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176
Probabilistic bounds on the coefficients of polynomials with only real zeros
, 1997
"... The work of Harper and subsequent authors has shown that finite sequences (a0,..., an) arising from combinatorial problems are often such that the polynomial A(z): = n k=0 akz k has only real zeros. Basic examples include rows from the arrays of binomial coefficients, Stirling numbers of the first a ..."
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The work of Harper and subsequent authors has shown that finite sequences (a0,..., an) arising from combinatorial problems are often such that the polynomial A(z): = n k=0 akz k has only real zeros. Basic examples include rows from the arrays of binomial coefficients, Stirling numbers of the first and second kinds, and Eulerian numbers. Assuming the ak are nonnegative, A(1)>0 and that A(z) is not constant, it is known that A(z) has only real zeros iff the normalized sequence (a0 A(1),..., an A(1)) is the probability distribution of the number of successes in n independent trials for some sequence of success probabilities. Such sequences (a0,..., an) are also known to be characterized by total positivity of the infinite matrix (ai & j) indexed by nonnegative integers i and j. This papers reviews inequalities and approximations for such sequences, called Polya frequency sequences which follow from their probabilistic representation. In combinatorial examples these inequalities yield a number of improvements of known estimates.
Bayesian nonparametric estimators derived from conditional Gibbs structures
 J. PHYS. A: MATH. GEN
, 2008
"... We consider discrete nonparametric priors which induce Gibbstype exchangeable random partitions and investigate their posterior behavior in detail. In particular, we deduce conditional distributions and the corresponding Bayesian nonparametric estimators, which can be readily exploited for predictin ..."
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Cited by 31 (7 self)
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We consider discrete nonparametric priors which induce Gibbstype exchangeable random partitions and investigate their posterior behavior in detail. In particular, we deduce conditional distributions and the corresponding Bayesian nonparametric estimators, which can be readily exploited for predicting various features of additional samples. The results provide useful tools for genomic applications where prediction of future outcomes is required.
The structure of the allelic partition of the total population for GaltonWatson processes with neutral mutations
"... We consider a (sub)critical Galton–Watson process with neutral mutations (infinite alleles model), and decompose the entire population into clusters of individuals carrying the same allele. We specify the law of this allelic partition in terms of the distribution of the number of clonechildren and ..."
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Cited by 29 (4 self)
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We consider a (sub)critical Galton–Watson process with neutral mutations (infinite alleles model), and decompose the entire population into clusters of individuals carrying the same allele. We specify the law of this allelic partition in terms of the distribution of the number of clonechildren and the number of mutantchildren of a typical individual. The approach combines an extension of Harris representation of Galton–Watson processes and a version of the ballot theorem. Some limit theorems related to the distribution of the allelic partition are also given. 1. Introduction. We consider a Galton–Watson process, that is, a population model with asexual reproduction such that at every generation, each individual gives birth to a random number of children according to a fixed distribution and independently of the other individuals in the population. We are interested in the situation where a child can be either a clone, that
PoissonKingman Partitions
 of Lecture NotesMonograph Series
, 2002
"... This paper presents some general formulas for random partitions of a finite set derived by Kingman's model of random sampling from an interval partition generated by subintervals whose lengths are the points of a Poisson point process. These lengths can be also interpreted as the jumps of a sub ..."
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Cited by 25 (3 self)
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This paper presents some general formulas for random partitions of a finite set derived by Kingman's model of random sampling from an interval partition generated by subintervals whose lengths are the points of a Poisson point process. These lengths can be also interpreted as the jumps of a subordinator, that is an increasing process with stationary independent increments. Examples include the twoparameter family of PoissonDirichlet models derived from the Poisson process of jumps of a stable subordinator. Applications are made to the random partition generated by the lengths of excursions of a Brownian motion or Brownian bridge conditioned on its local time at zero.
Random Walks on Trees and Matchings
 Electron. J. Probab
, 2002
"... We give sharp rates of convergence for a natural Markov chain on the space of phylogenetic trees and dually for the natural random walk on the set of perfect matchings in the complete graph on 2n vertices. Roughly, the results show that n log n steps are necessary and su#ce to achieve randomness. ..."
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Cited by 23 (6 self)
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We give sharp rates of convergence for a natural Markov chain on the space of phylogenetic trees and dually for the natural random walk on the set of perfect matchings in the complete graph on 2n vertices. Roughly, the results show that n log n steps are necessary and su#ce to achieve randomness. The proof depends on the representation theory of the symmetric group and a bijection between trees and matchings.
The Infinite GammaPoisson Feature Model
"... We present a probability distribution over nonnegative integer valued matrices with possibly an infinite number of columns. We also derive a stochastic process that reproduces this distribution over equivalence classes. This model can play the role of the prior in nonparametric Bayesian learning sc ..."
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We present a probability distribution over nonnegative integer valued matrices with possibly an infinite number of columns. We also derive a stochastic process that reproduces this distribution over equivalence classes. This model can play the role of the prior in nonparametric Bayesian learning scenarios where multiple latent features are associated with the observed data and each feature can have multiple appearances or occurrences within each data point. Such data arise naturally when learning visual object recognition systems from unlabelled images. Together with the nonparametric prior we consider a likelihood model that explains the visual appearance and location of local image patches. Inference with this model is carried out using a Markov chain Monte Carlo algorithm. 1
Arlequin ver 3.1  An Integrated Software Package for Population Genetics Data Analysis
, 2006
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Prediction Rules for Exchangeable Sequences Related to Species Sampling
 IN PROCESSOR DESIGN. MASTER’S THESIS. LM ERICSSON 2000
, 1998
"... Suppose an exchangable sequence with values in a nice measurable space S admits a prediction rule of the following form: given the first n terms of the sequence, the next term equals the jth distinct value observed so far with probability pj;n , for j = 1; 2; : : :, and otherwise is a new value wit ..."
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Cited by 22 (2 self)
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Suppose an exchangable sequence with values in a nice measurable space S admits a prediction rule of the following form: given the first n terms of the sequence, the next term equals the jth distinct value observed so far with probability pj;n , for j = 1; 2; : : :, and otherwise is a new value with distribution for some probability measure on S with no atoms. Then the pj;n depend only on the partitition of the first n integers induced by the first n values of the sequence. All possible distributions for such an exchangeable sequence are characterized in terms of constraints on the pj;n and in terms of their de Finetti representations.
Limit theorems for combinatorial structures via discrete process approximations
 RANDOM STRUCTURES AND ALGORITHMS
, 1992
"... Discrete functional limit theorems, which give independent process approximations for the joint distribution of the component structure of combinatorial objects such as permutations and mappings, have recently become available. In this article, we demonstrate the power of these theorems to provide e ..."
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Cited by 21 (2 self)
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Discrete functional limit theorems, which give independent process approximations for the joint distribution of the component structure of combinatorial objects such as permutations and mappings, have recently become available. In this article, we demonstrate the power of these theorems to provide elementary proofs of a variety of new and old limit theorems, including results previously proved by complicated analytical methods. Among the examples we treat are Brownian motion limit theorems for the cycle counts of a random permutation or the component counts of a random mapping, a Poisson limit law for the core of a random mapping, a generalization of the ErdosTurin Law for the logorder of a random permutation and the smallest component size of a random permutation, approximations to the joint laws of the smallest cycle sizes of a random mapping, and a limit distribution for the difference between the total number of cycles and the number of