that are valid for any finite population. Our
bounds are explicit, nonasymptotic and valid for a wide class of nonhomogeneous
branching Markov processes that we describe in detail. In the setting of
polymerase chain reactions, our results imply that enlarging the size of the
sample becomes useless for surprisingly small sizes. Establishing confidence
intervals requires precise estimates of the second moment of random samples.
The proof of these estimates is more involved than the proofs that allowed us,
in a previous paper, to deal with the first moment. On the other hand, our
method uses various, seemingly new, monotonicity properties of the harmonic
moments of sums of exchangeable random variables.Comment: Published at http://dx.doi.org/10.1214/009117904000000775 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org

### Harmonic moments of non homogeneous branching processes

We study the harmonic moments of Galton-Watson processes, possibly non
homogeneous, with positive values. Good estimates of these are needed to
compute unbiased estimators for non canonical branching
Markov processes, which occur, for instance, in the modeling of the
polymerase chain reaction. By convexity, the ratio of the harmonic mean to the
mean is at most 1. We prove that, for every square integrable branching
mechanisms, this ratio lies between 1-A/k and 1-B/k for every initial
population of size k greater than A. The positive constants A and B, such that
B is at most A, are explicit and depend only on the generation-by-generation
branching mechanisms. In particular, we do not use the distribution of the
limit of the classical martingale associated to the Galton-Watson process.
Thus, emphasis is put on non asymptotic bounds and on the dependence of the
harmonic mean upon the size of the initial population. In the Bernoulli case,
which is relevant for the modeling of the polymerase chain reaction, we prove
essentially optimal bounds that are valid for every initial population.
Finally, in the general case and for large enough initial populations, similar
techniques yield sharp estimates of the harmonic moments of higher degrees

### Coupling times with ambiguities for particle systems and applications to context-dependent DNA substitution models

We define a notion of coupling time with ambiguities for interacting particle
systems, and show how this can be used to prove ergodicity and to bound the
convergence time to equilibrium and the decay of correlations at equilibrium. A
motivation is to provide simple conditions which ensure that perturbed particle
systems share some properties of the underlying unperturbed system. We apply
these results to context-dependent substitution models recently introduced by
molecular biologists as descriptions of DNA evolution processes. These models
take into account the influence of the neighboring bases on the substitution
probabilities at a site of the DNA sequence, as opposed to most usual
substitution models which assume that sites evolve independently of each other.Comment: 33 page

### Coupling from the past times with ambiguities and perturbations of interacting particle systems

We discuss coupling from the past techniques (CFTP) for perturbations of
interacting particle systems on the d-dimensional integer lattice, with a
finite set of states, within the framework of the graphical construction of the
dynamics based on Poisson processes. We first develop general results for what
we call CFTP times with ambiguities. These are analogous to classical coupling
(from the past) times, except that the coupling property holds only provided
that some ambiguities concerning the stochastic evolution of the system are
resolved. If these ambiguities are rare enough on average, CFTP times with
ambiguities can be used to build actual CFTP times, whose properties can be
controlled in terms of those of the original CFTP time with ambiguities. We
then prove a general perturbation result, which can be stated informally as
follows. Start with an interacting particle system possessing a CFTP time whose
definition involves the exploration of an exponentially integrable number of
points in the graphical construction, and which satisfies the positive rates
property. Then consider a perturbation obtained by adding new transitions to
the original dynamics. Our result states that, provided that the perturbation
is small enough (in the sense of small enough rates), the perturbed interacting
particle system too possesses a CFTP time (with nice properties such as an
exponentially decaying tail). The proof consists in defining a CFTP time with
ambiguities for the perturbed dynamics, from the CFTP time for the unperturbed
dynamics. Finally, we discuss examples of particle systems to which this result
can be applied. Concrete examples include a class of neighbor-dependent
nucleotide substitution model, and variations of the classical voter model,
illustrating the ability of our approach to go beyond the case of weakly
interacting particle systems.Comment: This paper is an extended and revised version of an earlier
manuscript available as arXiv:0712.0072, where the results were limited to
perturbations of RN+YpR nucleotide substitution model

### Solvable models of neighbor-dependent nucleotide substitution processes

We prove that a wide class of models of Markov neighbor-dependent
substitution processes on the integer line is solvable. This class contains
some models of nucleotide substitutions recently introduced and studied
empirically by molecular biologists. We show that the polynucleotide
frequencies at equilibrium solve explicit finite-size linear systems. Finally,
the dynamics of the process and the distribution at equilibrium exhibit some
stringent, rather unexpected, independence properties. For example, nucleotide
sites at distance at least three evolve independently, and the sites, if
encoded as purines and pyrimidines, evolve independently.Comment: 47 pages, minor modification

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