Recent Asymptotic Results Research Articles

Fast-slow wave transitions induced by a random mean flow.

Motivated by recent asymptotic results in atmosphere-ocean fluid dynamics, we present an idealized numerical and theoretical study of two-dimensional dispersive waves propagating through a small-amplitude random mean flow. The objective is to delineate clearly the conditions under which the cumulative Doppler shifting and refraction by the mean flow can change the group velocity of the waves not only in direction, but also in magnitude. The latter effect enables a possible transition from fast to slow waves, which behave very differently. Within our model we find the conditions on the dispersion relation and the mean flow amplitude that allow or rule out such fast-slow transitions. For steady mean flows we determine a finite mean flow amplitude threshold below which such transitions can be ruled out indefinitely. For unsteady mean flows a sufficiently rapid rate of change means that this threshold goes to zero, i.e., in this scenario all waves eventually undergo a fast-slow transition regardless of mean flow amplitude, with corresponding implications for the long-term fate of these waves.

On the Oberwolfach problem for single-flip 2-factors via graceful labelings

Let F be a 2-regular graph of order v. The Oberwolfach problem OP(F), posed in 1967 and still open, asks for a decomposition of Kv into copies of F. In this paper we show that OP(F) has a solution whenever F has a sufficiently large cycle which meets a given lower bound and, in addition, has a single-flip automorphism, which is an involutory automorphism acting as a reflection on exactly one of the cycles of F. Furthermore, we prove analogous results for the minimum covering version and the maximum packing version of the problem. We also show a similar result when the edges of Kv have multiplicity 2, but in this case we do not require that F be single-flip.Our approach allows us to explicitly construct solutions to the Oberwolfach Problem with well-behaved automorphisms, in contrast with some recent asymptotic results, based on probabilistic methods, which are nonconstructive and do not provide a lower bound on the order of F that guarantees the solvability of OP(F).Our constructions are based on a doubling construction which applies to graceful labelings of 2-regular graphs with a vertex removed. We show that this class of graphs is graceful as long as the length of the path-component is sufficiently large. A much better lower bound on the length of the path is given for an α-labeling of such graphs to exist.

Statistical Inference on Tree Swallow Migrations with Random Forests

SummaryBird species’ migratory patterns have typically been studied through individual observations and historical records. In recent years, the eBird citizen science project, which solicits observations from thousands of bird watchers around the world, has opened the door for a data-driven approach to understanding the large-scale geographical movements. Here, we focus on the North American tree swallow (Tachycineta bicolor) occurrence patterns throughout the eastern USA. Migratory departure dates for this species are widely believed by both ornithologists and casual observers to vary substantially across years, but the reasons for this are largely unknown. In this work, we present evidence that maximum daily temperature is predictive of tree swallow occurrence. Because it is generally understood that species occurrence is a function of many complex, high order interactions between ecological covariates, we utilize the flexible modelling approach that is offered by random forests. Making use of recent asymptotic results, we provide formal hypothesis tests for predictive significance of various covariates and also develop and implement a permutation-based approach for formally assessing interannual variations by treating the prediction surfaces that are generated by random forests as functional data. Each of these tests suggest that maximum daily temperature is important in predicting migration patterns.

On asymptotic finite-time ruin probability of a renewal risk model with subexponential main claims and delayed claims

This paper considers a non-standard renewal risk model with constant force of interest, where each main claim may derive a delayed claim occurring after a random period of time. By means of recent asymptotic results for randomly weighted sums of subexponential random variables, we obtain some precise asymptotic expansions for the finite-time ruin probability when the main claims have a common subexponential tail.

Asymptotic properties of high‐speed waves in atomic chains

AbstractWe summarize some recent asymptotic results on the appoximation and the stability of high‐speed waves in FPUT chains.

A powerful wild bootstrap diagnosis of panel unit roots under linear trends and time-varying volatility

Noting that standard panel unit root tests (PURTs) are not reliable in the presence of time varying volatility, recent papers have suggested a few heteroskedasticity–robust PURTs. These tests do not remain pivotal, however, if they are applied on detrended data. Building on recent asymptotic results for bootstrap approximations (Smeekes and Urbain in A multivariate invariance principle for modified wild bootstrap methods with an application to unit root testing, 2014), we provide simulation evidence that recursive detrending followed by a wild bootstrap correction leads to very good size precision for a variety of PURTs designed for testing against panel stationarity formalized with a cross sectionally homogenous autoregressive parameter. With respect to power, the bootstrap variant of the original nonrobust test of Levin et al. (J Econom 108(1):1–24, 2002) turns out to be the most effective among alternative tests. At the implementation side, performing the wild bootstrap correction using the so-called Rademacher distribution is found to be the most effective for both size and power properties of panel unit root inference by means of wild bootstrap based critical values. The empirical illustration shows that GDP per capita is best characterized as a unit root process.

A more efficient second order blind identification method for separation of uncorrelated stationary time series

The classical second order source separation methods use approximate joint diagonalization of autocovariance matrices with several lags to estimate the unmixing matrix. Based on recent asymptotic results, we propose a novel unmixing matrix estimator which selects the best lag set from a finite set of candidate sets specified by the user. The theory is illustrated by a simulation study.

A note on the stability of swirling flows with radius-dependent density with respect to infinitesimal azimuthal disturbances

We study the stability of inviscid, incompressible swirling flows of variable density with respect to azimuthal, normal mode disturbances. We prove that the wave velocity of neutral modes is bounded. A further refinement of Fung’s semi-elliptical instability region is given. This new instability region depends not only on the minimum Richardson number, and the lower and upper bounds for the angular velocity like Fung’s semi-ellipse, but also on the azimuthal wave number and the radii of the inner and outer cylinders. An estimation for the growth rate of unstable disturbances is obtained and it is compared to some of the recent asymptotic results. doi:10.1017/S1446181115000036

A NOTE ON THE STABILITY OF SWIRLING FLOWS WITH RADIUS-DEPENDENT DENSITY WITH RESPECT TO INFINITESIMAL AZIMUTHAL DISTURBANCES

We study the stability of inviscid, incompressible swirling flows of variable density with respect to azimuthal, normal mode disturbances. We prove that the wave velocity of neutral modes is bounded. A further refinement of Fung’s semi-elliptical instability region is given. This new instability region depends not only on the minimum Richardson number, and the lower and upper bounds for the angular velocity like Fung’s semi-ellipse, but also on the azimuthal wave number and the radii of the inner and outer cylinders. An estimation for the growth rate of unstable disturbances is obtained and it is compared to some of the recent asymptotic results.

Outlier Detection Algorithms for Least Squares Time Series Regression

We review recent asymptotic results on some robust methods for multiple regression. The regressors include stationary and non-stationary time series as well as polynomial terms. The methods include the Huber-skip M-estimator, 1-step Huber-skip M-estimators, in particular the Impulse Indicator Saturation, iterated 1-step Huber-skip M-estimators and the Forward Search. These methods classify observations as outliers or not. From the asymptotic results we establish a new asymptotic theory for the gauge of these methods, which is the expected frequency of falsely detected outliers. The asymptotic theory involves normal distribution results and Poisson distribution results. The theory is applied to a time series data set.

On the oscillatory Marangoni instability in a thin film heated from below

We consider the classical problem of the Marangoni instability in a liquid layer with a deformable free surface atop a substrate heated from below. The linear stability analysis is performed numerically in order to extend the recent asymptotic results [S. Shklyaev, M. Khenner, and A. A. Alabuzhev, “Oscillatory and monotonic modes of long-wave Marangoni convection in a thin film,” Phys. Rev. E 82, 025302 (2010)] to finite-wavenumber perturbations. In spite of detailed analyses by many researchers, we have obtained novel computational results confirming the existence of the oscillatory mode for heating from below. Moreover, numerical simulations indicate that the oscillatory mode is critical in a wider range of parameters, than it is predicted by the asymptotic analysis. Additionally, we provide guiding data for the experimental observation of the oscillatory regime.

Almost Giant Clusters for Percolation on Large Trees with Logarithmic Heights

This paper is based on works presented at the 2012 Applied Probability Trust Lecture in Sheffield; its main purpose is to survey the recent asymptotic results of Bertoin (2012a) and Bertoin and Uribe Bravo (2012b) about Bernoulli bond percolation on certain large random trees with logarithmic height. We also provide a general criterion for the existence of giant percolation clusters in large trees, which answers a question raised by David Croydon.

Almost Giant Clusters for Percolation on Large Trees with Logarithmic Heights

This paper is based on works presented at the 2012 Applied Probability Trust Lecture in Sheffield; its main purpose is to survey the recent asymptotic results of Bertoin (2012a) and Bertoin and Uribe Bravo (2012b) about Bernoulli bond percolation on certain large random trees with logarithmic height. We also provide a general criterion for the existence of giant percolation clusters in large trees, which answers a question raised by David Croydon.

Random bistochastic matrices

Ensembles of random stochastic and bistochastic matrices are investigated. While all columns of a random stochastic matrix can be chosen independently, the rows and columns of a bistochastic matrix have to be correlated. We evaluate the probability measure induced into the Birkhoff polytope of bistochastic matrices by applying the Sinkhorn algorithm to a given ensemble of random stochastic matrices. For matrices of order N = 2 we derive explicit formulae for the probability distributions induced by random stochastic matrices with columns distributed according to the Dirichlet distribution. For arbitrary N we construct an initial ensemble of stochastic matrices which allows one to generate random bistochastic matrices according to a distribution locally flat at the center of the Birkhoff polytope. The value of the probability density at this point enables us to obtain an estimation of the volume of the Birkhoff polytope, consistent with recent asymptotic results.

Lattice vortices induced by noncommutativity

We show that the Moyal ⋆-product on the algebra of fields induces an effective lattice structure on vortex dynamics which can be explicitly constructed using recent asymptotic results.

The Relative Contribution of Jumps to Total Price Variance

We examine tests for jumps based on recent asymptotic results; we interpret the tests as Hausman-type tests. Monte Carlo evidence suggests that the daily ratio z-statistic has appropriate size, good power, and good jump detection capabilities revealed by the confusion matrix comprised of jump classification probabilities. We identify a pitfall in applying the asymptotic approximation over an entire sample. Theoretical and Monte Carlo analysis indicates that microstructure noise biases the tests against detecting jumps, and that a simple lagging strategy corrects the bias. Empirical work documents evidence for jumps that account for seven percent of stock market price variance.

A Note on Further Improvements of the TVZ-Bound

In this note, we present improvements of recent asymptotic results on constructions of nonlinear codes using algebraic geometry. The main constructions involve an adaptation of recent constructions by Niederreiter and Oumlzbudak; and Stichtenoth and Xing

Recent asymptotic results in testing for mixtures

Inferences concerning the number of components in a mixture distribution are often required. These can be performed under the framework of the generalized likelihood ratio test. The classical result giving a chi-squared asymptotic distribution in general does not apply, indeed the limiting distribution of the corresponding test statistic has long remained a mystery. The characterization of the asymptotic distribution in a general setting has been previously derived under a separation condition. The relaxation of the separation condition, the calculation of the percentile points and asymptotic power, both in the case of a bounded and an unbounded parameter set can be obtained. An overview of the very recent asymptotic results in the problem of testing homogeneity against a two-component mixture is provided. Illustrations of new and known results are presented.

The Relative Contribution of Jumps to Total Price Variance

We examine tests for jumps based on recent asymptotic results; we interpret the tests as Hausman-type tests. Monte Carlo evidence suggests that the daily ratio z-statistic has appropriate size, good power, and good jump detection capabilities revealed by the confusion matrix comprised of jump classification probabilities. We identify a pitfall in applying the asymptotic approximation over an entire sample. Theoretical and Monte Carlo analysis indicates that microstructure noise biases the tests against detecting jumps, and that a simple lagging strategy corrects the bias. Empirical work documents evidence for jumps that account for 7% of stock market price variance. Copyright 2005, Oxford University Press.

Modeling genetic algorithms with interacting particle systems

We present in this work a natural Interacting Particle System (IPS) approach for modeling and studying the asymptotic behavior of Genetic Algorithms (GAs). In this model, a population is seen as a distribution (or measure) on the search space, and the Genetic Algorithm as a measure valued dynamical system. This model allows one to apply recent convergence results from the IPS literature for studying the convergence of genetic algorithms when the size of the population tends to infinity.
 We first review a number of approaches to Genetic Algorithms modeling and related convergence results. We then describe a general and abstract discrete time Interacting Particle System model for GAs, and we propose a brief review of some recent asymptotic results about the convergence of the N -IPS approximating model (of finite N -sized-population GAs) towards the IPS model (of infinite population GAs), including law of large number theorems, ILp-mean and exponential bounds as well as large deviations principles.
 Finally, the impact of modeling Genetic Algorithms with our interacting particle system approach is detailed on different classes of generic genetic algorithms including mutation, cross-over and proportionate selection. We explore the connections between Feynman-Kac distribution flows and the simple genetic algorithm. This Feynman-Kac representation of the infinite population model is then used to develop asymptotic stability and uniform convergence results with respect to the time parameter.