S. N. Ethier

Fleming-Viot processes in population genetics

Fleming and Viot [Indiana Univ. Math. J., 28 (1979), pp. 817–843] introduced a class of probability-measure-valued diffusion processes that has attracted the interest of both pure and applied probabilists. This paper surveys the subject of Fleming–Viot processes as it relates to population genetics. Topics include:1. Introduction.2. Some measure-valued Markov chains.2.1. A diploid model. 2.2. The Wright–Fisher model.2.3. A Moran model.3. The Fleming–Viot process: characterization.4. Convergence.5. Ergodicity.6. An infinite particle system.7. Bounded mutation operators.8. Reversibility.9. Examples.9.1. Continuous-state stepwise-mutation model.9.2. Infinitely-many-neutral-alleles model.9.3. Infinitely-many-neutral-alleles model with ages.9.4. Two-locus model with recombination.9.5. n-locus model with gene conversion.9.6. Infinitely-many-sites model without recombination.9.7. Infinitely-many-neutral-alleles model with allelic genealogies.

Convergence to Fleming-Viot processes in the weak atomic topology

Stochastic models for gene frequencies can be viewed as probability-measure-valued processes. Fleming and Viot introduced a class of processes that arise as limits of genetic models as the population size and the number of possible genetic types tend to infinity. In general, the topology on the process values in which these limits exist is the topology of weak convergence; however, convergence in the weak topology is not strong enough for many genetic applications. A new topology on the space of finite measures is introduced in which convergence implies convergence of the sizes and locations of atoms, and conditions are given under which genetic models converge in this topology. As an application, Kingmans Poisson-Dirichlet limit is extended to models with selection.

On the two-locus sampling distribution

Two methods are discussed for evaluating the distribution of the configuration of unlabeled gametic types in a random sample of size n from the two-locus infinitely-many-neutral-alleles diffusion model at stationarity. Both involve finding systems of linear equations satisfied by the desired probabilities. The first approach, which is due to Golding, is to include additional probabilities in the system that allow some members of the sample to be specified at only one locus. The second approach, which is new, considers the joint distribution of the sample configuration and the number of recombination events since the time of the most recent common ancestor. The first approach is used for numerical computation, whereas the second approach is used to derive a two-locus version of Hoppes urn model. The latter permits efficient simulation of the two-locus sampling distribution, provided the recombination parameter is not too large.

THE PROPORTIONAL BETTOR'S RETURN ON INVESTMENT

Suppose you repeatedly play a game of chance in which you have the advantage. Your return on investment is your net gain divided by the total amount that you have bet. It is shown that the ratio of your return on investment under optimal proportional betting to your return on investment under constant betting converges to an exponential distribution with mean 2 as your advantage tends to 0. The case of non-optimal proportional betting is also treated.

A generalized likelihood ratio test to identify differentially expressed genes from microarray data

MOTIVATION Microarray technology emerges as a powerful tool in life science. One major application of microarray technology is to identify differentially expressed genes under various conditions. Currently, the statistical methods to analyze microarray data are generally unsatisfactory, mainly due to the lack of understanding of the distribution and error structure of microarray data. RESULTS We develop a generalized likelihood ratio (GLR) test based on the two-component model proposed by Rocke and Durbin to identify differentially expressed genes from microarray data. Simulation studies show that the GLR test is more powerful than commonly used methods, like the fold-change method and the two-sample t-test. When applied to microarray data, the GLR test identifies more differentially expressed genes than the t-test, has a lower false discovery rate and shows more consistency over independently repeated experiments. AVAILABILITY The approach is implemented in software called GLR, which is freely available for downloading at http://www.cc.utah.edu/~jw27c60

Testing for Favorable Numbers on a Roulette Wheel

Abstract We are forced to accept as [an] alternative that the random spinning of a roulette manufactured and daily readjusted with extraordinary care is not obedient to the laws of chance, but is chaotic in its manifestations! (Karl Pearson 1894) Given integers k ≥ k 0 ≥ 2, a test of fixed sample size and a sequential test are constructed for the purpose of testing the null hypothesis that max1≤i≤k p i ≤ 1/k 0, against the alternative that max1≤i≤k p i > 1/k 0, where p 1, …, pk are the parameters of a multinomial distribution with k cells. Results are applied to the problem of testing for favorable numbers on a roulette wheel, in which case k = 38 and k 0 = 36, thereby providing a partial solution to a problem posed by Wilson (1965).

Diffusion approximations of the two-locus Wright-Fisher model

Diffusion approximations are established for the multiallelic, two-locus Wright-Fisher model for mutation, selection, and random genetic drift in a finite, panmictic, monoecious, diploid population. All four combinations of weak or strong selection and tight or loose linkage are treated, though the proof in the case of strong selection and loose linkage is incomplete. Under certain conditions, explicit formulas are obtained for the stationary distributions of the two diffusions with loose linkage.

The Doctrine of Chances

reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Preface I have found many thousands more readers than I ever looked for. I have no right to say to these, You shall not find fault with my art, or fall asleep over my pages; but I ask you to believe that this person writing strives to tell the truth. If there is not that, there is nothing. This is a monograph/textbook on the probabilistic aspects of gambling, intended for those already familiar with probability at the post-calculus, pre-measure-theory level. Gambling motivated much of the early development of probability theory (David 1962). Gambling also had a major influence on 20th-century probability theory, as it provided the motivation for the concept of a martingale. Thus, gambling has contributed to probability theory. Conversely, probability theory has contributed much to gambling, from the gamblers ruin formula of Blaise Pascal [1623–1662] to the optimality of bold play due to Section 2) for a different point of view. v vi Preface the first evaluation of the bankers advantage at trente et quarante due to Siméon-Denis Poisson [1781–1840] to the first published card-counting system at twenty-one due to Edward O. Thorp [1932–]. Topics such as these are the principal focus of this book. Is gambling a subject worthy of academic study? Let us quote an authority from the 18th century on this question. In the preface to The Doctrine of Chances, De Moivre (1718, p. iii) wrote, Another use to be made of this Doctrine of Chances is, that it may serve in Conjunction with the other parts of the Mathematicks, as a fit introduction to the Art of Reasoning; it being known by experience that nothing can contribute more to the attaining of that Art, than the consideration of a long Train of Consequences, rightly deduced from undoubted Principles, of which this Book affords …

Bounds on gambler's ruin probabilities in terms of moments

Consider a wager that is more complicated than simply winning or losing the amount of the bet. For example, a pass line bet with double odds is such a wager, as is a bet on video poker using a specified drawing strategy. We are concerned with the probability that, in an independent sequence of identical wagers of this type, the gambler loses L or more betting units (i.e., the gambler is “ruined”) before he wins W or more betting units. Using an idea of Markov, Feller established upper and lower bounds on the probability of ruin, bounds that are often very close to each other. However, his formulation depends on finding a positive nontrivial root of the equation φ (ρ )=1, where φ is the probability generating function for the wager in question. Here we give simpler bounds, which rely on the first few moments of the specified wager, thereby making such gamblers ruin probabilities more easily computable.

The neutral two-locus model as a measure-valued diffusion.

The neutral two-locus model in population genetics is reformulated as a measure-valued diffusion process and is shown under certain conditions to have a unique stationary distribution and be weakly ergodic. The limits of the process and its stationary distribution as the recombination parameter tends to infinity are found. Genealogies are incorporated into the model, and it is shown that a random sample of size n from the population at stationarity has a common ancestor.