Asymptotic Estimates Research Articles

Verification of frequency-dependent impedance boundaries for the Finite-Volume Time-Domain scheme

Numerical simulation methods are widely used to approximate solutions to the wave-equation, taking into account the initial condition ('initial sources') and boundary conditions ('wall materials'). Trustworthy simulations are essential; a primary step is to ensure the correctness of the implementation through a process known as code verification. In this work, we do code verification by studying the convergence of the known finite-volume time-domain (FVTD) method with a frequency-dependent boundary condition modeled as an inductance/resistance/capacitance (LRC) termination. Due to the difficulties of obtaining an analytical solution for frequency-dependent impedance boundaries, we thoroughly derive and implement The Method of Manufactured Solutions tailored to the wave-equation inside a rectangular room for uniform frequency- dependent wall impedance. Asymptotic estimates are found empirically based on a convergence study compared against the manufactured ground truth solution. We investigate the impact of the frequency-dependent boundaries on the convergence affected by the total sum of numerical errors and validate the theoretical properties of the FVTD scheme for various LRC model configurations. The method should also serve as a useful resource for others to verify the correctness of their code implementations.

A Radio-loud Semiregular Variable

As a byproduct of our search for Galactic stellar systems with gamma-ray emission, we have identified an unrelated cool and evolved star (IRC-10412) that attracted our attention due to its strong radio emission level with a spectral index matching, almost perfectly, the canonical +0.6 value expected from an ionized stellar wind. A follow-up observational analysis was undertaken given that these two properties are hard to reconcile as originating in the same stellar object. As a result, IRC-10412 has been classified as a new semiregular variable of SRb type in the asymptotic giant branch, and different but consistent estimates of its mass-loss parameter are reported. We propose that its unusually high radio emission arises from a ∼10−5 M ⊙ yr−1 stellar wind exposed to an external source of ionizing photons, possibly coming from nearby OB associations.

Multivariate moment least-squares variance estimators for reversible Markov chains

Markov chain Monte Carlo (MCMC) is a commonly used method for approximating expectations with respect to probability distributions. Uncertainty assessment for MCMC estimators is essential in practical applications. Moreover, for multivariate functions of a Markov chain, it is important to estimate not only the auto-correlation for each component but also to estimate cross-correlations, in order to better assess sample quality, improve estimates of effective sample size, and use more effective stopping rules. Berg and Song (2023) introduced the moment least squares (momentLS) estimator, a shape-constrained estimator for the autocovariance sequence from a reversible Markov chain, for univariate functions of the Markov chain. Based on this sequence estimator, they proposed an estimator of the asymptotic variance of the sample mean from MCMC samples. In this study, we propose novel autocovariance sequence and asymptotic variance estimators for Markov chain functions with multiple components, based on the univariate momentLS estimators from Berg and Song (2023). We demonstrate strong consistency of the proposed auto(cross)-covariance sequence and asymptotic variance matrix estimators. We conduct empirical comparisons of our method with other state-of-the-art approaches on simulated and real-data examples, using popular samplers including the random-walk Metropolis sampler and the No-U-Turn sampler from STAN. Supplemental materials for this article are available online.

Approximation of operators on real line

There are exponential operators which are defined on the whole real line, namely, the Ismail–May operators , connected to , and the Gauss–Weierstrass operators , connected with 1. Here we consider the composition of these two operators and observe that the composition operator is not of exponential type operator. We find moment‐generating functions and moments of these operators and estimate some point‐wise convergence of these operators using the characteristic function. We also estimate quantitative Voronovskaya type asymptotic formula and error estimation. Also, the difference of the composition operators with its components and is estimated in terms of modulus of continuity. Finally, we provide some graphs to see visualize the convergence by considering the function .

On the number of prime factors with a given multiplicity over h-free and h-full numbers

Let k and n be natural numbers. Let ωk(n) denote the number of distinct prime factors of n with multiplicity k as studied by Elma and the third author [5]. We obtain asymptotic estimates for the first and the second moments of ωk(n) when restricted to the set of h-free and h-full numbers. We prove that ω1(n) has normal order log⁡log⁡n over h-free numbers, ωh(n) has normal order log⁡log⁡n over h-full numbers, and both of them satisfy the Erdős-Kac Theorem. Finally, we prove that the functions ωk(n) with 1<k<h do not have normal order over h-free numbers and ωk(n) with k>h do not have normal order over h-full numbers.

An asymptotic equality of Cartan's Second Main Theorem and some generalizations

An asymptotic equality of Cartan's Second Main Theorem and some generalizations

Asymptotic estimates for entire functions of minimal growth with given zeros

Let $\zeta=(\zeta_n)$ be an arbitrary complex sequence such that $0&lt;|\zeta_1|\le|\zeta_2|\le\dots$ and $\zeta_n\to\infty$ as $n\to\infty$, let $n_\zeta(r)$ and $N_\zeta(r)$ be the counting function and the integrated counting function of this sequence, respectively. By $\mathcal{E}_\zeta$ we denote the class of all entire functions whose zeros are precisely the $\zeta_n$, where a complex number that occurs $m$ times in the sequence $\zeta$ corresponds to a zero of multiplicity $m$. Suppose that $\Phi$ is a convex function on $\mathbb{R}$ such that $\Phi(\sigma)/\sigma\to+\infty$ as $\sigma\to+\infty$. It is proved that there exists an entire function $f\in\mathcal{E}_\zeta$ such that$$\varlimsup_{r\to+\infty}\frac{\ln\ln M_f(r)}{\Phi(\ln r)}\le\varlimsup_{r\to+\infty}\frac{\ln n_\zeta( r)}{\Phi(\ln r)},$$where $M_f(r)$ denotes the maximum modulus of the function $f$, and it is shown that the above inequality implies the inequality$$\varlimsup_{r\to+\infty}\frac{\ln\ln M_f(r)}{\Phi(\ln r)}\le\varlimsup_{r\to+\infty}\frac{\ln N_\zeta( r)}{\Phi(\ln r)}+\varlimsup_{\sigma\to+\infty}\frac{\ln\Phi'_+(\sigma)}{\Phi(\sigma)}.$$The formulated result is a consequence of the following more general statement: if the right-hand derivative $\Phi'_+$ of the function $\Phi$ assumes only integer values and $\sum_{n=1}^\infty e^{-\Phi(\ln|\zeta_n|)}&lt;+\infty$, then there exists an entire function $f\in\mathcal{E}_\zeta$ such that $\ln M_f(r)=o(e^{\Phi(\ln r)})$ as $r \to+\infty$.

Tail risk driven by investment losses and exogenous shocks

AbstractConsider a company whose business carries the potential for investment losses and is additionally vulnerable to exogenous shocks. The unpredictability of the shocks makes it challenging for both the company and the regulator to accurately assess their impact, potentially leading to an underestimation of solvency capital when employing traditional approaches. In this paper, we utilize a stylized model to conduct an extreme value analysis of the tail risk of the company under a Fréchet-type and a Gumbel-type shock. Our main results explicitly demonstrate the different roles of investment risk and shock risk in driving large losses. Furthermore, we derive asymptotic estimates for the value at risk and expected shortfall of the total loss. Numerical studies are conducted to examine the accuracy of the obtained estimates.

Tyranny-of-the-Minority Regression Adjustment in Randomized Experiments

Regression adjustment is widely used in the analysis of randomized experiments to improve the estimation efficiency of the treatment effect. This article reexamines a weighted regression adjustment method termed tyranny-of-the-minority (ToM), wherein units in the minority group are given greater weights. We demonstrate that ToM regression adjustment is more robust than Lin’s regression adjustment with treatment-covariate interactions, even though these two regression adjustment methods are asymptotically equivalent in completely randomized experiments. Moreover, ToM regression adjustment can be easily extended to stratified randomized experiments and completely randomized survey experiments. We obtain the design-based properties of the ToM regression-adjusted average treatment effect estimator under such designs. In particular, we show that the ToM regression-adjusted estimator improves the asymptotic estimation efficiency compared to the unadjusted estimator, even when the regression model is misspecified, and is optimal in the class of linearly adjusted estimators. We also study the asymptotic properties of various heteroscedasticity-robust standard errors and provide recommendations for practitioners. Simulation studies and real data analysis demonstrate ToM regression adjustment’s superiority over existing methods. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Finite-time ruin probability of a risk model with perturbation and subexponential main claims and by-claims

The paper considers a nonstandard risk model with stochastic return and perturbation, in which the price process of the investment portfolio is described as a geometric Lévy process and each main claim may induce a delayed by-claim. When the main claims and by-claims have subexponential distributions, we obtain some asymptotic estimations of the finite-time ruin probability.

Distribution of Sarcophagidae (Diptera, Oestroidea) in Brazilian biomes: richness, endemism, and sampling gaps

ABSTRACT Sarcophagid experts have made several efforts to associate biodiversity data and comprehend where each species occurs, but comprehensive faunal inventories remain scarce. Our aim was to provide a list of distributional patterns and endemic species and allow assessment of the sampling effort conducted within Brazilian biomes. We produced a dataset of Brazilian sarcophagids and overlaid with a biome map, to investigate distributional patterns, endemism and to build species accumulation curves. Additionally, we calculated nonparametric asymptotic species richness estimators and extrapolation of species diversity (Hill numbers). Our dataset comprised 288 sarcophagid species, which 21 were identified as endemic. The biomes with the highest species richness were the Atlantic Rainforest and the Amazon Forest, and no biome exhibited a stabilized asymptotic curve. This is the first proposal of listing Sarcophagidae species by biomes and essential to understand the spatial distribution of this family in Brazil. We present maps and richness estimators that allow identifying gaps and guiding survey planning.

Asymptotic estimates for ruin probabilities in a bidimensional delay-claim risk model with subexponential claims

This article studies the asymptotic behaviors for some common ruin probabilities of a bidimensional delay-claim risk model, in which each claim follows a subexponential distribution and each claim-size inherits some dependence structure. Meanwhile, the asymptotic behavior for sum ruin probability of the bidimensional delay-claim risk model is investigated when the two components of each claim inter-arrival time vector are arbitrarily dependent.

Human-in-the-Loop Consensus Control for Multiagent Systems With External Disturbances.

In this article, the human-in-the-loop leader-follower consensus control problem is addressed for multiagent systems (MASs) with unknown external disturbances. A human operator is deployed to monitor the MASs' team by transmitting an execution signal to a nonautonomous leader in response to any hazard detected, with the control input of the leader unknown to all followers. For each follower, a full-order observer, in which the observer error dynamic system decouples the unknown disturbance input, is designed for asymptotic state estimation. Then, an interval observer is constructed for the consensus error dynamic system, where the unknown disturbances and control inputs of its neighbors and its disturbance are treated as unknown inputs (UIs). To process the UIs, a new asymptotic algebraic UI reconstruction (UIR) scheme is proposed based on the interval observer, and one of the significant features of the UIR is the capacity to decouple the control input of the follower. The subsequent human-in-the-loop asymptotic convergence consensus protocol is developed by applying an observer-based distributed control strategy. Finally, the proposed control scheme is validated through two simulation examples.

Exogeneity tests and weak identification in IV regressions: Asymptotic theory and point estimation

This paper provides new insights on exogeneity tests in linear IV models and their use for estimation, when identification fails or may not be strong. We make two main contributions. First, we show that Durbin-Wu-Hausman (DWH) and Revankar-Hartley (RH) exogeneity tests have correct level asymptotically, even when the first-stage coefficient matrix (which controls identification) is rank-deficient. We provide necessary and sufficient conditions under which these tests are consistent. In particular, we show that test consistency can hold even when identification fails, provided at least one component of the structural parameter vector is identifiable. Second, we study point estimation after estimator (or model) selection, when the outcome of a DWH/RH test determines whether OLS or an IV method is employed in the second-stage. For this purpose, we use (non-local) concepts of asymptotic bias, asymptotic mean squared error (AMSE), and asymptotic relative efficiency (ARE), which remain applicable even when the estimators considered do not have moments (as can happen for 2SLS) or may be inconsistent. We study the asymptotic properties of OLS, 2SLS, and pretest estimators which select OLS or 2SLS based on the outcome of a DWH/RH test. We show that: (i) OLS typically dominates 2SLS estimator asymptotically for MSE across a broad spectrum of cases, including weak identification and moderate endogeneity; (ii) exogeneity-pretest estimators exhibit consistently good performance and asymptotically dominate both OLS and 2SLS. The proposed theoretical findings are documented by Monte Carlo simulations.

Regular Expressions Avoiding Absorbing Patterns and the Significance of Uniform Distribution

Although regular expressions do not correspond univocally to regular languages, it is still worthwhile to study their properties and algorithms. For the average case analysis one often relies on the uniform random generation using a specific grammar for regular expressions, that can represent regular languages with more or less redundancy. Generators that are uniform on the set of expressions are not necessarily uniform on the set of regular languages. Nevertheless, it is not straightforward that asymptotic estimates obtained by considering the whole set of regular expressions are different from those obtained using a more refined set that avoids some large class of equivalent expressions. In this paper we study a set of expressions that avoid a given absorbing pattern. It is shown that, although this set is significantly smaller than the standard one, the asymptotic average estimates for the size of the Glushkov automaton for these expressions does not differ from the standard case. Furthermore, for this set the asymptotic density of [Formula: see text]-accepting expressions is also the same as for the set of all standard regular expressions.

Asymptotic estimates of solution to damped fractional wave equation

It is known that the damped fractional wave equation has the diffusive structure as t→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$t\\rightarrow \\infty $\\end{document}. Let u(t,x)=e−tcosh(tL)f(x)+e−tsinh(tL)L(f(x)+g(x))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$u(t,x)=e^{-t}\\cosh (t\\sqrt{L})f(x)+e^{-t} \\frac{\\sinh (t\\sqrt{L})}{\\sqrt{L}}(f(x)+g(x))$\\end{document} be the solution of the Cauchy problem for the damped fractional wave equation, where L\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\sqrt{L}$\\end{document} involves the fractional Laplacian (−△)α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(-\ riangle )^{\\alpha}$\\end{document} on the space variable. We can study the decay estimate of the solution u(t,x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$u(t,x)$\\end{document} over the time t by means of the Cauchy problem for the parabolic equation. In this paper, we consider, for 0<α<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$0<\\alpha <1$\\end{document}, the Cauchy problem in the two- and three-dimensional spaces for the damped fractional wave equation and the corresponding parabolic equation and obtain the Triebel–Lizorkin space estimate of the difference of solutions. At the same time, we also consider, for α=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\alpha =1$\\end{document}, the case of the Cauchy problem in the four-dimensional space and obtain a Triebel–Lizorkin space estimate.

The Wasserstein Metric between a Discrete Probability Measure and a Continuous One

This paper examines the Wasserstein metric between the empirical probability measure of n discrete random variables and a continuous uniform measure in the d-dimensional ball, providing an asymptotic estimation of their expectations as n approaches infinity. Furthermore, we investigate this problem within a mixed process framework, where n discrete random variables are generated by the Poisson process.

Expanding the Areas of Attraction of Solutions to Singularly Perturbed Equations

In this paper, a singularly perturbed equation of the first order is considered, the concept of a region of attraction (DO) of the solution of a singularly perturbed equation to the solution of an unperturbed equation is introduced, and the existence of an OA is proved. The problem has been set about the possibility of expanding the areas of attraction of VCA solutions. It has been proven that if there is a region of attraction, then it can be expanded to the boundary of the region under consideration. In the proof, geometric constructions were used, using the level line of conjugate-harmonic functions, the method of successive approximations and methods of asymptotic estimates.

Asymptotical convergence of solutions of boundary value problems for singularly perturbed higher-order integro-differential equations

The article considers a two-point boundary value problem for a linear integro-differential equation of the n + m order with small parameters for m higher derivatives, provided that the roots of the additional characteristic equation are negative. The aim of the study is to obtain asymptotic estimates of the solution, to find out the asymptotic behavior of solutions in the vicinity of points where additional conditions are set, as well as to construct a degenerate problem, the solution of which tend to the solution of the initial perturbed boundary value problem. The Cauchy function and boundary functions of a boundary value problem for a singularly perturbed homogeneous differential equation are constructed, and their asymptotic estimates are obtained. Using the Cauchy functions and boundary functions, an analytical formula for solutions to the boundary value problem is obtained. A theorem on an asymptotic estimate for the solution of the considered boundary value problem is proved. The asymptotic behavior of the solution with respect to a small parameter and the order of growth of its derivatives are established. It is shown that the solution of the boundary value problem under consideration at the left end of this segment has the phenomenon of an initial jump and the order of this jump is determined. A modified degenerate boundary value problem containing initial jumps of the solution and the integral term is constructed.

Volume density maps of the 862 nm DIB carrier and interstellar dust. Hints on the role of carbon-rich ejecta from AGB stars

The carbonaceous macromolecules imprinting the numerous absorptions called diffuse interstellar bands (DIBs) in astronomical spectra are omnipresent in the Galaxy and beyond. They represent a considerable reservoir of organic matter. However, their chemical formulae, formation, and destruction sites remain unknown. Their spatial distribution and the local relation to other interstellar species is key to tracing their role in the lifecycle of organic matter. Volume density maps bring local instead of line-of-sight distributed information and allow for new diagnostics to be captured. We present the first large-scale volume (3D) density map of a DIB carrier and compare it with an equivalent map of interstellar dust. The DIB carrier map was obtained through hierarchical inversion of sim 202,000 measurements of the 8621 nm DIB obtained with the Gaia -RVS instrument. It covers about 4000 pc around the Sun in the Galactic plane. We built a dedicated interstellar dust map based on the extinction towards the same target stars. At the simeq 50 pc resolution of the maps, the shape of the 3D DIB distribution is found to be remarkably similar to the 3D distribution of dust. On the other hand, the DIB-to-dust local density ratio increases in low-dust areas. It is also increasing away from the disk, however, the minimum ratio is found to be shifted above the Galactic plane to Z=simeq +50pc. Finally, the average ratio is also surprisingly found to increase away from the Galactic Center. We suggest that the three latter trends may be indications of a dominant contribution of material from the carbon-rich category of dying giant stars to the formation of the carriers. Our suggestion is based on recent catalogs of asymptotic giant branch (AGB) stars and estimates of the mass fluxes of their C-rich and O-rich ejecta.