Asymptotic Results Research Articles

Product of polynomial values being large power

Abstract Erdös and Selfridge first showed that the product of consecutive integers cannot be a perfect power. Later, this result was generalized to polynomial values by various authors. They demonstrated that the product of consecutive polynomial values cannot be the perfect power for a suitable polynomial. In this article, we consider a related problem to the product of consecutive integers. We consider all sequences of polynomial values from a given interval whose products are almost perfect powers. We study the size of these powers and give an asymptotic result. We also define a group theoretic invariant, which is a natural generalization of the Davenport constant. We provide a non-trivial upper bound of this group theoretic invariant.

Large coupling asymptotics for the Lyapunov exponents of some Schrödinger cocyles over strongly expanding circle endomorphisms

ABSTRACT We quantify the coupling asymptotics for the Lyapunov exponent of the Schrödinger cocycle over strongly expanding maps x ↦ bx ( mod 1 ) on T , for a large class of potential functions. We refine the previous lower bound results for these Lyapunov exponents, to get asymptotic results for all E ∈ R and for sufficiently expanding circle maps.

Conditional parametric bootstrap in GLARMA models

Due to situations where asymptotic results are not valid or when we do not know the true distributions of the data set, the bootstrap technique stands out as an effective alternative to make inferences about model parameters. In this context, we introduce the conditional parametric bootstrap, applied to time series data, specifically using the Generalized Linear Autoregressive Moving Average (GLARMA) model. Our interest is to use the procedure to correct possible biases and construct confidence intervals and hypothesis testing for the parameters of the GLARMA model. We conducted simulation studies to evaluate the performance of the procedure in finite samples. The results show that the proposed technique is promising and can be applied in practical situations. An example is the analysis of the monthly number of chronic obstructive pulmonary disease (COPD) cases, which illustrates the potential of the proposed approach in a real situation.

UOWC spatial diversity techniques over hostile maritime environments: an approach under imperfect CSI and per-source power constraints

Optical communication in submarine environments has emerged as a novel technology that enables high bandwidth and high data rate links. However, the unique characteristics of the underwater channel impose new challenges, such as mitigating the remarkable absorption and scattering of hostile maritime environments. For the first time, we consider a per-source optical power constraint based on eye-safety regulations, which has never before been taken into account in Multiple-Input/Single-Output (MISO) systems within underwater optical wireless communication (UOWC) scenarios. Hence, we introduce an innovative spatial repetition coding (SRC) system model, which enables the analysis of an SRC scheme operating under either a per-source or a per-transmitter power constraint. In addition, a tractable generalized transmit laser selection (GTLS) model is presented in order to consider the impact of erroneous selections of the best laser source due to imperfect channel state information (CSI) at the transmitter, prevalent in underwater scenarios with dynamic fluctuations from water currents. Novel bit error rate closed-form expressions and asymptotic results are derived. The presented results demonstrate that an SRC system, when appropriately designed under a per-source power constraint, outperforms the TLS system by effectively mitigating the adverse effects of underwater links. Conversely, in situations where compact transmitters necessitate constraints that significantly modify eye-safety, TLS schemes are superior.

Analytical expression for continuum–continuum transition amplitude of hydrogen-like atoms with angular-momentum dependence

Abstract Attosecond chronoscopy typically utilises interfering two-photon transitions to access the phase information. Simulating these two-photon transitions is challenging due to the continuum–continuum transition term. The hydrogenic approximation within second-order perturbation theory has been widely used due to the existence of analytical expressions of the wave functions. So far, only (partially) asymptotic results have been derived, which fail to correctly describe the low-kinetic-energy behaviour, especially for high angular-momentum states. Here, we report an analytical expression that overcomes these limitations. It is based on the Appell’s F 1 function and uses the confluent hypergeometric function of the second kind as the intermediate state. We show that the derived formula quantitatively agrees with the numerical simulations using the time-dependent Schrödinger equation for various angular-momentum states, which improves the accuracy compared to the other analytical approaches that were previously reported. Furthermore, we give an angular-momentum-dependent asymptotic form of the outgoing wavefunction and the corresponding continuum–continuum dipole transition amplitudes.

Multi-sample means comparisons for imprecise interval data

In recent years, interval data have become an increasingly popular tool to solving modern data problems. Intervals are now often used for dimensionality reduction, data aggregation, privacy censorship, and quantifying awareness of various uncertainties. Among many statistical methods that are being studied and developed for interval data, significance tests are of particular importance due to their fundamental value both in theory and practice. The difficulty in developing such tests mainly lies in the fact that the concept of normality does not extend naturally to intervals, making the exact tests hard to formulate. As a result, most existing works have relied on bootstrap methods to approximate null distributions. However, this is not always feasible given limited sample sizes or other intrinsic characteristics of the data. In this paper, we propose a novel asymptotic test for comparing multi-sample means with interval data as a generalization of the classic ANOVA. Based on the random sets theory, we construct the test statistic in the form of a ratio of between-group interval variance and within-group interval variance. The limiting null distribution is derived under usual assumptions and mild regularity conditions. Simulation studies with various data configurations validate the asymptotic result, and show promising small sample performances. Finally, a real interval data ANOVA analysis is presented that showcases the applicability of our method.

Asymptotic Results for the Estimation of the Quadratic Score of a Clustering

In cluster analysis one often finds several partitions of a data set using different clustering methods and algorithms set with a variety of hyperparameters and tunings. The number of clusters K is one of the most relevant of such hyperparameters. Cluster selection is the task of choosing the desired partitions. The Bootstrap Quadratic Scoring is a recently introduced method where the cluster selection is performed by optimizing a score attached to a partition that is based on the quadratic discriminant function. Previously, we proposed the estimation of this cluster score via bootstrap resampling and investigated the proposed estimator based on numerical experiments and real data applications. However, that earlier work did not provide theoretical guarantees. In this paper, we fill that gap. We study the asymptotic behavior of the scoring method and show that the proposed estimator converges to well-defined population counterparts.

Low-frequency propagating and evanescent waves in strongly inhomogeneous sandwich plates

The paper aims at studying dispersion of elastic waves in a sandwich plate with the parameters, characteristic of aerogel core and hard skin layers, typical for aerospace applications including optimal design of fuselage structural components. The proposed approach relies on multiparametric analysis, taking into account the effect of strong transverse inhomogeneity. It is demonstrated that both an additional low-frequency propagating wave and a slowly decaying evanescent one appear due to a high contrast in geometric and mechanical parameters of the layers. The key findings include the derivation of two-mode asymptotic expansions of the full dispersion relation at the low-frequency limit, as well as elucidation of the non-trivial link between long-wave evanescent and propagating modes. A sophisticated composite nature of the obtained expansions involving various shortened forms is investigated. The range of validity for each of these forms over frequency and wave-number domains is evaluated. Comparison of asymptotic results with the numerical solution of the full dispersion relation is presented.

On type I blowup of some nonlinear heat equations with a potential

In this paper, we are concerned with the following initial-boundary value problem{ut=Δu+Q(|x|)|u|p−1u,x∈BR,t>0u(x,t)=0,x∈∂BR,t>0u(x,0)=u0(x),x∈BR, where p≥ps:=N+2N−2, u0∈L∞(BR), and Q(r)∈C1([0,R]), 0<C_≤Q(r)≤C‾<∞,Q′(r)≤0. We extend the asymptotic behavior results, which is well-known when Q is constant according to Matano-Merle (cf. [25]), for the blow-up solutions. More precisely, we show that when ps≤p<p⁎, the blowup of radial solution to this problem is always of Type I. This result partially generalizes the conclusions in [25] for Q≡1. This extension is nontrivial due to the appearance of Q. The quasi-monotonicity formula established by the third author and Cheng in [8] allows us to use an energy method to get a priori estimates on the rescaled solutions. The contraction mapping principle shows the existence of singular stationary solutions to an associated elliptic equation with a potential. In the end, the properties of zero number for solutions lead to the nonexistence of type II singularity for the problem.

Which shapes can appear in a curve shortening flow singularity?*

Abstract We study possible tangles that can occur in singularities of solutions to plane curve shortening flow. We exhibit solutions in which more complicated tangles with more than one self-intersection disappear into a singular point. It seems that there are many examples of this kind and that a complete classification presents a problem similar to the problem of classifying all knots in R 3 . As a particular example, we introduce the so-called n-loop curves, which generalize Matt Grayson’s figure-eight curve, and we conjecture a generalization of the Coiculescu–Schwartz asymptotic bow-tie result, namely, a vanishing n-loop, when re-scaled anisotropically to fit a square bounding box, converges to a ‘squeezed bow-tie,’ i.e. the curve { ( x , y ) : | x | ⩽ 1 , y = ± x n − 1 } ∪ { ( ± 1 , y ) : | y | ⩽ 1 } . As evidence in support of the conjecture, we provide a formal asymptotic analysis on one hand, and a numerical simulation for the cases n = 3 and n = 4 on the other.

The variance of integers without small prime factors in short intervals

AbstractThe variance of primes in short intervals relates to the Riemann Hypothesis, Montgomery’s Pair Correlation Conjecture and the Hardy–Littlewood Conjecture. In regards to its asymptotics, very little is known unconditionally. We study the variance of integers without prime factors below y, in short intervals. We use complex analysis and sieve theory to prove an unconditional asymptotic result in a range for which we give evidence is qualitatively best possible. We find that this variance connects with statistics of y-smooth numbers, and, as with primes, is asymptotically smaller than the naive probabilistic prediction once the length of the interval is at least a power of y.

Interior vertices and edges in integer partitions

This paper characterises the shape of the Young diagram associated with integer partitions in terms of two newly invented statistics which are defined in the text: namely interior vertices and horizontal edges. Generating functions and associated asymptotic results are developed for both interior vertices and horizontal edges.

Efficient Sampling From the Watson Distribution in Arbitrary Dimensions

In this paper, we present two efficient methods for sampling from the Watson distribution in arbitrary dimensions. The first method adapts the rejection sampling algorithm from Kent et al. (2018), originally designed for Bingham distributions, using angular central Gaussian envelopes. For the Watson distribution, we derive a closed-form expression for the parameters that maximize sampling efficiency, which is further investigated and bounded by asymptotic results. This approach avoids the curse of dimensionality through a smart matrix inversion, enabling fast runtimes even in high dimensions. The second method, based on Saw (1978), employs adaptive rejection sampling from a projected distribution. This algorithm is also effective in all dimensions and offers rapid sampling capabilities. Finally, our simulation study compares the two main methods, revealing that each excels under different conditions: the first method is more efficient for small samples or large dimensions, while the second performs better with larger samples and more concentrated distributions. Both algorithms are available in the R package watson on CRAN. Supplementary materials for this article are available online.

Methods for estimating integrated variance: Jump robustness issues in high frequency time series

Integrated variance is a volatility measure of a process in continuous time, which is applicable in financial mathematics as a means to optimize a portfolio, project dynamics of a price of a financial asset. The consistency of an estimator for integrated variance of a random process is at the core of this article. A fundamental diffusion process is extended by adding a jump component as a means of improving the descriptive function of the process. It is activity of jumps that is a factor subject to which is the consistency of the estimator for integrated variance. Due to this fact, consistency is defined as an extent to which the estimator at hand is jump robust. Two main methods for estimating integrated variance are considered and the capacity of corresponding estimators to withstand the effect of jumps while converging is briefly analyzed. The arguments indicating a necessity for further research of the effect of jumps with reference to works of the authors who have established a ground for analysis of integrated variance and those works containing main asymptotic results for consistency of integrated variance estimators are elaborated. Based on this, avenue for further research and development of asymptotic theory for consistency of an estimator for integrated variance is identified.

Consequences of thermal conductivity and thermal density on heat and mass transfer in the nanofluid boundary layer flow toward a stretching sheet in the presence of a magnetic field

The term “thermal conductance” is used to describe a material’s ability to transport or conduct heat. Materials with high thermal conductivity are employed as heating elements, while those with poor thermal conductivity are used for insulation purposes. It is known that the thermal conductivity of pure metals decreases as temperature increases. In this study, the primary focus is on the physical assessment of thermal conductivity, entropy, and the improvement rate of thermal density in a magnetic nanofluid. To achieve this, nonlinear partial differential equations are transformed into ordinary differential equations. These equations are further solved using a computational method known as the Keller box technique. Various flow parameters, such as the Eckert number, density parameter, magnetic-force parameter, thermophoretic number, buoyancy number, and Prandtl parameter, are examined for their impact on velocity, temperature distribution, and concentration distribution. For the asymptotic results, the appropriate range of parameters, such as 1.0 ≤ ξ ≤ 5.0, 0.0 ≤ n ≤ 0.9, 0.1 ≤ Ec ≤ 2.0, 0.7 ≤ Pr ≤ 7.0, 0.1 ≤ Nt ≤ 0.5, and 0.1 ≤ Nb ≤ 0.9, is utilized. The key findings of this study are related to the assessment of heat transfer in a magnetic nanofluid considering thermal conductivity, entropy generation, and temperature density. It is observed that the temperature distribution increases as entropy generation increases. From a physical perspective, thermal conductivity acts as a facilitating factor in enhancing heat transfer. The study concludes by emphasizing the consistency achieved through a comparison of the latest findings with previously reported analyses.

Heterogeneous radio frequency/underwater optical wireless communication relaying systems with MIMO scheme

This paper studies a cooperative relay transmission system within the framework of Multiple-Input Multiple-Output Radio Frequency/Underwater Optical Wireless Communication (MIMO-RF/UOWC), aiming to establish sea-based heterogeneous networks. In this setup, the RF links obey κ-μ fading, while the UOWC links undergo the generalized Gamma fading with the pointing error impairments. The relay operates under an Amplify-and-Forward (AF) protocol. Additionally, the attenuation caused by the Absorption and Scattering (AaS) is considered in UOWC links. The work yields precise results for the Average Channel Capacity (ACC), Outage Probability (OP), and average Bit Error Rate (BER). Furthermore, to reveal deeper insights, bounds on the ACC and asymptotic results for the OP and average BER are derived. The findings highlight the superior performance of MIMO-RF/UOWC AF systems compared to Single-Input-Single-Output (SISO)-RF/UOWC AF systems. Various factors affecting the Diversity Gain (DG) of the MIMO-RF/UOWC AF system include the number of antennas/apertures, fading parameters of both links, and pointing error parameters. Moreover, while an increase in the AaS effect can result in significant attenuation, it does not determine the achievable DG of the proposed MIMO-RF/UOWC AF relaying system.

Spatio-Functional Nadaraya–Watson Estimator of the Expectile Shortfall Regression

The main aim of this paper is to consider a new risk metric that permits taking into account the spatial interactions of data. The considered risk metric explores the spatial tail-expectation of the data. Indeed, it is obtained by combining the ideas of expected shortfall regression with an expectile risk model. A spatio-functional Nadaraya–Watson estimator of the studied metric risk is constructed. The main asymptotic results of this work are the establishment of almost complete convergence under a mixed spatial structure. The claimed asymptotic result is obtained under standard assumptions covering the double functionality of the model as well as the data. The impact of the spatial interaction of the data in the proposed risk metric is evaluated using simulated data. A real experiment was conducted to measure the feasibility of the Spatio-Functional Expectile Shortfall Regression (SFESR) in practice.

Optimal two-time point longitudinal models for estimating individual-level change: Asymptotic insights and practical implications

Based on findings from a simulation study, Parsons and McCormick (2024) argued that growth models with exactly two time points are poorly-suited to model individual differences in linear slopes in developmental studies. Their argument is based on an empirical investigation of the increase in precision to measure individual differences in linear slopes if studies are progressively extended by adding an extra measurement occasion after one unit of time (e.g., year) has passed. They concluded that two-time point models are inadequate to reliably model change at the individual level and that these models should focus on group-level effects. Here, we show that these limitations can be addressed by deconfounding the influence of study duration and the influence of adding an extra measurement occasion on precision to estimate individual differences in linear slopes. We use asymptotic results to gauge and compare precision of linear change models representing different study designs, and show that it is primarily the longer time span that increases precision, not the extra waves. Further, we show how the asymptotic results can be used to also consider irregularly spaced intervals as well as planned and unplanned missing data. In conclusion, we like to stress that true linear change can indeed be captured well with only two time points if careful study design planning is applied before running a study.

Adaptive hazard estimator of the peak over threshold model under random censoring with application

Although the hazard rates sharing a number of similar aspects. The hazard functions are far less frequently used. The problem of hazard functions estimation has received much attention in the statistical literature. This paper studies the hazard rate estimation with the peak over threshold (POT) modeling within completed data. Next, we have to adapt this estimation with the censored data. Ditto, the asymptotic normality has been established, whereas the case of random threshold t under the uncensored and the censored data, In this dissertation, our procedure of our asymptotic result is based on POT context with the threshold t is random. Wherefore we study the asymptotic normality of the proportion of non-censored observation estimator with the threshold t is random. As an application, we present a new extreme value index (EVI) in terms of the hazard functions estimation with both full and censored data. Then, we establish the asymptotic normality of the new EVI estimator for the heavy tailed index associated with the hazard function without censored for POT. And we adapted this estimator index with censored data for POT where the threshold t is random.

A martingale approach to Gaussian fluctuations and laws of iterated logarithm for Ewens–Pitman model

The Ewens–Pitman model refers to a distribution for random partitions of [n]={1,…,n}, which is indexed by a pair of parameters α∈[0,1) and θ>−α, with α=0 corresponding to the Ewens model in population genetics. The large n asymptotic properties of the Ewens–Pitman model have been the subject of numerous studies, with the focus being on the number Kn of partition sets and the number Kr,n of partition subsets of size r, for r=1,…,n. While for α=0 asymptotic results have been obtained in terms of almost-sure convergence and Gaussian fluctuations, for α∈(0,1) only almost-sure convergences are available, with the proof for Kr,n being given only as a sketch. In this paper, we make use of martingales to develop a unified and comprehensive treatment of the large n asymptotic behaviours of Kn and Kr,n for α∈(0,1), providing alternative, and rigorous, proofs of the almost-sure convergences of Kn and Kr,n, and covering the gap of Gaussian fluctuations. We also obtain new laws of the iterated logarithm for Kn and Kr,n.