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Recent Progress in Coalescent Theory
"... Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such ..."
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Cited by 48 (3 self)
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Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
Gibbs sampling, exponential families and orthogonal polynomials
 Statistical Sciences
, 2008
"... Abstract. We give families of examples where sharp rates of convergence to stationarity of the widely used Gibbs sampler are available. The examples involve standard exponential families and their conjugate priors. In each case, the transition operator is explicitly diagonalizable with classical ort ..."
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Cited by 40 (10 self)
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Abstract. We give families of examples where sharp rates of convergence to stationarity of the widely used Gibbs sampler are available. The examples involve standard exponential families and their conjugate priors. In each case, the transition operator is explicitly diagonalizable with classical orthogonal polynomials as eigenfunctions. Key words and phrases: Gibbs sampler, running time analyses, exponential families, conjugate priors, location families, orthogonal polynomials, singular value decomposition. 1.
The SDE solved by local times of a Brownian excursion or bridge derived from the height profile of a random tree or forest
, 1997
"... Let B be a standard onedimensional Brownian motion started at 0. Let L t;v (jBj) be the occupation density of jBj at level v up to time t. The distribution of the process of local times (L t;v (jBj); v 0) conditionally given B t = 0 and L t;0 (jBj) = ` is shown to be that of the unique strong solu ..."
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Cited by 32 (8 self)
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Let B be a standard onedimensional Brownian motion started at 0. Let L t;v (jBj) be the occupation density of jBj at level v up to time t. The distribution of the process of local times (L t;v (jBj); v 0) conditionally given B t = 0 and L t;0 (jBj) = ` is shown to be that of the unique strong solution X of the Ito SDE dXv = n 4 \Gamma X 2 v \Gamma t \Gamma R v 0 Xudu \Delta \Gamma1 o dv + 2 p XvdBv on the interval [0; V t (X)), where V t (X) := inffv : R v 0 Xudu = tg, and Xv = 0 for all v V t (X). This conditioned form of the RayKnight description of Brownian local times arises from study of the asymptotic distribution as n !1 and 2k= p n ! ` of the height profile of a uniform rooted random forest of k trees labeled by a set of n elements, as obtained by conditioning a uniform random mapping of the set to itself to have k cyclic points. The SDE is the continuous analog of a simple description of a GaltonWatson branching process conditioned on its total progeny....
Urn models, replicator processes, and random genetic drift
 SIAM J. Appl. Math
"... Abstract. To understand the relative importance of natural selection and random genetic drift in finite but growing populations, the asymptotic behavior of a class of generalized Polya urns is studied using the method of ordinary differential equation (ODE). Of particular interest is the replicator ..."
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Abstract. To understand the relative importance of natural selection and random genetic drift in finite but growing populations, the asymptotic behavior of a class of generalized Polya urns is studied using the method of ordinary differential equation (ODE). Of particular interest is the replicator process: two balls (individuals) are chosen from an urn (the population) at random with replacement and balls of the same colors (strategies) are added or removed according to probabilities that depend only on the colors of the chosen balls. Under the assumption that the expected number of balls being added always exceeds the expected number of balls being removed whenever balls are in the urn, the probability of nonextinction is shown to be positive. On the event of nonextinction, three results are proven: (i) the number of balls increases asymptotically at a linear rate, (ii) the distribution x(n) of strategies at the nth update is a “noisy ” Cauchy–Euler approximation to the mean limit ODE of the process, and (iii) the limit set of x(n) is almost surely a connected internally chain recurrent set for the mean limit ODE. Under a stronger set of assumptions, it is shown that for any attractor of the mean limit ODE there is a positive probability that the limit set for x(n) lies in this attractor. Theoretical and numerical estimates for the probabilities of nonextinction and convergence to an attractor suggest that random genetic drift is more likely to overcome natural selection in small populations for which pairwise interactions lead to highly variable outcomes, and is less likely to overcome natural selection in large populations with the potential for rapid growth. Key words. Markov chains, random genetic drift, urn models, replicator equations
Damage segregation at fissioning may increase growth rates : a superprocess model
, 2006
"... A fissioning organism may purge unrepairable damage by bequeathing it preferentially to one of its daughters. Using the mathematical formalism of superprocesses, we propose a flexible class of analytically tractable models that allow quite general effects of damage on death rates and splitting rate ..."
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Cited by 22 (1 self)
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A fissioning organism may purge unrepairable damage by bequeathing it preferentially to one of its daughters. Using the mathematical formalism of superprocesses, we propose a flexible class of analytically tractable models that allow quite general effects of damage on death rates and splitting rates and similarly general damage segregation mechanisms. We show that, in a suitable regime, the effects of randomness in damage segregation at fissioning are indistinguishable from those of randomness in the mechanism of damage accumulation during the organism’s lifetime. Moreover, the optimal population growth is achieved for a particular finite, nonzero level of combined randomness from these two sources. In particular, when damage accumulates deterministically, optimal population growth is achieved by a moderately unequal division of damage between the daughters. Too little or too much division is suboptimal. Connections are drawn both to recent experimental results on inheritance of damage in protozoans, to theories of the evolution of aging, and to models of resource division between siblings.
Stochastic models of evolution in genetics, ecology and linguistics
, 2007
"... Abstract. We give a overview of stochastic models of evolution that have found applications in genetics, ecology and linguistics for an audience of nonspecialists, especially statistical physicists. In particular, we focus mostly on neutral models in which no intrinsic advantage is ascribed to a par ..."
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Cited by 21 (3 self)
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Abstract. We give a overview of stochastic models of evolution that have found applications in genetics, ecology and linguistics for an audience of nonspecialists, especially statistical physicists. In particular, we focus mostly on neutral models in which no intrinsic advantage is ascribed to a particular type of the variable unit, for example a gene, appearing in the theory. In many cases these models are exactly solvable and furthermore go some way to describing observed features of genetic, ecological and linguistic systems. Stochastic Models of Evolution in Genetics, Ecology and Linguistics 2 1.
Inference for Observations of Integrated Diffusion Processes
"... Estimation of parameters in diusion models is usually based on observations of the process at discrete time points. Here we investigate estimation when a sample of discrete observations is not available, but, instead, observations of a running integral of the process with respect to some weight ..."
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Cited by 20 (7 self)
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Estimation of parameters in diusion models is usually based on observations of the process at discrete time points. Here we investigate estimation when a sample of discrete observations is not available, but, instead, observations of a running integral of the process with respect to some weight function. This type of observations is, for example, obtained when a realization of the process is observed after passage through an electronic lter. Another example is provided by the icecore data on oxygen isotopes used to investigate paleotemperatures. Finally, such data play a role in connection with the stochastic volatility models of nance. The integrated process is no longer a Markov process which render the use of martingale estimating functions dicult. Therefore, a generalization of the martingale estimating functions, namely the predictionbased estimating functions, is applied to estimate parameters in the underlying diusion process. The estimators are shown to be consistent and asymptotically normal. The method is applied to inference based on integrated data from OrnsteinUhlenbeck processes and from the CIRmodel for both of which an explicit estimating function can be found.
Evolution of discrete populations and the canonical diffusion of adaptive dynamics
, 2006
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HEIGHT PROCESS FOR SUPERCRITICAL CONTINUOUS STATE BRANCHING PROCESS
, 2006
"... Abstract. We define the height process for supercritical continuous state branching processes with quadratic branching mechanism. It appears as a projective limit of Brownian motions with positive drift reflected at 0 and a> 0 as a goes to infinity. Then we extend the pruning procedure of branch ..."
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Cited by 13 (5 self)
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Abstract. We define the height process for supercritical continuous state branching processes with quadratic branching mechanism. It appears as a projective limit of Brownian motions with positive drift reflected at 0 and a> 0 as a goes to infinity. Then we extend the pruning procedure of branching processes to the supercritical case. This give a complete duality picture between pruning and size proportional immigration for quadratic continuous state branching processes. 1.