Statistical method for testing the neutral mutation hypothesis by DNA polymorphism.

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Preliminaries.- Bell polynomials, composite structures and Gibbs partitions.- Exchangeable random partitions.- Sequential constructions of random partitions.- Poisson constructions of random partitions.- Coagulation and fragmentation processes.- Random walks and random forests.- The Brownian forest.- Brownian local times, branching and Bessel processes.- Brownian bridge asymptotics for random mappings.- Random forests and the additive coalescent.

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Combinatorial Stochastic Processes

All animals evaluate the salience of external stimuli and integrate them with internal physiological information into adaptive behavior. Natural and sexual selection impinge on these processes, yet our understanding of behavioral decision-making mechanisms and their evolution is still very limited. Insights from mammals indicate that two neural circuits are of crucial importance in this context: the social behavior network and the mesolimbic reward system. Here we review evidence from neurochemical, tract-tracing, developmental, and functional lesion/stimulation studies that delineates homology relationships for most of the nodes of these two circuits across the five major vertebrate lineages: mammals, birds, reptiles, amphibians, and teleost fish. We provide for the first time a comprehensive comparative analysis of the two neural circuits and conclude that they were already present in early vertebrates. We also propose that these circuits form a larger social decision-making (SDM) network that regulates adaptive behavior. Our synthesis thus provides an important foundation for understanding the evolution of the neural mechanisms underlying reward processing and behavioral regulation. J. Comp. Neurol. 519:3599–3639, 2011.

https://onlinelibrary.wiley.com/doi/pdf/10.1002/cne.22735

The Vertebrate mesolimbic reward system and social behavior network: A comparative synthesis

We consider a class of haploid population models with nonoverlapping generations and fixed population size $N$ assuming that the family sizes within a generation are exchangeable random variables. A weak convergence criterion is established for a properly scaled ancestral process as $N \to \infty$. It results in a full classification of the coalescent generators in the case of exchangeable reproduction. In general the coalescent process allows for simultaneous multiple mergers of ancestral lines.

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A Classification of Coalescent Processes for Haploid Exchangeable Population Models

We determine that the continuous-state branching processes for which the genealogy, suitably time-changed, can be described by an autonomous Markov process are precisely those arising from $\alpha$-stable branching mechanisms.  The random ancestral partition is then a time-changed $\Lambda$-coalescent, where $\Lambda$ is the Beta-distribution with parameters $2-\alpha$ and $\alpha$, and the time change is given by $Z^{1-\alpha}$, where $Z$ is the total population size. For $\alpha = 2$ (Feller's branching diffusion) and $\Lambda = \delta_0$ (Kingman's coalescent), this is in the spirit of (a non-spatial version of) Perkins' Disintegration Theorem.  For $\alpha =1$ and $\Lambda$ the uniform distribution on $[0,1]$, this is the duality discovered by Bertoin & Le Gall (2000) between the norming of Neveu's continuous state branching process and the Bolthausen-Sznitman coalescent. We present two approaches: one, exploiting the `modified lookdown construction', draws heavily on Donnelly & Kurtz (1999); the other is based on direct calculations with generators.

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Alpha-Stable Branching and Beta-Coalescents

A simple convergence theorem for sequences of Markov chains is presented in order to derive new ‘convergence-to-the-coalescent’ results for diploid neutral population models. For the so-called diploid Wright-Fisher model with selfing probability s and mutation rate θ, it is shown that the ancestral structure of n sampled genes can be treated in the framework of an n -coalescent with mutation rate θ := θ(1- s /2), if the population size N is large and if the time is measured in units of (2- s ) N generations.

A convergence theorem for markov chains arising in population genetics and the coalescent with selfing

We present a short probabilistic proof of a weak convergence result for the number of cuts needed to isolate the root of a random recursive tree. The proof is based on a coupling related to a certain random walk.

https://projecteuclid.org/download/pdf_1/euclid.ecp/1465224947

A probabilistic proof of a weak limit law for the number of cuts needed to isolate the root of a random recursive tree

One possible and widely used definition of the duality of Markov processes employs functions H relating one process to another in a certain way. For given processes X and Y the space U of all such functions H, called the duality space of X and Y, is studied in this paper. The algebraic structure of U is closely related to the eigenvalues and eigenvectors of the transition matrices of X and Y. Often as for example in physics (interacting particle systems) and in biology (population genetics models) dual processes arise naturally by looking forwards and backwards in time. In particular, time-reversible Markov processes are self-dual. In this paper, results on the duality space are presented for classes of haploid and two-sex population models. For example dim U = N+ 3 for the classical haploid Wright-Fisher model with fixed population size N.

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Martin Möhle

Papers

A Classification of Coalescent Processes for Haploid Exchangeable Population Models

Alpha-Stable Branching and Beta-Coalescents

A convergence theorem for markov chains arising in population genetics and the coalescent with selfing

A probabilistic proof of a weak limit law for the number of cuts needed to isolate the root of a random recursive tree

The concept of duality and applications to Markov processes arising in neutral population genetics models