We study a family of coalescent processes that undergo ``simultaneous multiple collisions,'' meaning many clusters particles can merge into single cluster at one time, and such mergers occur simultaneously. This processes, which we obtain from simple assumptions about the rates different types mergers, essentially coincides with Mohle Sagitov as limit scaled ancestral in population model exchangeable sizes. characterize possible merger terms measure, show how these coalescents be constructed...

We consider a system of particles which perform branching Brownian motion with negative drift and are killed upon reaching zero, in the near-critical regime where total population stays roughly constant approximately $N$ particles. show that characteristic time scale for evolution this is order $(\log N)^{3}$, sense when measured these units, scaled number converges to variant Neveu's continuous-state process. Furthermore, genealogy then governed by coalescent process known as...

Standard methods for computing prediction intervals in nonlinear regression can be effectively applied to neural networks when the number of training points is large. Simulations show, however, that these generate unreliable on smaller datasets network trained convergence. Stopping algorithm prior convergence, avoid overfitting, reduces effective parameters but lead are too wide. We present an alternative approach estimating using weight decay fit and show via a simulation study this method...

We determine that the continuous-state branching processes for which genealogy, suitably time-changed, can be described by an autonomous Markov process are precisely those arising from $\alpha$-stable mechanisms. The random ancestral partition is then a time-changed $\Lambda$-coalescent, where $\Lambda$ Beta-distribution with parameters $2-\alpha$ and $\alpha$, time change given $Z^{1-\alpha}$, $Z$ total population size. For $\alpha = 2$ (Feller's diffusion) $\Lambda \delta_0$ (Kingman's...

Coalescents with multiple collisions, also known as Λ-coalescents, were introduced by Pitman and Sagitov in 1999. These processes describe the evolution of particles that undergo stochastic coagulation such a way several blocks can merge at same time to form single block. In case measure Λ is Beta (2−α, α) distribution, they are genealogies large populations where individual produce number offspring. Here, we use recent result Birkner et al. prove Beta-coalescents be embedded continuous...

Let $\Pi_{\infty}$ be the standard $\Lambda$-coalescent of Pitman, which is defined so that $\Pi_{\infty}(0)$ partition positive integers into singletons, and, if $\Pi_n$ denotes restriction to $\{ 1,\ldots, n \}$, then whenever $\Pi_n(t)$ has $b$ blocks, each $k$-tuple blocks merging form a single block at rate $\lambda_{b,k}$, where $\lambda_{b,k} = \int_0^1 x^{k-2} (1-x)^{b-k} \Lambda(dx)$ for some finite measure $\Lambda$. We give necessary and sufficient condition ``come down from...

We consider a model of population fixed size $N$ undergoing selection. Each individual acquires beneficial mutations at rate $\mu _N$, and each mutation increases the individual’s fitness by $s_N$. dies one, when death occurs, an is chosen with probability proportional to give birth. Under certain conditions on parameters _N$ $s_N$, we show that genealogy can be described Bolthausen-Sznitman coalescent. This result confirms predictions Desai, Walczak, Fisher (2013), Neher Hallatschek (2013).

We consider the population genetics problem: how long does it take before some member of has $m$ specified mutations? The case $m=2$ is relevant to onset cancer due inactivation both copies a tumor suppressor gene. Models for larger are needed colon and other diseases where sequence mutations leads cells with uncontrolled growth.

When a beneficial mutation occurs in population, the new, favored allele may spread to entire population. This process is known as selective sweep. Suppose we sample n individuals at end of If focus on site chromosome that close location mutation, then many lineages will likely be descended from individual had while others different because recombination between two sites. We introduce approximations for effect The first one simple but not very accurate: flip independent coins with...

We consider a model of population fixed size $N$ in which each individual gets replaced at rate one and experiences mutation $\mu$. calculate the asymptotic distribution time that it takes before there is an with $m$ mutations. Several different behaviors are possible, depending on how $\mu$ changes $N$. These results have applications to problem determining waiting for regulatory sequences appear models cancer development.

Consider a population of fixed size that evolves over time. At each time, the genealogical structure can be described by coalescent tree whose branches are traced back to most recent common ancestor population. As time goes forward, genealogy evolves, leading what is known as an evolving coalescent. We will study for populations Bolthausen Sznitman obtain limiting behavior evolution and total length in tree. By similar methods, we also new result concerning number blocks Bolthausen-Sznitman

In mathematical population genetics, it is well known that one can represent the genealogy of a by tree, which indicates how ancestral lines individuals in coalesce as they are traced back time. As evolves over time, tree represents also changes, leading to tree-valued stochastic process evolving coalescent. Here we will consider coalescent for populations whose be described beta coalescent, give with very large family sizes. We show size tends infinity, evolution certain functionals such...

We consider a model of population fixed size $N$ undergoing selection. Each individual acquires beneficial mutations at rate $\mu _N$, and each mutation increases the individual’s fitness by $s_N$. dies one, when death occurs, an is chosen with probability proportional to give birth. Under certain conditions on parameters _N$ $s_N$, we obtain rigorous results for which accumulate in distribution fitnesses individuals given time. Our confirm predictions Desai Fisher (2007).

We consider one-dimensional branching Brownian motion in which particles are absorbed at the origin. assume that when a particle branches, offspring distribution is supercritical, but given critical drift towards origin so process eventually goes extinct with probability one. establish precise asymptotics for survives large time t, building on previous results by Kesten (1978) and Berestycki, Schweinsberg (2014). also prove Yaglom-type limit theorem behavior of conditioned to survive an...

While evolutionary approaches to medicine show promise, measuring evolution itself is difficult due experimental constraints and the dynamic nature of body systems. In cancer evolution, continuous observation clonal architecture impossible, longitudinal samples from multiple timepoints are rare. Increasingly available DNA sequencing datasets at single-cell resolution enable reconstruction past using mutational history, allowing for a better understanding dynamics prior detectable disease....

A cladogram is an unrooted tree with labeled leaves and unlabeled internal branchpoints of degree 3. Aldous has studied a Markov chain on the set n-leaf cladograms in which each transition consists removing random leaf its incident edge from then reattaching to remaining tree. Using coupling methods, showed that relaxation time (i.e., inverse spectral gap) for this O(n3). Here, we use method based distinguished paths prove O(n2) bound time, establishing conjecture Aldous.

We consider a model of population fixed size N in which each individual gets replaced at rate one and experiences mutation μ. calculate the asymptotic distribution time that it takes before there is an with m mutations. Several different behaviors are possible, depending on how μchanges N. These results have applications to problem determining waiting for regulatory sequences appear models cancer development.