arXiv subjects

Serik Sagitov

Publications and source records attributed to Serik Sagitov.

36 records · Page 2Linked to original sources

arXiv2015

Nonparametric estimation of infinitely divisible distributions based on variational analysis on measures

The paper develops new methods of non-parametric estimation a compound Poisson distribution. Such a problem arise, in particular, in the inference of a Levy process recorded at equidistant time intervals. Our key estimator is based on series decomposition of functionals of a measure and relies on the steepest descent technique recently developed in variational analysis of measures. Simulation studies demonstrate applicability domain of our methods and how they positively compare and complement the existing techniques. They are particularly suited for discrete compounding distributions, not necessarily concentrated on a grid nor on the positive or negative semi-axis. They also give good results for continuous distributions provided an appropriate smoothing is used for the obtained atomic measure.

math.ST↗

arXiv2015

Asymptotic results for the number of Wagner's solutions to a generalised birthday problem

We study two functionals of a random matrix $\boldsymbol A$ with independent elements uniformly distributed over the cyclic group of integers $\{0,1,\ldots, M-1\}$ modulo $M$. One of them, $V_0(\boldsymbol A)$ with mean $μ$, gives the total number of solutions for a generalised birthday problem, and the other, $W(\boldsymbol A)$ with mean $λ$, gives the number of solutions detected by Wagner's tree based algorithm. We establish two limit theorems. Theorem 2.1 describes an asymptotical behaviour of the ratio $λ/μ$ as $M\to\infty$. Theorem 2.2 suggests Chen-Stein bounds for the total variation distance between Poisson distribution and distributions of $V_{0}$ and $W$.

math.PR↗

arXiv2015

Extinction times for a birth-death process with weak competition

We consider a birth-death process with the birth rates $iλ$ and death rates $iμ+i(i-1)θ$, where $i$ is the current state of the process. A positive competition rate $θ$ is assumed to be small. In the supercritical case when $λ>μ$ this process can be viewed as a demographic model for a population with a high carrying capacity around $λ-μ\overθ$. The article reports in a self-contained manner on the asymptotic properties of the time to extinction for this logistic branching process as $θ\to0$. All three reproduction regimes $λ>μ$, $λ<μ$, and $λ=μ$ are studied.

math.PR↗

arXiv2014

Phylogenetic confidence intervals for the optimal trait value

We consider a stochastic evolutionary model for a phenotype developing amongst n related species with unknown phylogeny. The unknown tree is modelled by a Yule process conditioned on n contemporary nodes. The trait value is assumed to evolve along lineages as an Ornstein-Uhlenbeck process. As a result, the trait values of the n species form a sample with dependent observations. We establish three limit theorems for the sample mean corresponding to three domains for the adaptation rate. In the case of fast adaptation, we show that for large $n$ the normalized sample mean is approximately normally distributed. Using these limit theorems, we develop novel confidence interval formulae for the optimal trait value.

q-bio.PE↗

arXiv2014

Tail generating functions for Markov branching processes

We give a concise self-contained presentation of known and new limit theorems for the one-type Markov branching processes with continuous time. The new streamlined proofs are based on what we call, the tail generating function approach. Our analysis focuses on the singularity points of the master integral equation for the probability generating functions of the current population size.

math.PR↗

arXiv2014

A Consistent Estimator of the Evolutionary Rate

We consider a branching particle system where particles reproduce according to the pure birth Yule process with the birth rate L, conditioned on the observed number of particles to be equal n. Particles are assumed to move independently on the real line according to the Brownian motion with the local variance s2. In this paper we treat $n$ particles as a sample of related species. The spatial Brownian motion of a particle describes the development of a trait value of interest (e.g. log-body-size). We propose an unbiased estimator Rn2 of the evolutionary rate r2=s2/L. The estimator Rn2 is proportional to the sample variance Sn2 computed from n trait values. We find an approximate formula for the standard error of Rn2 based on a neat asymptotic relation for the variance of Sn2.

q-bio.PE↗

arXiv2013

Skeletons of near-critical Bienaymé-Galton-Watson branching processes

Skeletons of branching processes are defined as trees of lineages characterized by an appropriate signature of future reproduction success. In the supercritical case a natural choice is to look for the lineages that survive forever. In the critical case it was earlier suggested to distinguish the particles with the total number of descendants exceeding a certain threshold. These two definitions lead to asymptotic representations of the skeletons as either pure birth process (in the slightly supercritical case) or critical birth-death processes (in the critical case conditioned on the total number of particles exceeding a high threshold value). The limit skeletons reveal typical survival scenarios for the underlying branching processes. In this paper we consider near-critical Bienaymé-Galton-Watson processes and define their skeletons using marking of particles. If marking is rare, such skeletons are approximated by birth and death processes which can be subcritical, critical or supercritical. We obtain the limit skeleton for a sequential mutation model and compute the density distribution function for the time to escape from extinction.

math.PR↗

arXiv2012

A decomposable branching process in a Markovian environment

A population has two types of individuals, each occupying an island. One of those, where individuals of type 1 live, offers a variable environment. Type 2 individuals dwell on the other island, in a constant environment. Only one-way migration (1->2) is possible. We study the asymptotics of the survival probability in critical and subcritical cases.

math.PR↗

arXiv2012

Linear-fractional branching processes with countably many types

We study multi-type Bienaymé-Galton-Watson processes with linear-fractional reproduction laws using various analytical tools like contour process, spinal representation, Perron-Frobenius theorem for countable matrices, renewal theory. For this special class of branching processes with countably many types we present a transparent criterion for $R$-positive recurrence with respect to the type space. This criterion appeals to the Malthusian parameter and the mean age at childbearing of the associated linear-fractional Crump-Mode-Jagers process.

math.PR↗

arXiv2012

Decomposition of supercritical linear-fractional branching processes

It is well known that a supercritical single-type Bienyamé-Galton-Watson process can be viewed as a decomposable branching process formed by two subtypes of particles: those having infinite line of descent and those who have finite number of descendants. In this paper we analyze such a decomposition for the linear-fractional Bienyamé-Galton-Watson processes with countably many types.

math.PR↗

arXiv2012

Time to a single hybridization event in a group of species with unknown ancestral history

We consider a stochastic process for the generation of species which combines a Yule process with a simple model for hybridization between pairs of co-existent species. We assume that the origin of the process, when there was one species, occurred at an unknown time in the past, and we condition the process on producing n species via the Yule process and a single hybridization event. We prove results about the distribution of the time of the hybridization event. In particular we calculate a formula for all moments, and show that under various conditions, the distribution tends to an exponential with rate twice that of the birth rate for the Yule process.

q-bio.PE↗

arXiv2012

Statistical Inference of Allopolyploid Species Networks in the Presence of Incomplete Lineage Sorting

Polyploidy is an important speciation mechanism, particularly in land plants. Allopolyploid species are formed after hybridization between otherwise intersterile parental species. Recent theoretical progress has led to successful implementation of species tree models that take population genetic parameters into account. However, these models have not included allopolyploid hybridization and the special problems imposed when species trees of allopolyploids are inferred. Here, two new models for the statistical inference of the evolutionary history of allopolyploids are evaluated using simulations and demonstrated on two empirical data sets. It is assumed that there has been a single hybridization event between two diploid species resulting in a genomic allotetraploid. The evolutionary history can be represented as a network or as a multiply labeled tree, in which some pairs of tips are labeled with the same species. In one of the models (AlloppMUL), the multiply labeled tree is inferred directly. This is the simplest model and the most widely applicable, since fewer assumptions are made. The second model (AlloppNET) incorporates the hybridization event explicitly which means that fewer parameters need to be estimated. Both models are implemented in the BEAST framework. Simulations show that both models are useful and that AlloppNET is more accurate if the assumptions it is based on are valid. The models are demonstrated on previously analyzed data from the genus Pachycladon (Brassicaceae) and from the genus Silene (Caryophyllaceae).

q-bio.PE↗

arXiv2012

Interspecies correlation for neutrally evolving traits

A simple way to model phenotypic evolution is to assume that after splitting, the trait values of the sister species diverge as independent Brownian motions. Relying only on a prior distribution for the underlying species tree (conditioned on the number, n, of extant species) we study the random vector (X_1,...,X_n) of the observed trait values. In this paper we derive compact formulae for the variance of the sample mean and the mean of the sample variance for the vector (X_1,...,X_n). The key ingredient of these formulae is the correlation coefficient between two trait values randomly chosen from (X_1,...,X_n). This interspecies correlation coefficient takes into account not only variation due to the random sampling of two species out of n and the stochastic nature of Brownian motion but also the uncertainty in the phylogenetic tree. The latter is modeled by a (supercritical or critical) conditioned branching process. In the critical case we modify the Aldous-Popovic model by assuming a proper prior for the time of origin.

q-bio.PE↗

arXiv2011

Survival of branching processes in random environments

This review paper presents the known results on the asymptotics of the survival probability and limit theorems conditioned on survival of critical and subcritical branching processes in IID random environments. The key assumptions of the family of population models in question are: non-overlapping generations, independent reproduction of particles within a generation, independent reproduction laws between generations. This is a biologically important generalization of the time inhomogeneous branching processes. The assumption of IID (independent and identically distributed) random environments reflects uncertainty in the future (as well as historical) reproduction regimes in actual populations. This review focusses on a particular range of questions of prime interest for the authors. The reader should be aware of the fact that there are many very interesting papers covering other issues on branching processes in varying and random environments which are not mentioned here.

math.PR↗

arXiv2010

Coalescent approximation for structured populations in a stationary random environment

We establish convergence to the Kingman coalescent for the genealogy of a geographically - or otherwise - structured version of the Wright-Fisher population model with fast migration. The new feature is that migration probabilities may change in a random fashion. This brings a novel formula for the coalescent effective population size (EPS). We call it a quenched EPS to emphasize the key feature of our model - random environment. The quenched EPS is compared with an annealed (mean-field) EPS which describes the case of constant migration probabilities obtained by averaging the random migration probabilities over possible environments.

math.PR↗

arXiv2008

General branching processes in discrete time as random trees

The simple Galton--Watson process describes populations where individuals live one season and are then replaced by a random number of children. It can also be viewed as a way of generating random trees, each vertex being an individual of the family tree. This viewpoint has led to new insights and a revival of classical theory. We show how a similar reinterpretation can shed new light on the more interesting forms of branching processes that allow repeated bearings and, thus, overlapping generations. In particular, we use the stable pedigree law to give a transparent description of a size-biased version of general branching processes in discrete time. This allows us to analyze the $x\log x$ condition for exponential growth of supercritical general processes as well as relation between simple Galton--Watson and more general branching processes.

math.ST↗

arXiv2007

Reduced branching processes with very heavy tails

The reduced Markov branching process is a stochastic model for the genealogy of an unstructured biological population. Its limit behavior in the critical case is well studied for the Zolotarev-Slack regularity parameter $α\in(0,1]$. We turn to the case of very heavy tailed reproduction distribution $α=0$ assuming Zubkov's regularity condition with parameter $β\in(0,\infty)$. Our main result gives a new asymptotic pattern for the reduced branching process conditioned on non-extinction during a long time interval.

math.PR↗

arXiv2005

The coalescent effective size of age-structured populations

We establish convergence to the Kingman coalescent for a class of age-structured population models with time-constant population size. Time is discrete with unit called a year. Offspring numbers in a year may depend on mother's age.

math.PR↗