Bayesian estimators of diversity indexes on exchangeable random partitions
Bayesian diversity estimators are martingales converging almost surely and in mean with common limit and local behavior with plug in estimators.
math.PR↗
arXiv subjects
Servet Martinez
Publications and source records attributed to Servet Martinez.
A note on the convergence of the Bayesian entropy estimator for exchangeable partitions
We show that when the proportions of a countable set of species are organized as an exchangeable partition of the unit interval and we take a sample on it, then the Bayesian posterior entropy converges a.s. and in L^1 to the entropy of the species when the sample size diverges to infinity.
math.PR↗
Level numbers preserving transformations on excursionsdefined by random walks with state dependent jump laws
We study the number of individuals per level defined by excursions of random walks with state dependent jump law. These level numbersdetermine the probability of the excursion, and the set of transformations preserving the level numbers is, generically, the set of transformation that preserve the probability law of excursions.We compute the number of excursions having a fixed level numbers and we show that the class of shifts of excursions generate all the excursions having a fixed level numbers. We study the behaviorof the level numbers under the Vervaat transform and the Doob transform.
math.PR↗
Sterile versus prolific individuals pertaining to linear-fractional Bienaym{é}-Galton-Watson trees
In a Bienaymé-Galton-Watson process for which there is a positiveprobability for individuals of having no offspring, there is a subtlebalance and dependence between the sterile nodes (the dead nodes or leaves)and the prolific ones (the productive nodes) both at and up to the currentgeneration. We explore the many facets of this problem, especially in thecontext of an exactly solvable linear-fractional branching mechanism at allgeneration. Eased asymptotic issues are investigated. Relation of thisspecial branching process to skip-free to the left and simple random walks'excursions is then investigated. Mutual statistical information on theirshapes can be learnt from this association.
math.PR↗
Solving the migration-recombination equation from a genealogical point of view
We consider the discrete-time migration-recombination equation, a deterministic, nonlinear dynamical system that describes the evolution of the genetic type distribution of a population evolving under migration and recombination in a law of large numbers setting. We relate this dynamics (forward in time) to a Markov chain, namely a labelled partitioning process, backward in time. This way, we obtain a stochastic representation of the solution of the migration-recombination equation. As a consequence, one obtains an explicit solution of the nonlinear dynamics, simply in terms of powers of the transition matrix of the Markov chain. Finally, we investigate the limiting and quasi-limiting behaviour of the Markov chain, which gives immediate access to the asymptotic behaviour of the dynamical system. We finally sketch the analogous situation in continuous time.
math.PR↗
Inverse M-matrix, a new characterization
In this article we present a new characterization of inverse M-matrices, inverse row diagonally dominant M-matrices and inverse row and column diagonally dominant M-matrices, based on the positivity of certain inner products.
math.PR↗
Revisiting John Lamperti's maximal branching process
Lamperti's maximal branching process is revisited, with emphasis on the description of the shape of the invariant measures in both the recurrent and transient regimes. A truncated version of this chain is exhibited, preserving the monotonicity of the original Lamperti chain supported by the integers. The Brown theory of hitting times applies to the latter chain with finite state-space, including sharp strong time to stationarity. Additional information on these hitting time problems are drawn from the quasi-stationary point of view.
math.PR↗
Truncation in Duality and Intertwining Kernels
We study properties of truncations in the dual and intertwining process in the monotone case. The main properties are stated for the time-reversed process and the time of absorption of the truncated intertwining process.
math.PR↗
Diffusions from Infinity
In this paper we consider diffusions on the half line (0, $\infty$) such that the expectation of the arrival time at the origin is uniformly bounded in the initial point. This implies that there is a well defined diffusion process starting from infinity, which takes finite values at positive times. We study the behaviour of hitting times of large barriers and in a dual way, the behaviour of the process starting at infinity for small time. In particular we prove that the process coming down from infinity is in small time governed by a specific deterministic function. Suitably normalized fluctuations of the hitting times are asymptotically Gaussian. We also derive the tail of the distribution of the hitting time of the origin and a Yaglom limit for the diffusion starting from infinity. We finally prove that the distribution of this process killed at the origin is absolutely continuous with respect to the speed measure. The density is expressed in terms of the eigenvalues and eigenfunctions of the generator of the killed diffusion.
math.PR↗
A Note on the Quasi-Stationary Distribution of the Shiryaev Martingale on the Positive Half-Line
We obtain a closed-form formula for the quasi-stationary distribution of the classical Shiryaev martingale diffusion considered on the positive half-line $[A,+\infty)$ with $A>0$ fixed; the state space's left endpoint is assumed to be the killing boundary. The formula is obtained analytically as the solution of the appropriate singular Sturm-Liouville problem; the latter was first considered in Section 7.8.2 of Collet et al. (2013), but has heretofore remained unsolved.
math.PR↗
A probabilistic analysis of a discrete-time evolution in recombination II. (On partitions)
We study the discrete-time evolution of a transformation on a set of probability measures that is up-dated combining independently the marginals on the atoms of partitions. This model was recently introduced in Baake, Baake and Salamat (Discr. and contin. dynam. syst. 36, 2016) for continuous-time evolution and generalizes previous ones based upon dyadic partitions. We associate to the discrete-time evolution a natural Markov chain and describe its quasi-stationary behavior retrieving all the results we recently found for dyadic partitions.
math.PR↗
A probabilistic analysis of a discrete-time evolution in recombination
We give a closed form of the discrete-time evolution of a recombination transformation in population genetics. This decomposition allows to define a Markov chain in a natural way. We describe the geometric decay rate to the limit distribution, and the quasi-stationary behavior when conditioned to the event that the chain does not hit the limit distribution.
math.PR↗
Asymptotics for the heat kernel in multicone domains
A multi cone domain $Ω\subseteq \mathbb{R}^n$ is an open, connected set that resembles a finite collection of cones far away from the origin. We study the rate of decay in time of the heat kernel $p(t,x,y)$ of a Brownian motion killed upon exiting $Ω$, using both probabilistic and analytical techniques. We find that the decay is polynomial and we characterize $\lim_{t\to\infty} t^{1+α}p(t,x,y)$ in terms of the Martin boundary of $Ω$ at infinity, where $α>0$ depends on the geometry of $Ω$. We next derive an analogous result for $t^{κ/2}\mathbb{P}_x(T >t)$, with $κ= 1+α- n/2$, where $T$ is the exit time form $Ω$. Lastly, we deduce the renormalized Yaglom limit for the process conditioned on survival.
math.PR↗
On Moebius duality and Coarse-Graining
We study duality relations for zeta and Möbius matrices and monotone conditions on the kernels. We focus on the cases of family of sets and partitions. The conditions for positivity of the dual kernels are stated in terms of the positive Möbius cone of functions, which is described in terms of Sylvester formulae. We study duality under coarse-graining and show that an $h-$transform is needed to preserve stochasticity. We give conditions in order that zeta and Möbius matrices admit coarse-graining, and we prove they are satisfied for sets and partitions. This is a source of relevant examples in genetics on the haploid and multi-allelic Cannings models.
math.PR↗
A distribution weighting a set of laws whose initial states are grouped into classes
Let $I$ be a finite alphabet and $\aS\subset I$ be a nonempty strict subset. The sequences in $I^\ZZ$ are organized into connected regions which always start with a symbol in $\aS$. The regions are labelled by types $C(s)$, thus a region starting at $s'\in C(s)$ has the same type as one starting at $s$. Let $(\aP_s: s\in \aS)$ be a family of distributions on $I^\NN$ where each $\aP_s$ charges sequences starting with the symbol $s$. We can define a natural distribution $\PP$ on $I^\NN$, that counts the number of visits to the states from $\aP_s$, properly weighted. A dynamics of interest is such that at the first occurrence of $s'\in \aS\setminus C(s)$ the law regenerates with distribution $\aP_{s'}$. In this case we are able to find simple conditions for $\PP$ to be stationary. In addition, we study the following more complex model: once a symbol $s'\in \aS\setminus C(s)$ has been encountered, there is a decision to be made, either a new region of type $C(s')$ governed by $\aP_{s'}$ starts or the region continues to be a $C(s)$ region. This decision is modeled as random and depends on $s'$. In this setting a similar distribution to $\PP$ can be constructed and the conditions for stationarity are supplied. These models are inspired by genomic sequences where $I$ is the set of codons, the classes $(C(s): s\in \aS)$ group codons defining similar genomic classes, e.g. in bacteria there are two classes corresponding to the start and stop codons, and the random decision to continue a region or to begin a new region of a different class reflects the well-known fact that not every appearance of a start codon marks the beginning of a new coding region.
math.PR↗
Occupancy distributions arising in sampling from Gibbs-Poisson abundance models
Estimating the number $n$ of unseen species from a $k-$sample displaying only $p\leq k$ distinct sampled species has received attention for long. It requires a model of species abundance together with a sampling model. We start with a discrete model of iid stochastic species abundances, each with Gibbs-Poisson distribution. A $k-$sample drawn from the $n-$species abundances vector is the one obtained while conditioning it on summing to $k$% . We discuss the sampling formulae (species occupancy distributions, frequency of frequencies) in this context. We then develop some aspects of the estimation of $n$ problem from the size $k$ of the sample and the observed value of $P_{n,k}$, the number of distinct sampled species. It is shown that it always makes sense to study these occupancy problems from a Gibbs-Poisson abundance model in the context of a population with infinitely many species. From this extension, a parameter $γ$ naturally appears, which is a measure of richness or diversity of species. We rederive the sampling formulae for a population with infinitely many species, together with the distribution of the number $P_{k}$ of distinct sampled species. We investigate the estimation of $γ$ problem from the sample size $k$ and the observed value of $P_{k}$. We then exhibit a large special class of Gibbs-Poisson distributions having the property that sampling from a discrete abundance model may equivalently be viewed as a sampling problem from a random partition of unity, now in the continuum. When $n$ is finite, this partition may be built upon normalizing $% n$ infinitely divisible iid positive random variables by its partial sum. It is shown that the sampling process in the continuum should generically be biased on the total length appearing in the latter normalization. A construction with size-biased sampling from the ranked normalized jumps of a subordinator is also supplied, would the problem under study present infinitely many species. We illustrate our point of view with many examples, some of which being new ones.
stat.ME↗
Existence and uniqueness of a quasi-stationary distribution for Markov processes with fast return from infinity
We study the long time behaviour of a Markov process evolving in $\mathbb{N}$ and conditioned not to hit 0. Assuming that the process comes back quickly from infinity, we prove that the process admits a unique quasi-stationary distribution (in particular, the distribution of the conditioned process admits a limit when time goes to infinity). Moreover, we prove that the distribution of the process converges exponentially fast in total variation norm to its quasi-stationary distribution and we provide an explicit rate of convergence. As a first application of our result, we bring a new insight on the speed of convergence to the quasi-stationary distribution for birth and death processes: we prove that these processes converge exponentially fast to a quasi-stationary distribution if and only if they have a unique quasi-stationary distribution. Also, considering the lack of results on quasi-stationary distributions for non-irreducible processes on countable spaces, we show, as a second application of our result, the existence and uniqueness of a quasi-stationary distribution for a class of possibly non-irreducible processes.
math.PR↗
Stochastic models for a chemostat and long time behavior
We introduce two stochastic chemostat models consisting in a coupled population-nutrient process reflecting the interaction between the nutrient and the bacterias in the chemostat with finite volume. The nutrient concentration evolves continuously but depending on the population size, while the population size is a birth and death process with coefficients depending on time through the nutrient concentration. The nutrient is shared by the bacteria and creates a regulation of the bacterial population size. The latter and the fluctuations due to the random births and deaths of individuals make the population go almost surely to extinction. Therefore, we are interested in the long time behavior of the bacterial population conditioned to the non-extinction. We prove the global existence of the process and its almost sure extinction. The existence of quasi-stationary distributions is obtained based on a general fixed point argument. Moreover, we prove the absolute continuity of the nutrient distribution when conditioned to a fixed number of individuals and the smoothness of the corresponding densities.
math.PR↗