Non-standard Limit Research Articles

Limit analysis and design-oriented approach for out-of-plane loaded masonry walls strengthened by grouted anchors

Masonry structures without box-like behaviour are often prone to the complete or partial out-of-plane failure of their walls under horizontal actions, i.e. mainly the seismic ones. Steel grouted anchors or tie-rods have been widely used in the past to improve connections between orthogonal walls and avoid out-of-plane mechanisms. However, reliable analytical approaches to design such strengthening systems are lacking, whereas the evaluation of their effectiveness is still a matter of ongoing research. The paper presents a force-based design approach to assess the performance of three-wall masonry systems strengthened by grouted anchors, using a frictional macro-block model and a non-standard limit analysis, previously developed for unreinforced masonry structures. The extension of this model to strengthened masonry systems, with specific reference to the use of grouted anchors, allows assessing their horizontal capacity in terms of minimum load multiplier and expected failure. The novel aspects introduced in the updated model are: (1) a simplified closed-form formulation for the calculation of the stabilising moments provided by the grouted anchors at the onset of the failure mechanism, (2) the possibility of placing the grouted anchors with variable inclinations, which may improve their stabilising contribution and optimise the intervention. The effectiveness of the simplified formulation is firstly proved by the comparison with literature results; then, several parametric analyses are carried out to highlight the influence of some variables on the capacity of both the unreinforced and the strengthened three-wall system. Finally, a literature example related to a traditional intervention with steel tie-rods is used to apply the proposed procedure and investigate the performance of different design solutions with innovative grouted anchors.

Fully modified least squares cointegrating parameter estimation in multicointegrated systems

Multicointegration is traditionally defined as a particular long run relationship among variables in a parametric vector autoregressive model that introduces additional cointegrating links between these variables and partial sums of the equilibrium errors. This paper departs from the parametric model, using a semiparametric formulation that reveals the explicit role that singularity of the long run conditional covariance matrix plays in determining multicointegration. The semiparametric framework has the advantage that short run dynamics do not need to be modeled and estimation by standard techniques such as fully modified least squares (FM-OLS) on the original I1 system is straightforward. The paper derives FM-OLS limit theory in the multicointegrated setting, showing how faster rates of convergence are achieved in the direction of singularity and that the limit distribution depends on the distribution of the conditional one-sided long run covariance estimator used in FM-OLS estimation. Wald tests of restrictions on the regression coefficients have nonstandard limit theory which depends on nuisance parameters in general. The usual tests are shown to be conservative when the restrictions are isolated to the directions of singularity and, under certain conditions, are invariant to singularity otherwise. Simulations show that approximations derived in the paper work well in finite samples. The findings are illustrated empirically in an analysis of fiscal sustainability of the US government over the post-war period.

Homogenisation of high-contrast brittle materials

This paper is an overview on some recent results concerning the variational analysis of static fracture in the so-called high-contrast brittle composite materials. The paper is divided into two main parts. The first part is devoted to establish a compactness result for a general class of free-discontinuity functionals with degenerate (or high-contrast) integrands. The second part is focussed on some specific examples which show that the degeneracy of the integrands may lead to non-standard limit effects, which are specific to this high-contrast setting.

Fully Modified Least Squares for Multicointegrated Systems

Multicointegration is traditionally defined as a particular long run relationship among variables in a parametric vector autoregressive model that introduces links between these variables and partial sums of the equilibrium errors. This paper departs from the parametric model, using a semiparametric formulation that reveals the explicit role that singularity of the long run conditional covariance matrix plays in determining multicointegration. The semiparametric framework has the advantage that short run dynamics do not need to be modeled and estimation by standard techniques such as fully modified least squares (FM-OLS) on the original I(1) system is straightforward. The paper derives FM-OLS limit theory in the multicointegrated setting, showing how faster rates of convergence are achieved in the direction of singularity and that the limit distribution depends on the distribution of the conditional one-sided long run covariance estimator used in FM-OLS estimation. Wald tests of restrictions on the regression coefficients have nonstandard limit theory which depends on nuisance parameters in general. The usual tests are shown to be conservative when the restrictions are isolated to the directions of singularity and, under certain conditions, are invariant to singularity otherwise. Simulations show that approximations derived in the paper work well in finite samples. We illustrate our findings by analyzing fiscal sustainability of the US government over the post-war period.

Testing for cointegration in nonlinear asymmetric smooth transition error correction models

The present paper proposes an alternative procedure to test for cointegration in asymmetric exponential smooth transition autoregressive (AESTAR) model. The proposed new F-statistics test the null hypothesis of no cointegration against the alternative hypothesis of cointegration with asymmetric ESTAR adjustment. We derive the nonstandard limit distributions analytically via Monte Carlo simulations. We also study the finite sample properties and the power of proposed tests with Monte Carlo simulations and find that the empirical sizes of two tests are quite close to the nominal one and the proposed tests do not lead to over-rejections of the null hypothesis of no cointegration.

A residual-based multivariate constant correlation test

We propose a new multivariate constant correlation test based on residuals. This test takes into account the whole correlation matrix instead of the considering merely marginal correlations between bivariate data series. In financial markets, it is unrealistic to assume that the marginal variances are constant. This motivates us to develop a constant correlation test which allows for non-constant marginal variances in multivariate time series. However, when the assumption of constant marginal variances is relaxed, it can be shown that the residual effect leads to nonstandard limit distributions of the test statistics based on residual terms. The critical values of the test statistics are not directly available and we use a bootstrap approximation to obtain the corresponding critical values for the test. We also derive the limit distribution of the test statistics based on residuals under the null hypothesis. Monte Carlo simulations show that the test has appealing size and power properties in finite samples. We also apply our test to the stock returns in Euro Stoxx 50 and integrate the test into a binary segmentation algorithm to detect multiple break points.

Identification-Robust Inference on Risk Premia of Mimicking Portfolios of Non-traded Factors

Abstract Mimicking portfolios of economic (non-traded) factors are commonly constructed by projecting the factors on a set of base assets. When these factors are associated with small betas, the beta-estimator using their mimicking portfolios has non-standard limit behavior. This jeopardizes inference on risk premia in the commonly used Fama and MacBeth (1973) two-pass procedure. Using sorting or the average excess returns on the mimicking portfolios to estimate the risk premia leads to similar non-standard behavior. We therefore propose a novel test for the risk premia on mimicking portfolios. Its validity does not depend on the magnitude of the betas. Simulation evidence suggests that it performs well in terms of size and power. We use it to analyze the risk premium on the leverage factor of Adrian, Etula, and Muir (2014). Our results indicate that the leverage factor is a weak factor which leads to substantially different results for its risk premium.

Scaling Limit of Symmetric Random Walk in High-Contrast Periodic Environment

The paper deals with the asymptotic properties of a symmetric random walk in a high contrast periodic medium in $\mathbb Z^d$, $d\geq 1$. We show that under proper diffusive scaling the random walk exhibits a non-standard limit behaviour. In addition to the coordinate of the random walk in $\mathbb Z^d$ we introduce an extra variable that characterizes the position of the random walk in the period and show that this two-component process converges in law to a limit Markov process. The components of the limit process are mutually coupled, thus we cannot expect that the limit behaviour of the coordinate process is Markov. We also prove the convergence in the path space for the said random walk.

TARGETING ESTIMATION OF CCC-GARCH MODELS WITH INFINITE FOURTH MOMENTS

As an alternative to quasi-maximum likelihood, targeting estimation is a much applied estimation method for univariate and multivariate GARCH models. In terms of variance targeting estimation, recent research has pointed out that at least finite fourth moments of the data generating process is required, if one wants to perform inference in GARCH models by relying on asymptotic normality of the estimator. Such moment conditions may not be satisfied in practice for financial returns, highlighting a potential drawback of variance targeting estimation. In this paper, we consider the large-sample properties of the variance targeting estimator for the multivariate extended constant conditional correlation GARCH model when the distribution of the data generating process has infinite fourth moments. Using nonstandard limit theory, we derive new results for the estimator stating that, under suitable conditions, its limiting distribution is multivariate stable. The rate of consistency of the estimator is slower than$\sqrt T$and depends on the tail shape of the data generating process. A simulation study illustrates the derived properties of the targeting estimator.

Numerical limit analysis of steel-reinforced concrete walls and slabs

A limit analysis based design methodology is hereafter proposed and applied for the peak load evaluation of steel-reinforced concrete large-scale prototypes of structural walls and slabs. The methodology makes use of nonstandard limit analysis and predicts the peak load multiplier of the analyzed structures by detecting an upper and a lower bound to it. Some useful hints on the collapse mechanism the structure will exhibit at its limit state is also attainable. To check the reliability of the numerically detected peak loads and failure modes a comparison with experimental laboratory findings, available for the large-scale specimens considered, is presented.

Near equivalence on metric spaces and a nonstandard central limit theorem

This article proves a nonstandard Central Limit Theorem (CLT) in the sense of Nelson’s Radically Elementary Probability Theory. The CLT proved here is obtained by establishing the near equivalence of standardized averages obtained from L2 IID random variables to the standardized average resulting from a binomial CLT. A nonstandard model for near equivalence on metric spaces replaces conventional results of weak convergence. Statements and proofs remain radically elementary without applying the full Internal Set Theory. A nonstandard notion of normality is discussed.

General Limit Distributions for Sums of Random Variables with a Matrix Product Representation

The general limit distributions of the sum of random variables described by a finite matrix product ansatz are characterized. Using a mapping to a Hidden Markov Chain formalism, non-standard limit distributions are obtained, and related to a form of ergodicity breaking in the underlying non-homogeneous Hidden Markov Chain. The link between ergodicity and limit distributions is detailed and used to provide a full algorithmic characterization of the general limit distributions.

Nonstandard limit theorems and large deviations for the Jacobi beta ensemble

In this paper, we show weak convergence of the empirical eigenvalue distribution and of the weighted spectral measure of the Jacobi ensemble, when one or both parameters grow faster than the dimension n. In these cases, the limit measure is given by the Marchenko–Pastur law and the semicircle law, respectively. For the weighted spectral measure, we also prove large deviation principles under this scaling, where the rate functions are those of the other classical ensembles.

3D macro and micro-block models for limit analysis of out-of-plane loaded masonry walls with non-associative Coulomb friction

In this paper, a masonry system composed of a facade wall connected with two sidewalls and subjected to out-of-plane loading is investigated within the framework of three-dimensional limit analysis. Two different modeling approaches, namely macro and micro-block models, are adopted. A rigid-perfectly plastic model with dry contact interfaces governed by Coulomb failure criterion is assumed for masonry walls with regular units and staggering (non-standard limit analysis). Three classes of failure modes are investigated, involving rocking, sliding, twisting failure and combinations of them. The macro-block model is based on the assumption that the failure involves a number of cracks which separate the structure into a few macro-blocks and all the possible relative motions among micro-blocks are concentrated along the cracks. Two limiting conditions for the ultimate load factor are kinematically computed by use of minimization routines. The micro-block model is based on a concave contact formulation in which contact points are located at the corners of interfaces, allowing failure modes involving opening and sliding to be simulated. An iterative solution procedure is used to solve the non-associative friction problem, with second order cone programming (SOCP) used to allow the conic yield function to be solved directly. Both models are validated against experimental outcomes from the literature. A parametric analysis is carried out in order to highlight the influence of each geometrical and mechanical parameter on the prevalence of a mechanism over the other. The presence of an unrestrained horizontal floor system with different orientations is also analyzed.

Statistics of sums of correlated variables described by a matrix product ansatz

We determine the asymptotic distribution of the sum of correlated variables described by a matrix product ansatz with finite matrices, considering variables with finite variances. In cases in which the correlation length is finite, the law of large numbers is obeyed, and the rescaled sum converges to a Gaussian distribution. In contrast, when the correlation extends over the system size, we observe either a breaking of the law of large numbers, with the onset of giant fluctuations, or a generalization of the central limit theorem with a family of non-standard limit distributions. The corresponding distributions are found as mixtures of delta functions for the generalized law of large numbers, and as mixtures of Gaussian distributions for the generalized central limit theorem. Connections with statistical physics models are emphasized.

Peak loads and failure modes of steel-reinforced concrete beams: Predictions by limit analysis

Steel-reinforced concrete beams are analyzed by a plasticity-based approach grounded on nonstandard limit analysis theory. Peak loads and failure modes are predicted by determination of upper and lower bound limits computed with two finite element based numerical procedures. A number of real beams prototypes, belonging to a standard benchmark, are studied and the obtained results are critically discussed outlining possible future developments, strengths and weaknesses of the proposed approach.

Renormalization flow for extreme value statistics of random variables raised to a varying power

Using a renormalization approach, we study the asymptotic limit distribution of the maximum value in a set of independent and identically distributed random variables raised to a power qn that varies monotonically with the sample size n. Under these conditions, a non-standard class of max-stable limit distributions, which mirror the classical ones, emerges. Furthermore, a transition mechanism between the classical and the non-standard limit distributions is brought to light. If qn grows slower than a characteristic function q*n, the standard limit distributions are recovered, while if qn behaves asymptotically as λq*n, non-standard limit distributions emerge.

POWER MAXIMIZATION AND SIZE CONTROL IN HETEROSKEDASTICITY AND AUTOCORRELATION ROBUST TESTS WITH EXPONENTIATED KERNELS

Using the power kernels of Phillips, Sun, and Jin (2006, 2007), we examine the large sample asymptotic properties of the t-test for different choices of power parameter (ρ). We show that the nonstandard fixed-ρ limit distributions of the t-statistic provide more accurate approximations to the finite sample distributions than the conventional large-ρ limit distribution. We prove that the second-order corrected critical value based on an asymptotic expansion of the nonstandard limit distribution is also second-order correct under the large-ρ asymptotics. As a further contribution, we propose a new practical procedure for selecting the test-optimal power parameter that addresses the central concern of hypothesis testing: The selected power parameter is test-optimal in the sense that it minimizes the type II error while controlling for the type I error. A plug-in procedure for implementing the test-optimal power parameter is suggested. Simulations indicate that the new test is as accurate in size as the nonstandard test of Kiefer and Vogelsang (2002a, 2002b), and yet it does not incur the power loss that often hurts the performance of the latter test. The results complement recent work by Sun, Phillips, and Jin (2008) on conventional and bT HAC testing.

UNIFORM ASYMPTOTIC NORMALITY IN STATIONARY AND UNIT ROOT AUTOREGRESSION

While differencing transformations can eliminate nonstationarity, they typically reduce signal strength and correspondingly reduce rates of convergence in unit root autoregressions. The present paper shows that aggregating moment conditions that are formulated in differences provides an orderly mechanism for preserving information and signal strength in autoregressions with some very desirable properties. In first order autoregression, a partially aggregated estimator based on moment conditions in differences is shown to have a limiting normal distribution that holds uniformly in the autoregressive coefficient ρ, including stationary and unit root cases. The rate of convergence is $\root \of n $ when $\left| \rho \right| < 1$ and the limit distribution is the same as the Gaussian maximum likelihood estimator (MLE), but when ρ = 1 the rate of convergence to the normal distribution is within a slowly varying factor of n. A fully aggregated estimator (FAE) is shown to have the same limit behavior in the stationary case and to have nonstandard limit distributions in unit root and near integrated cases, which reduce both the bias and the variance of the MLE. This result shows that it is possible to improve on the asymptotic behavior of the MLE without using an artificial shrinkage technique or otherwise accelerating convergence at unity at the cost of performance in the neighborhood of unity. Confidence intervals constructed from the FAE using local asymptotic theory around unity also lead to improvements over the MLE.

NONSTANDARD CENTRAL LIMIT THEOREMS FOR MARKOV CHAINS

We prove a nonstandard central limit theorem for strongly mixing stationary sequences using the "radically elementary" probability theory developed by Nelson (1987) and use this result to obtain a nonstandard Markov chain central limit theorem. An analogue of a Gaussian autoregressive process is used to illustrate how to obtain the rate of convergence. A Markov chain nonstandard functional central limit theorem is also proved under conditions similar to square integrability of the underlying martingale.