S. N. Ethier

S. N. Ethier
University of Utah | UOU · Department of Mathematics

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134
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7,534
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Citations since 2017
12 Research Items
1939 Citations
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2017201820192020202120222023050100150200250300350
2017201820192020202120222023050100150200250300350

Publications

Publications (134)
Preprint
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Joseph Bertrand [1822--1900], who is often credited with a model of duopoly that has a unique Nash equilibrium, made another significant contribution to game theory. Specifically, his 1888 analysis of baccarat was the starting point for Borel's investigation of strategic games in the 1920s. In this paper we show, with near certainty, that Bertrand'...
Chapter
Parrondo’s coin-tossing games comprise two games, A A and B B . The result of game A A is determined by the toss of a fair coin. The result of game B B is determined by the toss of a p 0 p_0 -coin if capital is a multiple of r r , and by the toss of a p 1 p_1 -coin otherwise. In either game, the player wins one unit with heads and loses one unit wi...
Preprint
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Consider gambler's ruin with three players, 1, 2, and 3, having initial capitals $A$, $B$, and $C$. At each round a pair of players is chosen (uniformly at random) and a fair coin flip is made resulting in the transfer of one unit between these two players. Eventually, one of the players is eliminated and the game continues with the remaining two....
Article
Courses on the mathematics of gambling have been offered by a number of colleges and universities, and for a number of reasons. In the past 15 years, at least seven potential textbooks for such a course have been published. In this article we objectively compare these books for their probability content, their gambling content, and their mathematic...
Preprint
Parrondo's coin-tossing games comprise two games, $A$ and $B$. The result of game $A$ is determined by the toss of a fair coin. The result of game $B$ is determined by the toss of a $p_0$-coin if capital is a multiple of $r$, and by the toss of a $p_1$-coin otherwise. In either game, the player wins one unit with heads and loses one unit with tails...
Article
Parrondo’s coin-tossing games were introduced as a toy model of the flashing Brownian ratchet in statistical physics but have emerged as a paradigm for a much broader phenomenon that occurs if there is a reversal in direction in some system parameter when two similar dynamics are combined. Our focus here, however, is on the original Parrondo games,...
Preprint
Full-text available
Courses on the mathematics of gambling have been offered by a number of colleges and universities, and for a number of reasons. In the past 15 years, at least seven potential textbooks for such a course have been published. In this article we objectively compare these books for their probability content, their gambling content, and their mathematic...
Preprint
Full-text available
Snackjack is a highly simplified version of blackjack that was proposed by Ethier (2010) and given its name by Epstein (2013). The eight-card deck comprises two aces, two deuces, and four treys, with aces having value either 1 or 4, and deuces and treys having values 2 and 3, respectively. The target total is 7 (vs. 21 in blackjack), and ace-trey i...
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If the parameters of the original Parrondo games $A$ and $B$ are allowed to be arbitrary, subject to a fairness constraint, and if the two (fair) games $A$ and $B$ are played in an arbitrary periodic sequence, then the rate of profit can not only be positive, it can be arbitrarily close to 1 (i.e., 100%).
Article
The flashing Brownian ratchet is a stochastic process that alternates between two regimes, a one-dimensional Brownian motion and a Brownian ratchet, the latter being a one-dimensional diffusion process that drifts towards a minimum of a periodic asymmetric sawtooth potential. The result is directed motion. In the presence of a static homogeneous fo...
Preprint
Full-text available
The flashing Brownian ratchet is a stochastic process that alternates between two regimes, a one-dimensional Brownian motion and a Brownian ratchet, the latter being a one-dimensional diffusion process that drifts towards a minimum of a periodic asymmetric sawtooth potential. The result is directed motion. In the presence of a static homogeneous fo...
Article
Full-text available
A Brownian ratchet is a one-dimensional diffusion process that drifts toward a minimum of a periodic asymmetric sawtooth potential. A flashing Brownian ratchet is a process that alternates between two regimes, a one-dimensional Brownian motion and a Brownian ratchet, producing directed motion. These processes have been of interest to physicists and...
Article
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There are 134,459 distinct initial hands at the video poker game Jacks or Better, taking suit exchangeability into account. A computer program can determine the optimal strategy (i.e., which cards to hold) for each such hand, but a complete list of these strategies would require a book-length manuscript. Instead, a hand-rank table, which fits on a...
Article
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The two-parameter Poisson-Dirichlet diffusion, recently introduced by Petrov, extends the infinitely-many-neutral-alleles diffusion model, related to Kingman's one-parameter Poisson-Dirichlet distribution and to certain Fleming-Viot processes. The additional parameter has been shown to regulate the clustering structure of the population, but is yet...
Article
Parrondo games with one-dimensional spatial dependence were introduced by Toral and extended to the two-dimensional setting by Mihailovi\'c and Rajkovi\'c. $MN$ players are arranged in an $M\times N$ array. There are three games, the fair, spatially independent game $A$, the spatially dependent game $B$, and game $C$, which is a random mixture or n...
Article
The game of baccarat has evolved from a parlor game played by French aristocrats in the first half of the 19th century to a casino game that generated over US$41 billion in revenue for the casinos of Macau in 2013. The parlor game was originally a three-person zero-sum game. Later in the 19th century it was simplified to a two-person zero-sum game....
Article
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Baccara banque is a three-person zero-sum game parameterized by $\theta\in(0,1)$. A study of the game by Downton and Lockwood claimed that the Nash equilibrium is of only academic interest. Their preferred alternative is what we call the independent cooperative equilibrium. But this solution exists only for certain $\theta$. A third solution, which...
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Petrov constructed a diffusion process in the Kingman simplex whose unique stationary distribution is the two-parameter Poisson-Dirichlet distribution of Pitman and Yor. We show that the subset of the simplex comprising vectors whose coordinates do not sum to 1 acts like an entrance boundary for the diffusion.
Article
We prove some combinatorial identities using the Polya urn and the closely related Hoppe urn.
Article
The casino game of baccara chemin de fer is a bimatrix game, not a matrix game, because the house collects a five percent commission on Banker wins. We generalize the game, allowing Banker's strategy to be unconstrained and assuming a 100alpha percent commission on Banker wins, where 0<=alpha<2/5. Assuming for simplicity that cards are dealt with r...
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Assuming that cards are dealt with replacement from a single deck and that each of Player and Banker sees the total of his own two-card hand but not its composition, baccara is a 2 x 2^88 matrix game, which was solved by Kemeny and Snell in 1957. Assuming that cards are dealt without replacement from a d-deck shoe and that Banker sees the compositi...
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A formula for the number of toroidal m x n binary arrays, allowing rotation of the rows and/or the columns but not reflection, is known. Here we find a formula for the number of toroidal m x n binary arrays, allowing rotation and/or reflection of the rows and/or the columns.
Article
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Let game B be Toral's cooperative Parrondo game with (one-dimensional) spatial dependence, parameterized by N (3 or more) and p_0,p_1,p_2,p_3 in [0,1], and let game A be the special case p_0=p_1=p_2=p_3=1/2. Let mu_B (resp., mu_(1/2,1/2)) denote the mean profit per turn to the ensemble of N players always playing game B (resp., always playing the r...
Article
Full-text available
Toral introduced so-called cooperative Parrondo games, in which there are N >= 3 players arranged in a circle. At each turn one player is randomly chosen to play. He plays either game A or game B. Game A results in a win or loss of one unit based on the toss of a fair coin. Game B results in a win or loss of one unit based on the toss of a biased c...
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Toral (2002) considered an ensemble of N\geq2 players. In game B a player is randomly selected to play Parrondo's original capital-dependent game. In game A' two players are randomly selected without replacement, and the first transfers one unit of capital to the second. Game A' is fair (with respect to total capital), game B is losing (or fair), a...
Article
Full-text available
We consider a collective version of Parrondo's games with probabilities parametrized by rho in (0,1) in which a fraction phi in (0,1] of an infinite number of players collectively choose and individually play at each turn the game that yields the maximum average profit at that turn. Dinis and Parrondo (2003) and Van den Broeck and Cleuren (2004) st...
Article
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The antique Mills Futurity slot machine has two unusual features. First, if a player loses 10 times in a row, the 10 lost coins are returned. Second, the payout distribution varies from coup to coup in a manner that is nonrandom and periodic with period 10. It follows that the machine is driven by a 100-state irreducible period-10 Markov chain. Her...
Chapter
Video poker is an electronic form of five-card draw poker that dates back to the late 1970s. The player is dealt five cards and is allowed to replace any number of them by an equal number of cards drawn from the unseen deck. The rank of the resulting hand (and the bet size) determines the amount paid out to the player. We focus on two specific vide...
Chapter
In Section8.1 we describe and compare six well-known betting systems, namely the martingale, Fibonacci, Labouchere, Oscar, d’Alembert, and Blundell systems. Assuming a house maximum betting limit, each of these systems provides the gambler with a small win with high probability or a large loss with low probability. In Section8.2 we show, under cert...
Chapter
The reader is assumed to be familiar with basic probability, and here we provide the definitions and theorems, without proofs, for easy reference. We restrict our attention to discrete random variables but not necessarily to discrete sample spaces. A number of examples are worked out in detail, and problems are provided for those who need additiona...
Chapter
This chapter is concerned with aspects of card games that are common to a number of games. Section11.1 treats shuffling; in particular, we justify the well-known result that seven riffle shuffles are both necessary and sufficient to adequately mix a deck of 52 distinct cards. Section11.2 deals with dealing and the concept of exchangeability, and Se...
Article
Trente et quarante (“thirty and forty” in French) is an elegant card game that dates at least as far back as the 17th century and is still played in Monte Carlo. It has been studied by Poisson, De Morgan, Bertrand, and Thorp, among others. In Section 20.1 we describe the four betting opportunities and evaluate the associated probabilities and house...
Article
Craps is a 19th-century American simplification of hazard, a dice game thought to be of Arab origin that dates back to the Crusades. Mathematically, the most interesting feature of craps is that the principal bets require a random number of rolls for resolution. Section 15.1 examines these wagers, the so-called line bets, together with certain asso...
Article
Roulette, the quintessential casino game for system players, is of French origin and dates back to about 1796. In Section 13.1 we discuss the many wagers available at roulette, observing that the game is unbeatable when the wheel is unbiased. Section 13.2 is concerned with biased wheels, and how to identify and exploit them. In particular, we descr...
Article
Since 1990 a number of new casino games have been introduced, and here we consider two of the most successful such games, Let It Ride and Three Card Poker (Sections 16.1 and 16.2, respectively). Each is house banked and poker based, and each offers a greater advantage to the house than do the traditional games of craps, faro, baccarat, trente et qu...
Article
This chapter is concerned with a betting system for superfair games, known as optimal proportional play or the Kelly system, that maximizes the long-term geometric rate of growth of the gambler’s fortune. Section 10.1 considers the case in which a single superfair betting opportunity is available at each coup. Section 10.2 assumes that multiple bet...
Article
A gambler with a fixed goal is said to use bold play if at each coup he bets his entire fortune or just enough to achieve his goal in the event of a win, whichever amount is smaller. Section 9.1 proves that bold play is optimal at subfair red-and-black, that is, at games of independent coups in which each coup is won, and paid even money, with prob...
Article
This chapter is concerned with finding the probability that, in an independent sequence of identical wagers, the gambler loses L or more units (that is, he is ruined) before he wins W or more units. In Section 7.1, we treat the case of even-money payoffs by deriving an explicit formula. In Section 7.2, we consider more-general integer-valued payoff...
Article
In Section 6.1 we define the notion of the house advantage of a wager, a numerical index of its unfavorability. The house advantage is the ratio of the gambler’s expected loss to his expected amount bet. Actually, there are at least three aspects of the definition that are arguable, the first concerning how pushes are accounted for, the second conc...
Article
Twenty-one, or blackjack, first appeared in 18th-century France and is today’s most popular casino table game. Its current level of popularity dates back only to the 1960s, when Thorp published Beat the Dealer: A Winning Strategy for the Game of Twenty-One. In Section 21.1 we specify the set of rules assumed, and we note the lack of symmetry of the...
Chapter
A martingale is a sequence of discrete random variables indexed by a time parameter with the property that the conditional expectation of a future term given the past and present terms is the present term. It can be thought of as a stochastic model for the gambler’s fortune in a fair game. In Section 3.1 we formalize and generalize this definition...
Chapter
Game theory is concerned with games of strategy, which may or may not be games of chance. In Section 5.1 we introduce matrix games, with emphasis on the games of le her and chemin de fer, and in Section 5.2 we prove the fundamental theorem concerning such games, the minimax theorem. Few house-banked casino games fit into this framework because typi...
Chapter
The three-reel slot machine, invented in San Francisco in 1898, underwent substantial evolution over the course of the 20th century. For example, while classical slots were purely mechanical or electro-mechanical, modern ones are electronic and controlled by microprocessors with random-number generators. Slot machines are the only casino games for...
Chapter
Baccarat is a 20th-century Argentinian descendant of the elegant card games chemin de fer (Example 5.1.4 on p. 167) and baccara en banque, which are of French origin and date back to the 19th century. Unlike its ancestors, baccarat offers no discretionary strategy decisions. This makes the game easier to analyze but less interesting mathematically...
Chapter
Conditional expectations are expectations with respect to conditional probabilities. The reader may already have some familiarity with this topic, but here we go into greater detail than is typical in an introductory probability course. This material is essential for much of what follows.
Chapter
A Markov chain is a sequence of discrete random variables indexed by a time parameter with the property that the conditional probability of a future event, given the present state and the past history, does not depend on the past history. Section 4.1 presents several examples, and Section 4.2 introduces the notions of transience and recurrence and...
Article
Full-text available
It was widely reported in the media that, on 23 May 2009, at the Borgata Hotel Casino & Spa in Atlantic City, Patricia DeMauro, playing craps for only the second time, rolled the dice for four hours and 18 minutes, finally sevening out at the 154th roll, a world record. Initial estimates of the probability of this event were erroneous, but consensu...
Article
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That there exist two losing games that can be combined, either by random mixture or by nonrandom alternation, to form a winning game is known as Parrondo's paradox. We establish a strong law of large numbers and a central limit theorem for the Parrondo player's sequence of profits, both in a one-parameter family of capital-dependent games and in a...
Article
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The duration of play formulae of De Moivre are generalized, using restricted Pascal triangles, to games in which the gambler wins μ units or loses ν units at each play, where μ and ν are positive integers. We treat the cases of both one and two barriers.
Chapter
Nondegenerate Diffusions Degenerate Diffusions Other Processes Problems Notes
Chapter
The Prohorov Metric Prohorov's Theorem Weak Convergence Separating and Convergence Determining Sets The Space DE[0, ∞), The Compact Sets of DE[0, ∞) Convergence in Distribution in DE[0, ∞) Criteria for Relative Compactness in DE[0, ∞) Further Criteria for Relative Compactness in DE[0, ∞) Convergence to a Process in EE[0, ∞) Problems Notes
Chapter
Galton-Watson Processes Two-Type Markov Branching Processes Branching Processes in Random Environments Branching Markov Processes Problems Notes
Chapter
The Martingale Central Limit Theorem Measures of Mixing Central Limit Theorems for Stationary Sequences Diffusion Approximations Strong Approximation Theorems Problems Notes
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Brownian Motion Stochastic Integrals Stochastic Integral Equations Problems Notes
Chapter
Markov Processes and Transition Functions Markov Jump Processes and Feller Processes The Martingale Problem: Generalities and Sample Path Properties The Martingale Problem: Uniqueness, the Markov Property, and Duality The Martingale Problem: Existence The Martingale Problem: Localization The Martingale Problem: Generalizations Convergence Theorems...
Chapter
Introduction Driving Process in a Compact State Space Driving Process in a Noncompact State Space Non-Markovian Driving Process Problems Notes
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Title Copyright Preface Table of Contents
Chapter
Stochastic Processes Martingales Local Martingales The Projection Theorem The Doob-Meyer Decomposition Square Integrable Martingales Semigroups of Conditioned Shifts Martingales Indexed by Directed Sets Problems Notes
Chapter
Examples Law of Large Numbers and Central Limit Theorem Diffusion Approximations Hitting Distributions Problems Notes
Chapter
Definitions and Basic Properties The Hille-Yosida Theorem Cores Multivalued Operators Semigroups on Function Spaces Approximation Theorems Perturbation Theorems Problems Notes
Chapter
One-Parameter Random Time Changes Multiparameter Random Time Changes Convergence Markov Processes Diffusion Processes Problems Notes
Article
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Consider the random walk on the set of nonnegative integers that takes two steps to the left (just one step from state 1) with probability p∈[1/3,1) and one step to the right with probability 1−p. State 0 is absorbing and the initial state is a fixed positive integer j0. Here we find the distribution of the absorption time. The absorption time is t...
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A simplified proof of Thorp and Walden's fundamental theorem of card counting is presented, and a corresponding central limit theorem is established. Results are applied to the casino game of trente et quarante, which was studied by Poisson and De Morgan.
Article
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A simplified proof of Thorp and Walden's fundamental theorem of card counting is presented, and a corresponding central limit theorem is established. Results are applied to the casino game of trente et quarante, which was studied by Poisson and De Morgan.
Article
It is well known that the Kelly system of proportional betting, which maximizes the long-term geometric rate of growth of the gambler's fortune, minimizes the expected time required to reach a specified goal. Less well known is the fact that it maximizes the median of the gambler's fortune. This was pointed out by the author in a 1988 paper, but on...
Article
It is well known that the Kelly system of proportional betting, which maximizes the long-term geometric rate of growth of the gambler's fortune, minimizes the expected time required to reach a specified goal. Less well known is the fact that it maximizes the median of the gambler's fortune. This was pointed out by the author in a 1988 paper, but on...
Article
Full-text available
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 str...
Article
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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 m...
Article
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Tachida (1991) proposed a discrete-time model of nearly neutral mutation in which the selection coefficient of a new mutant has a fixed normal distribution with mean 0. The usual diffusion approximation leads to a probability-measure-valued diffusion process, known as a Fleming-Viot process, with the unusual feature of an unbounded selection intens...
Article
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Using duality, an expansion is found for the transition function of the reversible $K$-allele diffusion model in population genetics. In the neutral case, the expansion is explicit but already known. When selection is present, it depends on the distribution at time $t$ of a specified $K$-type birth-and-death process starting at “infinity.” The latt...
Article
In a previous paper the authors studied a Fleming-Viot process with house-of-cards mutation and an unbounded haploid selection intensity function. Results included existence and uniqueness of solutions of an appropriate martingale problem, existence, uniqueness, and reversibility of stationary distributions, and a weak limit theorem for a correspon...
Article
It is argued that the ‘Contrebanque de Noirbourg’ episode in Thackeray’s 1850 Christmas book, ‘The Kickleburys on the Rhine’, was based on an actual event, and that the ‘infallible system for playing rouge et noir’ was in fact le montant belge, or the Belgian progression. This gambling system can be modeled by a three-dimensional Markov chain, whic...
Article
The definition of a (discrete-time) martingale is generalized, allowing the process to be nonadapted, and it is shown that the optional stopping theorem still holds. This result was motivated by and is applied to the study of gambling systems, especially at games such as craps in which bets are not immediately resolved.
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Fleming-Viot processes are probability-measure-valued diffusion processes that can be used as stochastic models in population genetics. Here we use duality methods to prove ergodic theorems for Fleming-Viot processes, including those with recombination. Coupling methods are also used to establish ergodicity of Fleming-Viot processes, first without...
Article
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"Oscar's system" is a gambling system in which the aim is to win one betting unit, at least with high probability, and then start over again. The system can be modeled by an irreducible Markov chain in a subset of the two-dimensional integer lattice. We show that the Markov chain, which depends on a parameter p representing the single-trial win pro...
Article
Fleming-Viot processes and Dawson-Watanabe processes are two classes of "superprocesses" that have received a great deal of attention in recent years. These processes have many properties in common. In this paper, we prove a result that helps to explain why this is so. It allows one to prove certain theorems for one class when they are true for the...
Article
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 topo...
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Let $S$ be a compact metric space, let $\theta \geq 0$, and let $\nu_0$ be a Borel probability measure on $S$. An explicit formula is found for the transition function of the Fleming-Viot process with type space $S$ and mutation operator $(Af)(x) = (1/2)\theta\int_S(f(\xi) - f(x))\nu_0(d\xi)$.
Article
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...
Article
Let S be a compact metric space, let θ ≥ 0, let ν 0 be a Borel probability measure on S, and let λ be real. An explicit formula is found for the transition function of the measure-valued branching diffusion with type space S, immigration intensity θ/2, immigrant-type distribution ν 0 , and criticality parameter λ. If λ > 0, the formula shows that t...
Article
The complete set of eigenvalues is found for the (unlabeled) infinitely-many-neutral-alleles diffusion model. The transition density for the process, originally derived by Griffiths, is rederived as an eigenfunction expansion.
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The complete set of eigenvalues is found for the (unlabeled) infinitely-many-neutral-alleles diffusion model. The transition density for the process, originally derived by Griffiths, is rederived as an eigenfunction expansion.
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There are many situations in which grain distributions resulting from in situ hybridization of radioactively labeled probes to unique genes should be subjected to a statistical analysis. However, the problems posed by analysis of in situ hybridization data are not straightforward, and no completely satisfying method is currently available. We have...
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It is shown that the two descriptions of the ages of alleles corresponding to the two formulations of the stationary infinitely-many-neutral-alleles diffusion model discussed by Ethier (1990a) are equivalent.
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It is shown that the two descriptions of the ages of alleles corresponding to the two formulations of the stationary infinitely-many-neutral-alleles diffusion model discussed by Ethier (1990a) are equivalent.
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Let $S$ be a compact metric space, let $\theta > 0$, and let $P(x,dy)$ be a one-step Feller transition function on $S \times \mathscr{B}(S)$ corresponding to a weakly ergodic Markov chain in $S$ with unique stationary distribution $\nu_0$. The neutral diffusion model, or Fleming-Viot process, with type space $S$, mutation intensity $\frac{1}{2}\the...
Article
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 incorpora...
Article
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 incorpora...
Article
We prove that the frequencies of the oldest, second-oldest, third- oldest,... alleles in the stationary infinitely-many-neutral-alleles diffusion model are distributed as X 1 ,(1-X 1 )X 2 ,(1-X 1 )(1-X 2 )X 3 ,···, where X 1 ,X 2 ,X 3 ,··. are independent beta (1,θ) random variables, θ being twice the mutation intensity; that is, the frequencies of...
Article
We prove that the frequencies of the oldest, second-oldest, third-oldest, … alleles in the stationary infinitely-many-neutral-alleles diffusion model are distributed as X 1 , (1 − X 1 ) X 2 , (1 − X 1 )(1 − X 2 ) X 3 , …, where X 1 , X 2, X 3 , … are independent beta (1, θ ) random variables, θ being twice the mutation intensity; that is, the frequ...
Article
We discuss two formulations of the infinitely-many-neutral-alleles diffusion model that can be used to study the ages of alleles. The first one, which was introduced elsewhere, assumes values in the set of probability distributions on the set of alleles, and the ages of the alleles can be inferred from its sample paths. We illustrate this approach...
Article
We discuss two formulations of the infinitely-many-neutral-alleles diffusion model that can be used to study the ages of alleles. The first one, which was introduced elsewhere, assumes values in the set of probability distributions on the set of alleles, and the ages of the alleles can be inferred from its sample paths. We illustrate this approach...
Article
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 Goldin...

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