An asymptotic sampling formula for the coalescent with Recombination

Large deviation principles are established for some random\ncombinatorial structures including the Ewens sampling formula and the Pitman\nsampling formula. A path-level large deviation principle is established for the\nformer on the cadlag space D$(o, 1], R)$ equipped with the uniform convergence\ntopology, and the rate function is the same as for a Poisson process justifying\nthe Poisson process approximation for the Ewens sampling formula at the large\ndeviation level. A large deviation principle for the total number of parts in a\npartition is obtained for the Pitman formula; here the rate function depends\nonly on one of the two parameters which display the different roles of the two\nparameters at different scales. In addition to these large deviation results,\nwe also provide an embedding scheme which gives the Pitman sampling formula. A\nproduct of this embedding is an intuitive alternate proof of a result of Pitman\non the limiting total number of parts.

Recurrence equations for the number of types and the frequency of each type in a random sample drawn from a finite population undergoing discrete, nonoverlapping generations and reproducing according to the Cannings exchangeable model are deduced under the assumption of a mutation scheme with infinitely many types. The case of overlapping generations in discrete time is also considered. The equations are developed for the Wright-Fisher model and the Moran model, and extended to the case of the limit coalescent with nonrecurrent mutation as the population size goes to ∞ and the mutation rate to 0. Computations of the total variation distance for the distribution of the number of types in the sample suggest that the exact Moran model provides a better approximation for the sampling formula under the exact Wright-Fisher model than the Ewens sampling formula in the limit of the Kingman coalescent with nonrecurrent mutation. On the other hand, this model seems to provide a good approximation for a Λ-coalescent with nonrecurrent mutation as long as the probability of multiple mergers and the mutation rate are small enough.

Since the highlights of our professional life are what makes it interesting and rewarding, I have decided to describe in this account the events surrounding the derivation of the so-called “Ewens sampling formula” of population genetics, which has formed a high point in my career as an applied probabilist. Before doing this I will briefly describe my background and early training in population genetics theory.

In this paper, we establish an upper bound for time to convergence to stationarity for the discrete time infinite alleles Moran model. If M is the population size and mu is the mutation rate, this bound gives a cutoff time of log(Mmu)/mu generations. The stationary distribution for this process in the case of sampling without replacement is the Ewens sampling formula. We show that the bound for the total variation distance from the generation t distribution to the Ewens sampling formula is well approximated by one of the extreme value distributions, namely, a standard Gumbel distribution. Beginning with the card shuffling examples of Aldous and Diaconis and extending the ideas of Donnelly and Rodrigues for the two allele model, this model adds to the list of Markov chains that show evidence for the cutoff phenomenon. Because of the broad use of infinite alleles models, this cutoff sets the time scale of applicability for statistical tests based on the Ewens sampling formula and other tests of neutrality in a number of population genetic studies.

The Ewens sampling formula in population genetics can be viewed as a probability measure on the group of permutations of a finite set of integers. Functional limit theory for processes defined through partial sums of dependent variables with respect to the Ewens sampling formula is developed. Techniques from probabilistic number theory are used to establish necessary and sufficient conditions for weak convergence of the associated dependent process to a process with independent increments. Not many results on the necessity part are known in the literature.

Ordered and unordered random partitions of an integer and the GEM distribution

An error bound for convergence to the Ewens sampling formula is given where the population size or mutation rate may vary from generation to generation, or the population is not yet at equilibrium. An application is given to a model of Hartl and Campbell about selectively-equivalent subtypes within a class of deleterious alleles, and a theorem is proven showing that the size of the deleterious class stays within bounds sufficient to apply the first result. Generalizations are discussed.

The merits of neutral theory

The adaptive weighted norm extrapolation (AWNE) procedure of Cabrera and Parks (1991) is generalized to non-uniform one-dimensional (1-D) and two-dimensional (2-D) sampling configurations. This extrapolation procedure which is based on iterative modification of the frequency weighting function in minimum weighted norm extrapolation produces high frequency-domain resolution. The approach presented is derived for totally arbitrary sampling configurations to include those that appear in inverse synthetic aperture radar (ISAR) imaging of a rotating object. The resulting procedure produces estimates of signal samples on rectangular coordinates in the neighborhood of the given samples (interpolation) as well as beyond (extrapolation). When applied to SAR/ISAR imaging, the procedure produces a complex reflectivity image in the slant-range cross-range domain from the samples of the object's Fourier transform, available at non-cartesian coordinate locations. Examples illustrate the performance of this interpolation/extrapolation procedure using samples of l-D and 2-D sums of complex sinusoids to simulate range profiles and images of a few point scatterers. >

We consider a Levy-driven tandem queue with an intermediate input assuming that its buffer content process obtained by a reflection mapping has the stationary distribution. For this queue, no closed form formula is known, not only for its distribution but also for the corresponding transform. In this paper, we consider only light-tailed inputs. For the Brownian input case, we derive exact tail asymptotics for the marginal stationary distribution of the second buffer content, while weaker asymptotic results are obtained for the general Levy input case. The results generalize those of Lieshout and Mandjes from the recent papers (Lieshout and Mandjes in Math. Methods Oper. Res. 66:275---298, 2007 and Queueing Syst. 60:203---226, 2008) for the corresponding tandem queue without an intermediate input.

Kernite, ideally Na2B4O6(OH)2∙3H2O, is a major constituent of borate deposits and one of the most important mineral commodities of B. The chemical composition and crystal structure of kernite from the Kramer Deposit (Kern County, California) were investigated by a suite of analytical techniques (i.e., titrimetric determination of B content, gravimetric method for Na, ion selective electrode for F, high-T mass loss for H2O content, inductively coupled plasma atomic emission spectroscopy for REE and other minor elements, elemental analysis for C, N, and H contents) and single-crystal X-ray (at 293 K) and neutron (at 20 K) diffraction. The concentrations of more than 50 elements were measured. The general experimental formula of the kernite sample used in this study is Na1.99B3.99O6(OH)2∙3.01H2O. The fraction of other elements is, overall, insignificant: excluding B, kernite from the Kramer Deposit does not act as geochemical trap of other technologically relevant elements (e.g., Li, Be, or REE). The X-ray and neutron structure model obtained in this study confirms that the structure of kernite is built up by: two (crystallographically independent) triangular BO2OH groups and two tetrahedral BO4 groups, which share corner-bridging O atoms to form threefold rings, giving chains running along [010], and NaO4(OH)(OH2) and NaO2(OH)(OH2)3 polyhedra. Positional disorder of two H sites of H2O molecules was observed by the neutron structure refinement and corroborated by the maximum-entropy method calculation, which consistently provided a model based on a static disorder, rather than a dynamic one. The H-bonding network in the structure of kernite is complex, pervasive, and plays a primary role on its structural stability: the majority of the oxygen sites are involved in H-bonding, as donors or as acceptors. The potential utilizations of kernite, as a source of B (B2O3 ~50 wt%), are discussed, on the basis of the experimental findings of this study.

As a primary branch of Operations Research, Queueing Theory models and analyzes engineering systems with random fluctuations. With the development of internet and computation techniques, the engineering systems today are much bigger in scale and more complicated in structure than 20 years ago, which raises numerous new problems to researchers in the field of queueing theory. The aim of this thesis is to explore new methods and tools, from both algorithmic and analytical perspectives, that are useful to solve such problems. In Chapter 1 and 2, we introduce some techniques of asymptotic analysis that are relatively new to queueing applications in order to give more accurate probabilistic characterization of queueing models with large scale and complicated structure. In particular, Chapter 1 gives the first functional large deviation result for infinite-server system with general inter-arrival and service times. The functional approach we use enables a nice description of the whole system over the entire time horizon of interest, which is important in real problems. In Chapter 2, we construct a queueing model for the so-called limit order book that is used in main financial markets worldwide. We use an asymptotic approach called multi-scale modeling to disentangle the complicated dependence among the elements in the trading system and to reduce the model dimensionality. The asymptotic regime we use is inspired by empirical observations and the resulting limit process explains and reproduces stylized features of real market data. Chapter 2 also provides a nice example of novel applications of queueing models in systems, such as the electronic trading system, that are traditionally outside the scope of queueing theory. Chapter 3 and 4 focus on stochastic simulation methods for performance evaluation of queueing models where analytic approaches fail. In Chapter 3, we develop a perfect sampling algorithm to generate exact samples from the stationary distribution of stochastic fluid networks in polynomial time. Our approach can be used for time-varying networks with general inter-arrival and service times, whose stationary distributions have no analytic expression. In Chapter 4, we focus on the stochastic systems with continuous random fluctuations, for instance, the workload arrives to the system in continuous flow like a Levy process. We develop a general framework of simulation algorithms featuring a deterministic error bound and an almost square root convergence rate. As an application, we apply this framework to estimate the stationary distributions of reflected Brownian motions and the performance of our algorithm is better than existing prevalent numeric methods.

On the tail behavior of queues with discrete autoregressive arrivals

ABSTRACTMoments have been traditionally used to characterize a probability distribution. Recently, linear moments (L-moments) and trimmed L-moments (TL-moments) are appealing alternatives to the conventional moments. This paper focuses on the computation of theoretical L-moments and TL-moments and emphasizes the use of combinatorial identities. We are able to derive new closed-form formulas of L-moments and TL-moments for continuous probability distributions. Finally, closed-form formulas for the L-moments for the exponential distribution and the uniform distribution are also obtained.

Critical velocities of a three-layer composite tube under a moving internal pressure are derived in the closed-form expressions. The formulation is based on a first-order shear deformation shell theory that accounts for the transverse shear, rotary inertia and material anisotropy. The composite tube consists of three perfectly bonded cylindrical layers of dissimilar materials, each of which can be orthotropic, transversely isotropic, cubic or isotropic. Closed-form formulas for four critical velocities are first obtained for the general case by considering the effects of transverse shear, rotary inertia, material orthotropy and radial stress. These general formulas are then reduced to specific ones for composite tubes without the transverse shear, rotary inertia or radial stress effect and for tubes with simpler anisotropy. In addition, it is shown that the current model for three-layer tubes can recover those for single- and two-layer tubes as special cases. To demonstrate the newly derived closed-form formulas, an example is given for a three-layer composite tube made of an isotropic inner layer, an orthotropic core and an isotropic outer layer. All four critical velocities of this three-layer composite tube are determined by directly applying the new model. Numerical results reveal that three values of the lowest critical velocity predicted by the current analytical model agree well with those computationally obtained in an existing study.