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The invariance principle and Berry–Esseen bound for an error variance estimator based on the residuals are established by using a Taylor expansion and the classical invariance principle and Berry–Esseen bound for independent random variables. Some examples are given to illustrate their applications. [ABSTRACT FROM AUTHOR]

A class of partial sum processes based on a sequence of observations having the structure of finite-order moving averages is studied. The random component of this sequence is formed using a heterogeneous process in discrete time, while the non-random component is formed using a regularly varying function at infinity. The heterogeneous process with discrete time is defined as a power transform of partial sums of a certain stationary sequence. An approximation of the random processes from the above-mentioned class is studied by random processes defined as the convolution of a power transform of the fractional Brownian motion with a power function. Sufficient conditions for -convergence in the Donsker invariance principle are obtained. [ABSTRACT FROM AUTHOR]

Let d be a positive integer, and let A be a set in Z d that contains finitely many points with integer coordinates. We consider a standard random walk X perturbed on the set A. This means that X is a Markov chain whose transition probabilities from the points outside A coincide with those of a standard random walk on Z d , whereas the transition probabilities from the points inside A are different. We investigate the impact of the perturbation on a scaling limit of X. It turns out that if d ⩾ 2, then in a typical situation the scaling limit of X coincides with that of the underlying standard random walk. This is unlike the case d = 1, in which the scaling limit of X is usually a skew Brownian motion, a skew stable Lévy process, or some other "skew" process. The distinction between the one-dimensional and multidimensional cases under comparable assumptions may simply be caused by transience of the underlying standard random walk in Z d for d ⩾ 3. More interestingly, in the situation where the standard random walk in Z 2 is recurrent, the preservation of its Donsker scaling limit is secured by the fact that the number of visits of X to the set A is much smaller than in the one-dimensional case. As a consequence, the influence of the perturbation vanishes upon the scaling. On the other edge of the spectrum, we have the situation in which the standard random walk admits a Donsker's scaling limit, whereas its locally perturbed version does not because of huge jumps from the set A, which occur early enough. [ABSTRACT FROM AUTHOR]

We study a counterbalanced random walk S ˇ n = X ˇ 1 + ⋯ + X ˇ n , which is a discrete time non-Markovian process and X ˇ n are given recursively as follows. For n ≥ 2, X ˇ n is a new independent sample from some fixed law μ ≠ 0 with a fixed probability p, and X ˇ n = − X ˇ v (n) with probability 1 − p, where v(n) is a uniform random variable on {1, ⋯, n − 1}. We apply martingale method to obtain a strong invariance principle for Šn. [ABSTRACT FROM AUTHOR]

The G-Brownian-motion-driven stochastic differential equations (G-SDEs) as well as the G-expectation, which were seminally proposed by Peng and his colleagues, have been extensively applied to describing a particular kind of uncertainty arising in real-world systems modeling. Mathematically depicting long-time and limit behaviors of the solution produced by G-SDEs is beneficial to understanding the mechanisms of system's evolution. Here, we develop a new G-semimartingale convergence theorem and further establish a new invariance principle for investigating the long-time behaviors emergent in G-SDEs. We also validate the uniqueness and the global existence of the solution of G-SDEs whose vector fields are only locally Lipschitzian with a linear upper bound. To demonstrate the broad applicability of our analytically established results, we investigate its application to achieving G-stochastic control in a few representative dynamical systems. [ABSTRACT FROM AUTHOR]

Copyright of Studies in Hispanic & Lusophone Linguistics is the property of De Gruyter and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we consider the convergence rate with respect to Wasserstein distance in the invariance principle for deterministic non-uniformly hyperbolic systems. Our results apply to uniformly hyperbolic systems and large classes of non-uniformly hyperbolic systems including intermittent maps, Viana maps, unimodal maps and others. Furthermore, as a non-trivial application to the homogenization problem, we investigate the Wasserstein convergence rate of a fast–slow discrete deterministic system to a stochastic differential equation. [ABSTRACT FROM AUTHOR]

In the form of a scholarly exchange with ChatGPT, we cover fundamentals of modeling stochastic dependence with copulas. The conversation is aimed at a broad audience and provides a light introduction to the topic of copula modeling, a field of potential relevance in all areas where more than one random variable appears in the modeling process. Topics covered include the definition, Sklar's theorem, the invariance principle, pseudo-observations, tail dependence and stochastic representations. The conversation also shows to what degree it can be useful (or not) to learn about such concepts by interacting with the current version of a chatbot. [ABSTRACT FROM AUTHOR]

This paper is devoted to an invariance principle for Kemperman's model of oscillating random walk on Z. This result appears as an extension of the invariance principal theorem for classical random walks on Z or reflected random walks on N0. Relying on some natural Markov sub-process which takes into account the oscillation of the random walks between Z and Z+, we first construct an aperiodic sequence of renewal operators acting on a suitable Banach space and then apply a powerful theorem proved by S. Gouëzel. [ABSTRACT FROM AUTHOR]

A given spatial distance can be measured using either a length or a time unit. A spatial-choice problem with given distances represented in a space frame and in a time frame is logically equivalent. Different representations of the same choice problem should yield the same preferences according to the invariance principle. To test invariance in the context of spatial choice, we used a constant velocity to construct six spatial (intertemporal) choice problems with single-placed (single-dated) outcomes in a space (time) frame and investigated whether invariance-violating behaviour could be detected under the two frames (Studies 1–3). If such behaviour existed, we then aimed to determine whether two models—the attribute-comparison model, which predicts a framing effect occurs if framing shifts people's judgement of the inequality relationship between the differences of two options in time/space dimension and that in outcome dimension, and the utility comparison model, which predicts that a framing effect occurs if framing shifts people's judgement of the inequality relationship between two options' overall utility—could account for it, and which of these two models is better supported by the data (Studies 2–3). The results indicate that a time–space framing effect existed, as people's preference orders were significantly changed by the different descriptions of spatial-choice problems, and this new time–space framing effect could be satisfactorily explained by the attribute-comparison rather than the utility comparison model. Our findings could support the creation of new forms of choice architecture that improve decisions about health, wealth, and happiness. [ABSTRACT FROM AUTHOR]

Several authors have studied convergence in distribution to the Brownian web under diffusive scaling of systems of Markovian random walks. In a paper by R. Roy, K. Saha and A. Sarkar, convergence to the Brownian web is proved for a system of coalescing random paths—the random directed forest—which are not Markovian. Paths in the random directed forest do not cross each other before coalescence. Here, we study a variation of the random directed forest where paths can cross each other and prove convergence to the Brownian web. This provides an example of how the techniques to prove convergence to the Brownian web for systems allowing crossings can be applied to non-Markovian systems. [ABSTRACT FROM AUTHOR]

We study moderate deviations and convergence rates for the optimal values and optimal solutions of sample average approximations. Firstly, we give an extension of the Delta method in large deviations. Then under Lipschitz continuity on the objective function, we establish a moderate deviation principle for the optimal value by the Delta method. When the objective function is twice continuously differentiable and the optimal solution of true optimization problem is unique, we obtain a moderate deviation principle for the optimal solution and a Cramér-type moderate deviation for the optimal value. Motivated by the Donsker invariance principle, we consider a functional form of stochastic programming problem and establish a Donsker invariance principle, a functional moderate deviation principle, and a Strassen invariance principle for the optimal value. [ABSTRACT FROM AUTHOR]

The segmentation of data into stationary stretches, also known as the multiple change point problem, is important for many applications in time series analysis and signal processing. Based on strong invariance principles, we analyze a data segmentation methodology using moving sum statistics for a class of regime-switching multivariate processes, where each switch results in a change in the drift. In particular, this framework includes the data segmentation of multivariate partial sum, integrated diffusion, and renewal processes, even if the distance between the change points is sublinear. We study the asymptotic behavior of the corresponding change point estimators, show their consistency, and derive the corresponding localization rates, which are minimax optimal in a variety of situations, including an unbounded number of changes in Wiener processes with drift. Furthermore, we derive the limit distribution of the change point estimators for local changes. This result can, in principle, be used to derive confidence intervals for the change points. [ABSTRACT FROM AUTHOR]

Given a random walk (S n) with typical step distributed according to some fixed law and a fixed parameter p ∈ (0 , 1) , the associated positively step-reinforced random walk is a discrete-time process which performs at each step, with probability 1 - p , the same step as (S n) while with probability p, it repeats one of the steps it performed previously chosen uniformly at random. The negatively step-reinforced random walk follows the same dynamics but when a step is repeated its sign is also changed. In this work, we shall prove functional limit theorems for the triplet of a random walk, coupled with its positive and negative reinforced versions when p < 1 / 2 and when the typical step is centred. The limiting process is Gaussian and admits a simple representation in terms of stochastic integrals, B (t) , t p ∫ 0 t s - p d B r (s) , t - p ∫ 0 t s p d B c (s) t ∈ R + for properly correlated Brownian motions B , B r , B c . The processes in the second and third coordinate are called the noise reinforced Brownian motion (as named in [1]), and the noise counterbalanced Brownian motion of B. Different couplings are also considered, allowing us in some cases to drop the centredness hypothesis and to completely identify for all regimes p ∈ (0 , 1) the limiting behaviour of step reinforced random walks. Our method exhausts a martingale approach in conjunction with the martingale functional CLT. [ABSTRACT FROM AUTHOR]

This paper is concerned with the uniform CLT for an empirical estimator of countable state space semi-Markov processes indexed by functions under the uniformly integrable entropy condition. The results of this paper generalise those of Limnios [(2004), 'A Functional Central Limit Theorem for the Empirical Estimator of a Semi-Markov Kernel', Journal of Nonparametric Statistics, 16(1–2), 13–18. The International Conference on Recent Trends and Directions in Nonparametric Statistics]. We apply our results to the following important topics: on the kernel-type estimator of the semi-Markov kernel, the weak convergence of the smoothed empirical semi-Markov processes, we introduce the copula estimators in the semi-Markov framework and we establish their limiting laws by using the functional delta methods. We prove the main results for the uniform CLT via a martingale difference sequence as in Bae et al. [(2010), 'The Uniform CLT for Martingale Difference Arrays Under the Uniformly Integrable Entropy', Bulletin of the Korean Mathematical Society, 47(1), 39–51]. [ABSTRACT FROM AUTHOR]

Two widely used boundary conditions at the interface between the free fluid and porous medium, prescribing either continuity of the normal stress or continuity of the pressure, are examined. It is shown, that the pressure continuity condition satisfies the principle of invariance with respect to the choice of coordinate system and has a clear physical meaning. For the problem of a 2D viscous flow of fluid around a porous body in a periodic Kuwabara cell, an analytic solution is obtained for both conditions within the framework of the macroscopic Stokes–Darcy model. The results obtained using the analytic solution are compared against the numerical results from the microscopic Stokes model. It is shown, that the condition of the continuity of pressure is more consistent with the microscopic Stokes model compared to the condition of continuity of the normal stress. [ABSTRACT FROM AUTHOR]

In this paper, we study the weak invariance principle for stationary ortho-martingales with values in 2-smooth or cotype 2 Banach spaces. Then, with the help of a suitable maximal ortho-martingale approximation, we derive the weak invariance principle for stationary random fields in L p , 1 ≤ p ≤ 2 , under a condition in the spirit of Hannan. As an application, we get an asymptotic result for the L p -distances (1 ≤ p ≤ 2) between the common distribution function and the corresponding empirical distribution function of stationary random fields. [ABSTRACT FROM AUTHOR]

Consider a stochastic process X , regenerative at a state x which is instantaneous and regular. Let L be a regenerative local time for X at x. Suppose furthermore that X can be approximated by discrete time regenerative processes X n for which x is accessible. We give conditions on X and X n so that the naturally defined local time of X n (which counts the quantity of visits to x) converges weakly to L. This limit theorem generalizes previous invariance principles that have appeared in the literature. Furthermore, it allows one to prove novel invariance principles for local times of regenerative processes converging to diffusions or to solutions of SDEs with jumps. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of systems with the Coulomb friction expressed as Lagrange's equations of the second kind with solutions in the sense of Filippov. The friction coefficients are assumed to have dependence on generalized states, velocities, and time which can arise for various reasons such as temperature change, surface roughness, and other characteristics of the rubbing bodies. Combining the direct method of Lyapunov functions with semidefinite derivatives and the method of limit equations stemming from the works of Sell (1967) and Artstein (1977, 1978) on the topological dynamics of nonautonomous systems, we construct the limit differential inclusions by the methods of set-valued analysis and the theory of discontinuous systems. [ABSTRACT FROM AUTHOR]

We investigate the invariance properties of a class of switched systems where the value of a switching signal determines the current mode of operation (among a finite number of them) and, for each fixed mode, its dynamics are described by a Differential-Algebraic Equation (DAE). Motivated by the lack of invariance principles of switched DAE systems, we develop such principles for switched DAE systems under arbitrary and dwell-time switching. By obtaining a hybrid system model that describes the switched DAE system, we build from invariance results, for hybrid systems, the invariance principles for such switched systems. Examples are included to illustrate the results.

In this paper, we study the central limit theorem (CLT) and its weak invariance principle (WIP) for sums of stationary random fields non-necessarily adapted, under different normalizations. To do so, we first state sufficient conditions for the validity of a suitable ortho-martingale approximation. Then, with the help of this approximation, we derive projective criteria under which the CLT as well as the WIP hold. These projective criteria are in the spirit of Hannan's condition and are well adapted to linear random fields with ortho-martingale innovations and which exhibit long memory. [ABSTRACT FROM AUTHOR]

Copyright of Automatisierungstechnik is the property of De Gruyter and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Sports economics is now 65 years old, since Rottenberg's "The Baseball Players' Labor Market". This milestone coincides with the 20th anniversary of the Journal of Sports Economics. Rottenberg's two main offerings, the match outcome uncertainty hypothesis and the invariance principle, are reviewed, including empirical verdicts from other recent literature reviews. In addition, perhaps less well-known issues in the literature suggest further investigation of these crucial hypotheses, and others that arise. Finally, the place of the Journal of Sports Economics in the evolution of sports economics is offered. [ABSTRACT FROM AUTHOR]

We study ballisticity conditions for d-dimensional random walks in strong mixing environments, with underlying dimension d ≥ 2 . Specifically, we introduce an effective polynomial condition similar to that given by Berger et al. (Comm. Pure Appl. Math. 77:1947–1973, 2014). In a mixing setup we prove that this condition implies the corresponding stretched exponential decay, and obtain an annealed functional central limit theorem for the random walk process centered at the limiting velocity. This paper complements previous work of Guerra (Ann. Probab. 47:3003–3054, 2019) and completes the answer about the meaning of condition (T ′) | ℓ in a mixing setting, an open question posed by Comets and Zeitouni (Ann. Probab. 32:880–914, 2004). [ABSTRACT FROM AUTHOR]

Invariance properties and convergence of solutions of set dynamical systems are studied. Using a framework for systems with set-valued states, notions of stability and detectability, similar to the existing results for classical dynamical systems, are defined and used to obtain information about the convergence properties of solutions. In particular, it is shown that local stability, detectability, and boundedness can be combined to conclude convergence of set-valued solutions. Under the assumption of bounded solutions and outer semicontinuity of the set-valued maps that define the system's dynamics, invariance properties for set dynamical systems are also presented along with an invariance principle. The invariance principle involves the use of Lyapunov-like functions to locate invariant sets. Examples illustrate the results.

We explore new connections between the dynamics of conservative partially hyperbolic systems and the geometric measure-theoretic properties of their invariant foliations. Our methods are applied to two main classes of volume-preserving diffeomorphisms: fibered partially hyperbolic diffeomorphisms and center-fixing partially hyperbolic systems. When the center is one-dimensional, assuming the diffeomorphism is accessible, we prove that the disintegration of the volume measure along the center foliation is either atomic or Lebesgue. Moreover, the latter case is rigid in dimension three (this does not require accessibility): the center foliation is actually smooth and the diffeomorphism is smoothly conjugate to an explicit rigid model. A partial extension to fibered partially hyperbolic systems with compact fibers of any dimension is also obtained. A common feature of these classes of diffeomorphisms is that the center leaves either are compact or can be made compact by taking an appropriate dynamically defined quotient. For volume-preserving partially hyperbolic diffeomorphisms whose center foliation is absolutely continuous, if the generic center leaf is a circle, then every center leaf is compact. [ABSTRACT FROM AUTHOR]

The presence of symmetries in physical theories implies a pernicious form of underdetermination. In order to avoid this theoretical vice, philosophers often espouse a principle called Leibniz Equivalence, which states that symmetry-related models represent the same state of affairs. Moreover, philosophers have claimed that the existence of non-trivial symmetries motivates us to accept the Invariance Principle, which states that quantities that vary under a theory's symmetries aren't physically real. Leibniz Equivalence and the Invariance Principle are often seen as part of the same package. I argue that this is a mistake: Leibniz Equivalence and the Invariance Principle are orthogonal to each other. This means that it is possible to hold that symmetry-related models represent the same state of affairs whilst having a realist attitude towards variant quantities. Various arguments have been presented in favour of the Invariance Principle: a rejection of the Invariance Principle is inter alia supposed to cause indeterminism, undetectability or failure of reference. I respond that these arguments at best support Leibniz Equivalence. [ABSTRACT FROM AUTHOR]

The asymptotic behavior of systems with Coulomb friction represented as Lagrange's equations of the second kind is investigated. Lyapunov's direct method is used in combination with the method of limiting equations, which goes back to the works by G.R. Sell (1967) and Z. Artstein (1977, 1978) on topological dynamics of nonautonomous systems. The results generalize LaSalle's principle of invariance. [ABSTRACT FROM AUTHOR]

Using the randomization approach, introduced by A. Tempelman in Randomized multivariate central limit theorems for ergodic homogeneous random fields, Stochastic Processes and their Applications. 143 (2022), 89–105, we prove: (a) a randomized version of the invariance principle (the functional CLT); (b) a version the Glivenko–Cantelli theorem; (c) a randomized theorem about convergence of empirical processes to the Brownian bridge. We also weaken the moment condition in the randomized CLTs, proved in the mentioned article. The main point of our work is that most of our theorems are valid for all ergodic homogeneous random fields on Z m and R m , m ≥ 1. [ABSTRACT FROM AUTHOR]

Ferromagnetic materials show magnetic structures with domains and walls. Thanks to decades of research regarding the origin and behaviour of magnetic domains, we now possess a general foundation which has been verified experimentally in single crystals and powders. The governing equations at the microscopic scale were built in the 1960 s when Brown published calculations of the magnetic moments distribution inside domain walls. This micromagnetic theory uses the so called LLG ' L andau- L ifshitz- G ilbert' equation and can include the damping effects. The LLG equation requires a coupling to the field derived from energy contributions: exchange, anisotropy, magnetostriction, stray-field ... and the anti-eddy field. At the macroscopic scale, such behaviours are lumped in a homogenized magnetization law for the electromagnetic field equations inside larger polycrystals. Therefore, the inhomogeneous magnetic material nature is always ignored. T he T ensor M agnetic P hase T heory (TMPT) describes the magnetic structure and the magnetization thanks to a tensor variable at a mesoscopic scale. The material structuring is then explained thanks to an energy balance which will be discussed. This paper presents the results of investigations on entangled relationships between the micromagnetic theory, the electromagnetic theory and what is called the Tensor Magnetic Phase Theory (TMPT), which statistically describes the magnetic structures of soft ferromagnetic materials. It examines a connection between the TMPT and the LLG by deriving the main energy terms. Then, the TMPT must stay compatible and coupled to the Maxwell equations at low frequencies with volume and surface connections. Additionally, this paper investigates the way to derive the domains structuring and magnetization laws through the Lagrange principle and the corresponding conservation laws with invariants linked to the Nœther theorem. Finally, the TMPT must be discussed while checking its coherence and formulation when changing the reference frame. [ABSTRACT FROM AUTHOR]

The system \begin{equation*} \ddot{q}_k+\gamma \dot{q}_k=V'(q_{k+1}-q_k)-V'(q_k-q_{k-1})\qquad (k=1,\ldots,N-2) \end{equation*} is considered, where $00$ (fixed endpoints – this is the original Fermi–Pasta–Ulam oscillator provided that the damping coefficient $\gamma$ equals zero); $q_1(t)-q_0(t)= L/(N-1)$, $q_{N-1}(t)-q_{N-2}(t)= L/(N-1)$ (free endpoints); $q_0(t)=-(K-q_{N-2}(t))$, $q_{N-1}(t)=q_1(t)+K$, $K=\hbox{const.}$ (cycle). We prove that the unique equilibrium state of the system with fixed endpoints is asymptotically stable. We also prove that the system with free endpoints and the cycle asymptotically stop at an equilibrium state along their arbitrary motion, i.e., for every motion there is $q_1^\infty\in\mathbb{R}$ such that $\lim_{t\to\infty}q_k(t)=q_1^\infty+(k-1)\overline{r}$, $\lim_{t\to\infty}\dot q_k(t)=0$ $(k=1,\ldots,N-2)$, where the constant $\overline{r}$ is defined by the equation $V'(\overline{r})=0$.

This article presents a novel controller design to ensure the prescribed performance of a vibrating flexible beam system that possesses boundary output constraints and external disturbances. First, boundary output constraints are visualized as the prescribed performance characteristics, and a smooth performance function along with a proper error transformation is introduced. Second, a boundary constrained controller with a disturbance observer is developed to achieve the present performance, cope with the external disturbance, and stabilize the vibration in the beam system. Third, the principle of invariance is used for rigorous analysis and to ensure the system’s asymptotic stability. Finally, the result of simulation research shows the effectiveness of the suggested approach. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the central limit theorem and the invariance principle for linear processes generated by a new notion of independently and identically distributed (IID) random variables for sub-linear expectations initiated by Peng [19]. It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov's central limit theorem and invariance principle to the case where probability measures are no longer additive. [ABSTRACT FROM AUTHOR]

In the framework of nonadditive probabilities or sublinear expectations, the Kolmogorov's strong law of large numbers (SLLN) states that for a sequence of independent and identically distributed (IID) random variables, limit points of its sample mean quasi-surely fall inside an interval given by a pair of lower and upper means. In this article, we will investigate a cluster set of limit points of a sequence of stochastic processes, which are given by linear interpolating of the sample mean of IID random variables under sublinear expectations, and show an invariance principle. The invariance principle will strengthen the Kolmogorov's SLLN under nonadditive probabilities in some extent. [ABSTRACT FROM AUTHOR]

We establish a quenched local central limit theorem for the dynamic random conductance model on Z d only assuming ergodicity with respect to space-time shifts and a moment condition. As a key analytic ingredient we show Hölder continuity estimates for solutions to the heat equation for discrete finite difference operators in divergence form with time-dependent degenerate weights. The proof is based on De Giorgi's iteration technique. In addition, we also derive a quenched local central limit theorem for the static random conductance model on a class of random graphs with degenerate ergodic weights. [ABSTRACT FROM AUTHOR]

In this paper, an approach to investigate switched affine system via matrix inequalities is presented. Particularly, an extension of LaSalle’s invariance principle for this class of systems under arbitrary dwell-time switching signal is presented. The proposed results employ a common auxiliary scalar function and also multiple auxiliary scalar functions to study the asymptotic behavior of switched solutions and estimate their attractors for any dwell-time switching signal. A specific feature of these results is that the derivative of the auxiliary scalar functions can assume positive values in some bounded sets. Moreover, a problem of constrained optimization is formulated to numerically determine the auxiliary scalar functions and minimize the volume of the estimated attractor. Numerical examples show the potential of the theoretical results in providing information on the asymptotic behavior of solutions of the switched affine systems under arbitrary dwell-time switching signals.