Matching random colored points with rectangles

Let S[0,1]2 be a set of n points, randomly and uniformly selected. Let RB be a random partition, or coloring, of S in which each point of S is included in R uniformly at random with probability 1/2. We study the random variable M(n) equal to the number of points of S that are covered by the rectangles of a maximum strong matching of S with axis-aligned rectangles. The matching consists of closed rectangles that cover exactly two points of S of the same color. A matching is strong if all its rectangles are pairwise disjoint. We prove that almost surely M(n)=0.83n for n large enough. Our approach is based on modeling a deterministic greedy matching algorithm, that runs over the random point set, as a Markov chain.Research supported by projects MTM2015-63791-R MINECO/FEDER and Gen. Cat. DGR 2017SGR1640Postprint (author's final draft

Dynamics of a Fleming-Viot type particle system on the cycle graph

International audienceWe study the Fleming-Viot particle process formed by N interacting continuous-time asymmetric random walks on the cycle graph, with uniform killing. We show that this model has a remarkable exact solvability, despite the fact that it is non-reversible with non-explicit invariant distribution. Our main results include quantitative propagation of chaos and exponential ergodicity with explicit constants, as well as formulas for covariances at equilibrium in terms of the Chebyshev polynomials. We also obtain a bound uniform in time for the convergence of the proportion of particles in each state when the number of particles goes to infinity

Uniform in time propagation of chaos for a Moran model

29 pagesThe goal of this article is to study the limit of the empirical distribution induced by a mutation-selection multi-allelic Moran model, whose dynamic is given by a continuous-time irreducible Markov chain. The rate matrix driving the mutation is assumed irreducible and the selection rates are assumed uniformly bounded. The paper is divided into two parts. The first one deals with processes with general selection rates. For this case we are able to prove the propagation of chaos in $\mathbb{L}^p$ over the compacts, with speed of convergence of order $1/\sqrt{N}$. Further on, we consider a specific type of selection that we call additive selection. Essentially, we assume that the selection rate can be decomposed as the sum of three terms: a term depending on the allelic type of the parent (which can be understood as selection at death), another term depending on the allelic type of the descendant (which can be understood as selection at birth) and a third term which is symmetric. Under this setting, our results include a uniform in time bound for the propagation on chaos in $\mathbb{L}^p$ of order $1/\sqrt{N}$, and the proof of the asymptotic normality with zero mean and explicit variance, for the approximation error between the empirical distribution and its limit, when the number of individuals tend towards infinity. Additionally, we explore the interpretation of the Moran model with additive selection as a particle process whose empirical distribution approximates a quasi-stationary distribution, in the same spirit as the Fleming\,--\,Viot particle systems. We then address the problem of minimising the asymptotic quadratic error, when the time and the number of particles go to infinity

Further results on stochastic orderings and aging classes in systems with age replacement

Reliability properties associated to the classic models of systems with age replacement have been a usual topic of research. Most previous works have checked aging properties of the lifetime of the working units using stochastic comparisons between the systems with age replacement at different times. However, from a practical point of view, it would be also interesting to deduce the belonging to aging classes of the lifetime of the system from the aging properties of the lifetime of its working units. The first part of this article deals with this problem. Later, stochastic orderings are established between the two systems with replacement at the same time using several stochastic comparisons among the lifetimes of their working units. In addition, the lifetimes of two systems with age replacement are also compared assuming stochastic orderings between the number of replacement until failure and the lifetimes of their working units conditioned to be less or equal than the replacement time. Similar comparisons are accomplished considering two systems with age replacement where the replacements occur at random time. This last case is very interesting for real-life applications. Illustrative examples are also presented

Exact calculation of the expected SFS in structured populations

The Site Frequency Spectrum (SFS), summary statistic of the distribution of derived allele frequencies in a sample of DNA sequences, provides information about genetic variation and can be used to make population inferences. The exact calculation of the expected SFS in panmictic population has been derived in the Markovian coalescent theory for decades, but its generalization to the structured coalescent is hampered by the almost exponential growth of the states space. We propose here a complete algorithmic procedure, from how to build a suitable state space and sort it, to how to take advantage of the sparsity of the rate matrix and to solve numerically the linear system by an iterative method. The simplest case of the symmetrical n-island is then processed to arrive at a ready-to-use demographic parameters inference framework

Matching random colored points with rectangles

Let S ¿ [0, 1]2 be a set of n points, randomly and uniformly selected. Let R ¿ B be a random partition, or coloring, of S in which each point of S is included in R uniformly at random with probability 1/2. We study the random number M(n) of points of S that are covered by the rectangles of a maximum strong matching of S with axis-aligned rectangles. The matching consists of closed rectangles that cover exactly two points of S of the same color. A matching is strong if all its rectangles are pairwise disjoint. We prove that almost surely M(n) = 0.83 n for n large enough. Our approach is based on modeling a deterministic greedy matching algorithm, that runs over the random point set, as a Markov chain

The IICR and the non-stationary structured coalescent: towards demographic inference with arbitrary changes in population structure

In the last years, a wide range of methods allowing to reconstruct past population size changes from genome-wide data have been developed. At the same time, there has been an increasing recognition that population structure can generate genetic data similar to those produced under models of population size change. Recently, Mazet et al. (Heredity 116:362-371, 2016) showed that, for any model of population structure, it is always possible to find a panmictic model with a particular function of population size changes, having exactly the same distribution of T 2 (the coalescence time for a sample of size two) as that of the structured model. They called this function IICR (Inverse Instantaneous Coalescence Rate) and showed that it does not necessarily correspond to population size changes under non-panmictic models. Besides, most of the methods used to analyse data under models of population structure tend to arbitrarily fix that structure and to minimise or neglect population size changes. Here, we extend the seminal work of Herbots (PhD thesis, University of London, 1994) on the structured coalescent and propose a new framework, the Non-Stationary Structured Coalescent (NSSC) that incorporates demographic events (changes in gene flow and/or deme sizes) to models of nearly any complexity. We show how to compute the IICR under a wide family of stationary and non-stationary models. As an example we address the question of human and Neanderthal evolution and discuss how the NSSC framework allows to interpret genomic data under this new perspective