Limit Theorem Research Articles

Correcting misinterpretations on the distribution of feasible solution lengths in the traveling salesman problem

The traveling salesman problem (TSP) is the canonical combinatorial optimization problem famous throughout literature. There exists an objective function associated with every feasible solution. However, the increase in the number of possible solutions makes this an NP-Hard problem. We show that the central limit theorem (CLT) applies to the problem. We then conduct extensive computational testing to show that the cycle lengths tend to a normal distribution as the problem grows large. When the size of the TSP problem exceeds computational power, better understanding solution distributions allows us to save resources. This is a non-trivial result as understanding solution distributions in huge TSP problems helps us to minimize computational effort that may not lead to significantly better results.

COINTEGRATING POLYNOMIAL REGRESSIONS: ROBUSTNESS OF FULLY MODIFIED OLS

Cointegrating polynomial regressions (CPRs) include deterministic variables, integrated variables, and their powers as explanatory variables. Based on a novel kernel-weighted limit result and a novel functional central limit theorem, this paper shows that the fully modified ordinary least squares (FM-OLS) estimator of Phillips and Hansen (1990, Review of Economic Studies 57, 99–125) is robust to being used in CPRs. Being used in CPRs refers to a widespread empirical practice that treats the integrated variables and their powers, incorrectly, as a vector of integrated variables and uses textbook FM-OLS. Robustness means that this “formal” FM-OLS practice leads to a zero mean Gaussian mixture limiting distribution that coincides with the limiting distribution of the Wagner and Hong (2016, Econometric Theory 32, 1289–1315) application of the FM estimation principle to the CPR case. The only restriction for this result to hold is that all integrated variables to power one are included as regressors. Even though simulation results indicate performance advantages of the Wagner and Hong (2016, Econometric Theory 32, 1289–1315) estimator, partly even in large samples, the results of the paper give an asymptotic foundation to “formal” FM-OLS and thus enlarge the usability of the Phillips and Hansen (1990, Review of Economic Studies 57, 99–125) estimator implemented in many software packages.

Stable laws for random dynamical systems

Abstract In this paper, we consider random dynamical systems formed by concatenating maps acting on the unit interval $[0,1]$ in an independent and identically distributed (i.i.d.) fashion. Considered as a stationary Markov process, the random dynamical system possesses a unique stationary measure $\nu $ . We consider a class of non-square-integrable observables $\phi $ , mostly of form $\phi (x)=d(x,x_0)^{-{1}/{\alpha }}$ , where $x_0$ is a non-recurrent point (in particular a non-periodic point) satisfying some other genericity conditions and, more generally, regularly varying observables with index $\alpha \in (0,2)$ . The two types of maps we concatenate are a class of piecewise $C^2$ expanding maps and a class of intermittent maps possessing an indifferent fixed point at the origin. Under conditions on the dynamics and $\alpha $ , we establish Poisson limit laws, convergence of scaled Birkhoff sums to a stable limit law, and functional stable limit laws in both the annealed and quenched case. The scaling constants for the limit laws for almost every quenched realization are the same as those of the annealed case and determined by $\nu $ . This is in contrast to the scalings in quenched central limit theorems where the centering constants depend in a critical way upon the realization and are not the same for almost every realization.

Size-dependence of fracture processes in intact rocks

In brittle heterogeneous materials, the ultimate stress carries information on the material’s load-bearing capacity and, for this reason, this stress is an important engineering quantity. However, this stress is just a point in the sequence of stress–strain responses, each of them reflects the current state of damage in the material. Initially, isolated cracks grow, sense each other, and form multiple crack networks. The networks enable dilatational shear and facilitate further growth of damage. Large samples contain a broad spectrum of the crack-enabling defects and, therefore, the sample size becomes an irrelevant factor. However, when the defect population is limited, the damage process is constrained and, as a result, the peak stress is rising. According to the central limit theorem, the defect’s propensity for generating cracks should obey the rules of normal distribution. It turns out that the normal distribution of defects becomes a Weibull-like distribution of strength, where shape of the distribution is affected by the sample size and, also, is sensitive to hydrostatic pressure. The size-dependence of fracture processes is the main objective of the study.

A Non-Symmetric Kesten Criterion and Ratio Limit Theorem for Random Walks on Amenable Groups

Abstract We consider random walks on countable groups. A celebrated result of Kesten says that the spectral radius of a symmetric walk (whose support generates the group as a semigroup) is equal to one if and only if the group is amenable. We give an analogue of this result for walks that are not symmetric. We also conclude a ratio limit theorem for amenable groups.

Statistical analysis of irregularly spaced spatial data in frequency domain

Central limit theorems (CLTs) for frequency‐domain statistics are fundamental tools in frequency‐domain analysis. However, for irregularly spaced data, they are still limited. In both the pure increasing domain and the mixed increasing domain asymptotic frameworks, three CLTs of frequency‐domain statistics are established for the observations at uniformly distributed sampling locations over a rectangular sampling region. One is for discrete Fourier transforms (DFTs), while the other two pertain to generalized spectral means (GSMs). The asymptotic joint normality and independence of the DFT at any finite number of standard frequencies are derived. Additionally, the asymptotic normalities of two GSMs are set up, with asymptotic variances given in different forms, according to the Gaussian or non‐Gaussian model assumption. Three established CLTs are very useful in investigating the sampling properties of many important frequency‐domain statistics, such as periodogram, non‐negative definite auto‐covariance estimator, spectral density estimator, and Whittle likelihood estimator as well.

Limit Theorems for the Cubic Mean-Field Ising Model

AbstractWe study a mean-field spin model with three- and two-body interactions. The equilibrium measure for large volumes is shown to have three pure states, the phases of the model. They include the two with opposite magnetization and an unpolarized one with zero magnetization, merging at the critical point. We prove that the central limit theorem holds for a suitably rescaled magnetization, while its violation with the typical quartic behavior appears at the critical point.

Fog, Friction, and Failure in Organized Conflict: A Formal Study

Organized conflict, while confined by the laws of physics—and, under profound strategic incompetence, by the Lanchester equations—is not a physical process but rather an extended exchange between cognitive entities that have been shaped by path-dependent historical trajectories and cultural traditions. Cognition itself is confined by the necessity of duality, with an underlying information source constrained by the asymptotic limit theorems of information and control theories. We introduce the concept of a ‘basic underlying probability distribution’ characteristic of the particular cognitive process studied. The dynamic behavior of such systems is profoundly different for ‘thin-tailed’ and ‘fat-tailed’ distributions. The perspective permits the construction of new probability models that may provide useful statistical tools for the analysis of observational and experimental data associated with organized conflict, and, in some measure, for its management.

On Rio’s proof of limit theorems for dependent random fields

This paper presents an exposition of Rio’s proof of the strong law of large numbers and extends his method to random fields. In addition to considering the rate of convergence in the Marcinkiewicz–Zygmund strong law of large numbers, we go a step further by establishing (i) the Hsu–Robbins–Erdös–Spitzer–Baum–Katz theorem, (ii) the Feller weak law of large numbers, and (iii) the Pyke–Root theorem on mean convergence for dependent random fields. These results significantly improve several particular cases in the literature. The proof is based on new maximal inequalities that hold for random fields satisfying a very general dependence structure.

Estimates of constants in the limit theorems for chaotic dynamical systems

In a vast area of probabilistic limit theorems for dynamical systems with chaotic behaviors always only functional form (exponential, power, etc.) of the asymptotic laws and of convergence rates were studied. However, for basically all applications, e.g., for computer simulations, development of algorithms to study chaotic dynamical systems numerically, as well as for design and analysis of real (e.g., in physics) experiments, the exact values (or at least estimates) of constants (parameters) of the functions, which appear in the asymptotic laws and rates of convergence, are of primary interest. In this paper, we provide such estimates of constants (parameters) in the central limit theorem, large deviations principle, law of large numbers and the rate of correlations decay for strongly chaotic dynamical systems.

The divide-and-conquer sequential Monte Carlo algorithm: Theoretical properties and limit theorems

The divide-and-conquer sequential Monte Carlo algorithm: Theoretical properties and limit theorems

Limit theorems for Fréchet mean sets

Limit theorems for Fréchet mean sets

Central limit theorem and near classical Berry-Esseen rate for self normalized sums in high dimensions

Central limit theorem and near classical Berry-Esseen rate for self normalized sums in high dimensions

Entrywise limit theorems for eigenvectors of signal-plus-noise matrix models with weak signals

Entrywise limit theorems for eigenvectors of signal-plus-noise matrix models with weak signals

Gaussian fluctuations of the elephant random walk with gradually increasing memory

The elephant random walk (ERW) is a discrete-time random walk introduced by Schütz and Trimper (2004) in order to investigate how long-range memory affects the behavior of the random walk. Its particularity is that the next step of the walker depends on its whole past through a parameter p∈[0,1] . In this work, we investigate the validity of the central limit theorem of the ERW when the walker has only a gradually increasing memory. Our contribution provides a positive answer to a conjecture raised in a recent work by Gut and Stadtmüller (2022 Stat. Probab. Lett. 189 109598).

Galton–Watson Theta-Processes in a Varying Environment

Abstract We consider a special class of Galton–Watson theta-processes in a varying environment fully defined by four parameters, with two of them ( θ , r ) {(\theta,r)} being fixed over time n, and the other two ( a n , c n ) {(a_{n},c_{n})} characterizing the altering reproduction laws. We establish a sequence of transparent limit theorems for the theta-processes with possibly defective reproduction laws. These results may serve as a stepping stone towards incisive general results for the Galton–Watson processes in a varying environment.

Sanov-type large deviations and conditional limit theorems for high-dimensional Orlicz balls

In this paper, we prove a Sanov-type large deviation principle for the sequence of empirical measures of vectors chosen uniformly at random from an Orlicz ball. From this level-2 large deviation result, in a combination with Gibbs conditioning, entropy maximization and an Orlicz version of the Poincaré-Maxwell-Borel lemma, we deduce a conditional limit theorem for high-dimensional Orlicz balls. In more geometric parlance, the latter shows that if V1 and V2 are Orlicz functions, then random points in the V1-Orlicz ball, conditioned on having a small V2-Orlicz radius, look like an appropriately scaled V2-Orlicz ball. In fact, we show that the limiting distribution in our Poincaré-Maxwell-Borel lemma, and thus the geometric interpretation, undergoes a phase transition depending on the magnitude of the V2-Orlicz radius.

Plasticity-based evaluation method for in-plane shear strength of steel-plate composite element

The steel-plate composite (SC) wall is considered an attractive structural type for the next generation nuclear power plants because of its efficient construction schedule, radiation shielding function, and excellent mechanical properties. During seismic action, the SC walls sustain large in-plane and out-of-plane loadings. So, it is essential for the evaluation of the in-plane and out-of-plane load carrying capacity of SC walls in design. Up to date, several equations that are developed based on elastic theory are proposed and adopted by relevant design codes. In the previous study by the authors, a lower bound equation was present for evaluation of the in-plane shear strength of SC elements based on the lower limit theorem of plastic limit theory. As a counterpart to that paper, an upper bound equation for evaluation of in-plane shear strength of SC elements based on the upper limit theorem of plastic limit theory was developed, and its minimum value solution which is the closest one to the exact solution was derived. Meanwhile, considering that concrete is not a perfectly plastic material, another equation is present to prevent the concrete between the steel faceplates from crushing prior to the steel yielding. A comparison between the results predicted by the proposed equation in this study and the test results of the SC panel specimens suggested the adequacy and accuracy of the proposed equation. In addition, comparisons of results evaluated by the proposed equations and the equations in codes JEAG 4618 and AISC N690 were also performed.

ARE UNOBSERVABLES SEPARABLE?

It is common to assume in empirical research that observables and unobservables are additively separable, especially when the former are endogenous. This is because it is widely recognized that identification and estimation challenges arise when interactions between the two are allowed for. Starting from a nonseparable IV model, where the instrumental variable is independent of unobservables, we develop a novel nonparametric test of separability of unobservables. The large-sample distribution of the test statistics is nonstandard and relies on a Donsker-type central limit theorem for the empirical distribution of nonparametric IV residuals, which may be of independent interest. Using a dataset drawn from the 2015 U.S. Consumer Expenditure Survey, we find that the test rejects the separability in Engel curves for some commodities.

Gauge-Invariant Quantum Thermodynamics: Consequences for the First Law.

The universality of classical thermodynamics rests on the central limit theorem, due to which, measurements of thermal fluctuations are unable to reveal detailed information regarding the microscopic structure of a macroscopic body. When small systems are considered and fluctuations become important, thermodynamic quantities can be understood in the context of classical stochastic mechanics. A fundamental assumption behind thermodynamics is therefore that of coarse graining, which stems from a substantial lack of control over all degrees of freedom. However, when quantum systems are concerned, one claims a high level of control. As a consequence, information theory plays a major role in the identification of thermodynamic functions. Here, drawing from the concept of gauge symmetry-essential in all modern physical theories-we put forward a new possible intermediate route. Working within the realm of quantum thermodynamics, we explicitly construct physically motivated gauge transformations which encode a gentle variant of coarse graining behind thermodynamics. As a first application of this new framework, we reinterpret quantum work and heat, as well as the role of quantum coherence.