Asymptotic Results Research Articles

Birth–death dynamics for sampling: global convergence, approximations and their asymptotics

Motivated by the challenge of sampling Gibbs measures with nonconvex potentials, we study a continuum birth–death dynamics. We improve results in previous works (Liu et al 2023 Appl. Math. Optim. 87 48; Lu et al 2019 arXiv:1905.09863) and provide weaker hypotheses under which the probability density of the birth–death governed by Kullback–Leibler divergence or by χ 2 divergence converge exponentially fast to the Gibbs equilibrium measure, with a universal rate that is independent of the potential barrier. To build a practical numerical sampler based on the pure birth–death dynamics, we consider an interacting particle system, which is inspired by the gradient flow structure and the classical Fokker–Planck equation and relies on kernel-based approximations of the measure. Using the technique of Γ-convergence of gradient flows, we show that on the torus, smooth and bounded positive solutions of the kernelised dynamics converge on finite time intervals, to the pure birth–death dynamics as the kernel bandwidth shrinks to zero. Moreover we provide quantitative estimates on the bias of minimisers of the energy corresponding to the kernelised dynamics. Finally we prove the long-time asymptotic results on the convergence of the asymptotic states of the kernelised dynamics towards the Gibbs measure.

NEW MODULI SPACES OF ONE-DIMENSIONAL SHEAVES ON

AbstractWe define a one-dimensional family of Bridgeland stability conditions on $\mathbb {P}^n$ , named “Euler” stability condition. We conjecture that the “Euler” stability condition converges to Gieseker stability for coherent sheaves. Here, we focus on ${\mathbb P}^3$ , first identifying Euler stability conditions with double-tilt stability conditions, and then we consider moduli of one-dimensional sheaves, proving some asymptotic results, boundedness for walls, and then explicitly computing walls and wall-crossings for sheaves supported on rational curves of degrees $3$ and $4$ .

On the asymptotic stability of N-soliton solution for the short pulse equation with weighted Sobolev initial data

In this work, we are devoted to studying the Cauchy problem of the short pulse (SP) equation with weighted Sobolev initial data. By developing the ∂¯-generalization of Deift-Zhou nonlinear steepest descent method, we derive the leading order approximation to the solution q(x,t) in solitonic region of space-time, (yt)=ξ for any fixed ξ=ξ0∈(0,+∞), and give bounds for the error decaying as |t|→∞. Based on the resulting asymptotic behavior, the asymptotic approximation of the SP equation is characterized with the soliton term confirmed by N(I)-soliton on discrete spectrum with residual error up to O(t−1). Combining with previous results presented by Yang and Fan, that is the long time asymptotic expansion of the solution q(x,t) in solitonic region ξ=yt∈(−∞,0), our results show that the soliton resolution conjecture of the SP equation is fully solved in all space-time solitonic regions. Moreover, considering properties of the scattering map of SP equation, we derive an asymptotic stability result for a class of initial values.

A numerical method for solving snapping out Brownian motion in 2D bounded domains

Diffusion in heterogeneous media partitioned by semi-permeable interfaces has a wide range of applications in the physical and life sciences, including gas permeation in soils, diffusion magnetic resonance imaging (dMRI), drug delivery, thermal conduction in composite media, synaptic receptor trafficking, and intercellular gap junctions. At the single particle level, diffusion across a semi-permeable interface can be formulated in terms of so-called snapping out Brownian motion (SNOBM). The latter sews together successive rounds of reflected BM, each of which is restricted to one side of the interface. Each round of reflected BM is killed when the local time at the interface exceeds an independent, exponentially distributed random variable. (The local time specifies the amount of time a reflected Brownian particle spends in a neighborhood of the interface.) The particle then immediately resumes reflected BM on the same side or the other side of the interface according to a stochastic switch, and the process is iterated. In this paper, we develop a Monte Carlo algorithm for simulating a two-dimensional version of SNOBM, which is used to solve a first passage time (FPT) problem for diffusion in a domain with semi-permeable partially absorbing traps. Our method combines a walk-on-spheres (WOS) method with an efficient algorithm for computing the boundary local time that uses a Skorokhod integral representation of the latter. We validate our algorithm by comparing the Monte Carlo estimates of the mean first passage time (MFPT) to the exact solution for a single circular trap, and show that our simulations are consistent with asymptotic results obtained for the 2D narrow capture problem involving multiple small circular targets. We also discuss extensions to higher dimensions.

Eigenvalue Tests for the Number of Latent Factors in Short Panels

Abstract This article studies new tests for the number of latent factors in a large cross-sectional factor model with small time dimension. These tests are based on the eigenvalues of variance–covariance matrices of (possibly weighted) asset returns and rely on either an assumption of spherical errors, or instrumental variables for factor betas. We establish the asymptotic distributional results using expansion theorems based on perturbation theory for symmetric matrices. Our framework accommodates semi-strong factors in the systematic components. We propose a novel statistical test for weak factors against strong or semi-strong factors. We provide an empirical application to U.S. equity data. Evidence for a different number of latent factors according to market downturns and market upturns is statistically ambiguous in the considered subperiods. In particular, our results contradict the common wisdom of a single-factor model in bear markets.

Robust estimators of functional single index models for longitudinal data

Functional single index model is a powerful model for longitudinal data, which effectively reduces the dimension of covariates. Inspired by robust calibration of computer models, we propose a robust estimator based on Huber loss for functional single index model with fixed dimension parameter. Furthermore, we also consider the case of high dimension parameter, and proposed a robust L 1 estimator of the unknown parameter based on adaptive lasso penalty. Theoretical properties including the asymptotic and non asymptotic results have also been investigated. Numerical studies including simulated experiments and an application to AIDS data verify the validity of the proposed estimators.

On asymptotic properties of spacings

In this work, we investigate spacings based on order statistics obtained from continuous distribution functions. At the beginning of the paper, we present distributional results for spacings and a method of classification of distributions according to their tails. Then we use this method to derive asymptotic results for spacings. By applying some special versions of Borel–Cantelli lemma, we obtain their strong limit results. At the end of the paper, we present some illustrative examples.

RIS Partitioning Based Scalable Beamforming Design for Large-Scale MIMO: Asymptotic Analysis and Optimization

In next-generation wireless networks, reconfigurable intelligent surface (RIS)-assisted multiple-input multiple-output (MIMO) systems are foreseeable to support a large number of antennas at the transceiver as well as a large number of reflecting elements at the RIS. To fully unleash the potential of RIS, the phase shifts of RIS elements should be carefully designed, resulting in a high-dimensional non-convex optimization problem that is hard to solve with affordable computational complexity. In this paper, we address this scalability issue by partitioning RIS into sub-surfaces, so as to optimize the phase shifts in sub-surface levels to reduce complexity. Specifically, each sub-surface employs a linear phase variation structure to anomalously reflect the incident signal to a desired direction, and the sizes of sub-surfaces can be adaptively adjusted according to channel conditions. We formulate the achievable rate maximization problem by jointly optimizing the transmit covariance matrix and the RIS phase shifts. Under the RIS partitioning framework, the RIS phase shifts optimization reduces to the manipulation of the sub-surface sizes, the phase gradients of sub-surfaces, as well as the common phase shifts of sub-surfaces. Then, we characterize the asymptotic behavior of the system with an infinitely large number of transceiver antennas and RIS elements. The asymptotic analysis provides useful insights on the understanding of the fundamental performance-complexity tradeoff in RIS partitioning design. We show that in the asymptotic domain, the achievable rate maximization problem has a rather simple form with an explicit physical meaning of optimization variables. We develop an efficient algorithm to find an approximately optimal solution to the asymptotic problem via a one-dimensional (1D) grid search. Moreover, we discuss the insights and impacts of the asymptotic result on finite-size system design. By applying the asymptotic result to a finite-size system with necessary modifications, we show by numerical results that the proposed design achieves a favorable tradeoff between system performance and computational complexity.

Funnel-Based Asymptotic Control of Leader–Follower Nonholonomic Robots Subject to Formation Constraints

This work studies control of a leader-follower pair of nonholonomic systems in the presence of formation constraints like communication, feasibility and performance constraints. In such constrained configuration, communication constraints require to maintain limited-range connectivity, feasibility constraints on bearing angles are meant to avoid singularity in the control, and performance constraints include collision avoidance and pre-specified transient behaviour. Different from proposed solutions based on approximation-based prescribed performance or barrier Lyapunov techniques, this work provides an asymptotic result via a properly designed funnel technique. Stability analysis is provided and a comparative study with the state of the art illustrates the benefits of the proposed method.

Comparing the spectrum of Schrödinger operators on quantum graphs

We study Schrödinger operators on compact finite metric graphs subject to δ \delta -coupling and standard boundary conditions. We compare the n n -th eigenvalues of those self-adjoint realizations and derive an asymptotic result for the mean value of deviations. By doing this, we generalize recent results from Rudnick et al. [Comm. Math. Phys. 388 (2021), pp. 1603–1635] obtained for domains in R 2 \mathbb {R}^2 to the setting of quantum graphs. This also leads to a generalization of related results previously and independently obtained by Sofer [Spectral curves of quantum graphs with δ s \delta _s type vertex conditions, arXiv:2212.09143, 2022] and Band et al. [Differences between Robin and Neumann eigenvalues on metric graphs, arXiv:2212.12531, 2022] for metric graphs. In addition, based on our main result, we introduce some surface measures for a (quantum) graph which might prove useful in the future.

Large-Separation Behavior of the Casimir–Polder Force from Real Graphene Sheet Deposited on a Dielectric Substrate

The Casimir–Polder force between atoms or nanoparticles and graphene-coated dielectric substrates is investigated in the region of large separations. Graphene coating with any value of the energy gap and chemical potential is described in the framework of the Dirac model using the formalism of the polarization tensor. It is shown that the Casimir–Polder force from a graphene-coated substrate reaches the limit of large separations at approximately 5.6 μm distance between an atom or a nanoparticle and graphene coating independently of the values of the energy gap and chemical potential. According to our results, however, the classical limit, where the Casimir–Polder force no longer depends on the Planck constant and the speed of light, may be attained at much larger separations depending on the values of the energy gap and chemical potential. In addition, we have found a simple analytic expression for the Casimir–Polder force from a graphene-coated substrate at large separations and determined the region of its applicability. It is demonstrated that the asymptotic results for the large-separation Casimir–Polder force from a graphene-coated substrate are in better agreement with the results of numerical computations for the graphene sheets with larger chemical potential and smaller energy gap. Possible applications of the obtained results in nanotechnology and bioelectronics are discussed.

Linear panel regressions with two-way unobserved heterogeneity

We study linear panel regression models in which the unobserved error term is an unknown smooth function of two-way unobserved fixed effects. In standard additive or interactive fixed effect models the individual specific and time specific effects are assumed to enter with a known functional form (additive or multiplicative). In this paper, we allow for this functional form to be more general and unknown. We discuss two different estimation approaches that allow consistent estimation of the regression parameters in this setting as the number of individuals and the number of time periods grow to infinity. The first approach uses the interactive fixed effect estimator in Bai (2009), which is still applicable here, as long as the number of factors in the estimation grows asymptotically. The second approach first discretizes the two-way unobserved heterogeneity (similar to what Bonhomme et al., 2021 are doing for one-way heterogeneity) and then estimates a simple linear fixed effect model with additive two-way grouped fixed effects. For both estimation methods we obtain asymptotic convergence results, perform Monte Carlo simulations, and employ the estimators in an empirical application to UK house price data.

Eigenvalues and eigenfunctions of nonuniform beams with a new method based on perturbation

By using a new method based on perturbation method and Fredholm theory, the eigenvalues and eigenfunctions of nonuniform Euler-Bernoulli beam vibration are solved analytically. The eigenvalues and eigenfunctions under boundary conditions are given for the tension beam problem which considers the variation of tension in vertical marine riser from top to bottom caused by gravity effect. The asymptotic analytical results are compared with classical results to verify the effectiveness of the new method based on perturbation method. The influence of the change of the bottom tension on the perturbation method is analyzed. The results show that: for each order of natural frequency of riser, the natural frequency will gradually increase with the increase of bottom tension, and the obtained natural frequency will become more and more accurate; when the bottom tension is constant, the new method based on perturbation method in this paper can give good accuracy for higher order natural frequency rather than lower order.

Maxwell–Bloch equations without spectral broadening: the long-time asymptotics of an input pulse in a long two-level laser amplifier

We study the problem of propagation of an input electromagnetic pulse through a long two-level laser amplifier under trivial initial conditions. In this paper, we consider an unstable model described by the Maxwell–Bloch equations without spectral broadening. Previously, this model was studied by Manakov in (1982 Zh. Eksp. Teor. Fiz. 83 68–75) and together with Novokshenov in (1986 Teor. Mat. Fiz. 69 40–54). We consider this model in a more natural formulation as an initial-boundary (mixed) problem using a modern version of the inverse scattering transform method in the form of a suitable Riemann–Hilbert (RH) problem. The RH problem arises as a result of applying the Fokas–Its method of simultaneous analysis of the corresponding spectral problems for the Ablowitz–Kaup–Newell–Segur equations. This approach makes it possible to obtain rigorous asymptotic results at large times, which differ significantly from the previous ones. Differences take place both near the light cone and in the tail region, where a new type of solitons is found against an oscillating background. These solitons are physically relevant, their velocities are smaller than the speed of light. The number of such solitons can be either finite or infinite (in the latter case, the set of zeros has a condensation point at infinity). Such solitons cannot be reflectionless, they are generated by zeros of the reflection coefficient of the input pulse (and not by poles of the transmission coefficient). Thus our approach shows the presence of a new phenomenon in soliton theory, namely, the boundary condition (input pulse) of a mixed problem under trivial initial conditions can generate solitons due to the zeros of the reflection coefficient, while the poles of the transmission coefficient do not contribute to the asymptotics of the solution.

Testing symmetry of model errors for non linear multiplicative distortion measurement error models

To study the symmetry and asymmetry of the model error under multiplicative distortion measurement errors setting, we propose a correlation coefficient-based measure between the distribution function and the root of density function. The unknown distribution function and density function are estimated from four kinds of residuals: the conditional mean calibration-based residuals, the conditional absolute mean calibration-based residuals, the conditional variance calibration-based residuals, and the conditional absolute logarithmic calibration-based residuals. We study the asymptotic results of the estimators of correlation coefficient-based measure under four calibrations. Next, we consider statistical inference of the correlation coefficient-based measure by using the empirical likelihood method. The empirical likelihood statistics are shown to be an asymptotically standard chi-squared distribution. Simulation studies demonstrate the performance of the proposed estimators and test statistics. A real example is analyzed to illustrate its practical usage.

Marginal Distributions at the Tip of a Grafted Stiff Polymer: Analytical and Monte Carlo Investigations

AbstractThis study investigates the marginal distributions of grafted stiff polymer tips in a 2D embedding space using both analytical methods and Monte Carlo simulations. By mapping active Brownian particle (ABP) trajectories in the short‐time regime, analytical expressions for the elongation of the free end of the polymer under horizontal and vertical forces are derived and these expressions are validated using Monte Carlo simulations. These results indicate that the theoretical predictions match well with the simulation results when the chain length is short or the force is large. However, a slight discrepancy is observed between the theoretical and simulation results when the chain is extremely long, although the qualitative asymptotic results remain valid. Additionally, expressions are provided for the horizontal and vertical force versus displacement for the wormlike chain under the weakly bending approximation. This research provides insights into how the trajectories of an ABP correspond to the equilibrium configuration of a semiflexible polymer. These findings have potential applications in various fields, including biophysics and materials science, where understanding the behavior of grafted polymers is crucial.

R-primitive k-normal elements in arithmetic progressions over finite fields

Let be a finite field with qn elements. For a positive divisor r of , the element is called r-primitive if its multiplicative order is . Also, for a non-negative integer k, the element is k-normal over if in has degree k. In this paper we discuss the existence of elements in arithmetic progressions with being ri -primitive and at least one of the elements in the arithmetic progression being k-normal over . We obtain asymptotic results for general and concrete results when for .

On the limit distribution of the power function induced by a design prior

The hybrid frequentist-Bayesian approach to sample size determination is based on the expectation of the power function of a test with respect to a design prior for the unknown parameter value. In clinical trials this quantity is often called probability of success (PoS). Determination of the limiting value of PoS as the number of observations tends to infinity, that is crucial for well defined sample size criteria, has been considered in previous articles. Here, we focus on the asymptotic behavior of the whole distribution of the power function induced by the design prior. Under mild conditions, we provide asymptotic results for the three most common classes of hypotheses on a scalar parameter. The impact of the design parameters choice on the distribution of the power function and on its limit is discussed.

Unified semicompeting risks analysis of hepatitis natural history through mediation modeling.

Natural history of hepatitis B or C is comprised of multiple milestones such as liver cirrhosis and liver cancer. To fully characterize its natural course, semicompeting risks represent a common problem where liver cirrhosis and liver cancer are both of interest, but only the former may be censored by the latter. Copula, frailty and multistate models serve as well-established analytics for semicompeting risks. Here, we cast the semicompeting risks in a mediation framework, with liver cirrhosis as a mediator and liver cancer as an outcome. We define the indirect and direct effects as the effects of an exposure on the liver cancer incidence mediated and not mediated through liver cirrhosis, respectively. With the estimands derived as conditional probabilities, we derive respective expressions under the copula, frailty, and multistate models. Next, we propose estimators based on nonparametric maximum likelihood or U-statistics and establish their asymptotic results. Numerical studies demonstrate that the efficiency of copula models leads to potential bias due to model misspecification. Moreover, the robustness of frailty models is accompanied by a loss in efficiency, and multistate models balance the efficiency and robustness. We demonstrate the utility of the proposed methods by a hepatitis study, showing that hepatitis B and C lead to a higher incidence of liver cancer by increasing liver cirrhosis incidence. Thus, mediation modeling provides a unified framework that accommodates various semicompeting risks models.

Infinite horizon fuzzy optimal control: optimality does not imply asymptotic stability

In this paper, novel stability result for discrete-time infinite horizon optimal control using fuzzy objective functions is presented. For this class of control, the fuzzy goals and the fuzzy constraints introduced in the fuzzy objective function handle the constraints placed on both the state and the control vectors. We analyze the asymptotic stability of the equilibrium for the infinite horizon fuzzy optimal control law using the minimum aggregation operator. We show that the infinite horizon control with the minimum aggregation operator does not guarantee the asymptotic stability of the equilibrium in general. This is done by deriving an analytical solution of the control law for a simple linear system using a fuzzy dynamic programming approach. An example that shows the novel asymptotic stability result of the equilibrium for discrete-time infinite horizon optimal control with fuzzy objective function problem is given.